GPT-3.5, GPT-4, or BARD? Evaluating LLMs Reasoning Ability in Zero-Shot Setting and Performance Boosting Through Prompts

1. Introduction

• Researchers disagree on whether LLMs show satisfactory zero-shot reasoning: some report reasoning capabilities, while others argue language ability is not reasoning and cite poor planning performance.Prior work also links impressive reasoning to very large models and reports success with Chain-of-Thought prompting.Source: §1 · p. 3
• The paper evaluates ChatGPT-3.5, ChatGPT-4, and BARD on question-answering tasks spanning deductive, inductive, abductive, analogical, causal, and multi-hop reasoning.The stated aim is to investigate these models’ reasoning capabilities.Source: §1 · p. 4
• The authors propose engineered prompts intended to improve zero-shot LLM performance and report empirical evidence for their effectiveness.The paper presents prompt engineering as a contribution to the debate over LLM reasoning.Source: §1 · p. 4
• The authors make their samples publicly available and their test suite reproducible on all three evaluated models, addressing missing protocol and prompt details in Bang et al. (2023).They state that reproducibility can support fair experimentation and further research.Source: §1 · p. 4

2. Related Work

• Prior work offers conflicting views: some argue LLMs reason satisfactorily in zero-shot settings, while others report inadequate planning and reasoning, with stronger results associated with very large models.The cited discussion describes models over 100 billion parameters as showing especially impressive reasoning in some contexts.Source: §2 · p. 4
• Chain-of-Thought prompting adds intermediate reasoning demonstrations, with variants including “Let’s think step by step” and iterative CoT.The cited literature reports CoT improvements on arithmetic and commonsense tasks and describes prompt-based learning for few-shot or zero-shot adaptation.Source: §2 · p. 4
• Rationale engineering develops or improves CoT through rationale refinement, exploration, and verification.Examples include complexity-based prompting, Auto-CoT, self-consistency, multiple demonstrations, and trained verification.Source: §2 · p. 4
• Hybrid approaches combine prompting with methods such as fine-tuning on CoT data or bootstrapping to strengthen reasoning.The cited work presents bootstrapping as a way for LLMs to improve without fine-tuning.Source: §2 · p. 5
• The review describes ChatGPT-3.5 as an InstructGPT-based model and ChatGPT-4 as a multimodal model, while noting reported GPT-4 limitations and human-level performance on many assessments.GPT-4’s cited limitations include false information, harmful advice, a restricted context window, and no ability to learn from experience.Source: §2 · p. 5
• Prior ChatGPT-3.5 evaluations report stronger deductive than inductive reasoning and difficulty with mathematics, while BARD is described as an experimental, feedback-refined LaMDA version.The review also lists BARD’s stated aims of improving accuracy and reducing bias, false positives, false negatives, personal opinions, and vulnerability to adversarial prompting.Source: §2 · p. 5

3.1. Logical reasoning

• Logical reasoning includes deduction from general principles, induction from observations, and abduction that forms explanations from incomplete information.Deduction treats a conclusion as necessary when its premises are true; induction allows a conclusion to be false even when its premises are true.Source: §3.1 · pp. 5–6

3.2. Non-textual semantic reasoning

• Mathematical reasoning uses logical and mathematical principles, breaking complex problems into manageable components and expressing ideas precisely.The section places mathematical reasoning under non-textual semantic reasoning.Source: §3.2 · p. 6

3.3. Commonsense reasoning

• Commonsense reasoning draws on knowledge and understanding of the world to support informed decisions and judgments.The examples include recognizing that putting a hand in fire causes harm and inferring rain from an umbrella.Source: §3.3 · p. 6

3.4. Casual reasoning

• Causal reasoning examines relationships among variables or events to identify cause-and-effect relationships.The section notes that machines may not make the intuitive human choice between a chocolate-related headache and an unrelated sports hypothesis.Source: §3.4 · p. 7

3.5. Multi-hop reasoning

• Multi-hop reasoning integrates information from multiple sources through several logical steps to reach a supported answer.The example links the Nobel Committee’s 1921 decision and prize-postponement rules to Einstein receiving the 1921 Physics prize in 1922.Source: §3.5 · p. 7

• The section presents dataset categories in Table 1, then describes the datasets’ individual characteristics.Source: §4 · p. 7

4.1.1. Deductive reasoning

• EntailmentBank tests multi-step entailment with explanations in tree form, while bAbI task 15 asks models to deduce conclusions from premises.EntailmentBank has seven fields and three tasks of increasing difficulty.Source: §4.1.1 · p. 7

4.1.2. Inductive reasoning

• CLUTRR tests inference of unstated family relationships from semi-artificial stories, while bAbI task 16 tests inducing general rules from premises.Source: §4.1.2 · p. 7

4.1.3. Abductive reasoning

• αNLI tests abductive reasoning by asking which of two hypotheses better explains a pair of observations.It contains about 80,000 human-annotated examples split into training, validation, and test sets.Source: §4.1.3 · p. 7

4.2.1. Mathematical reasoning

• Math contains over 100,000 high-school-level problems, including multiple-choice, free-response, and proof-based questions across subjects such as algebra, geometry, calculus, and statistics.Source: §4.2.1 · p. 7

4.3. Commonsense reasoning

• CommonsenseQA has 12,247 five-option questions covering everyday commonsense, including cause and effect, spatial reasoning, social norms, and sequencing.The dataset also covers physical laws.Source: §4.3 · p. 7
• PiQA contains 2,000 multiple-choice questions across topics including physics, biology, and social behavior, split into 1,500 training and 500 test questions.Source: §4.3 · p. 8
• Pep-3k tests whether physical events are plausible or implausible and was constructed using Amazon Mechanical Turk.Source: §4.3 · p. 8

4.4. Causal reasoning

• e-Care is a human-annotated dataset for causal reasoning.Source: §4.4 · p. 8
• For each question, the task is to choose which of two hypotheses better explains the cause or effect of a premise.Source: §4.4 · p. 8
• The dataset contains 21, 324 causal questions and 13, 048 unique explanations, with train, development, and test splits reported for both.Source: §4.4 · p. 8

4.5. Multi-hop reasoning

• HotpotQA contains over 100,000 question-answer pairs derived from Wikipedia paragraphs.Source: §4.5 · p. 8
• Each question comes with supporting paragraphs that provide context to assist in answering.Source: §4.5 · p. 8
• The questions span subjects such as science, history, and general culture.Source: §4.5 · p. 8

5.1. Data selection

• Thirty samples were randomly selected from each evaluation dataset, using available train, development, or test sets; the exact samples are listed in Appendix A.Selection examples include 10 samples from each Math training difficulty and 30 from specified splits for other datasets.Source: §5.1 · p. 8

5.2. Prompt Engineering

• The prompt set combines established high-performing prompts with the authors’ proposals, marked by a dagger symbol.Source: §5.2 · p. 8
• For bAbI tasks 15 and 16, prompts frame questions as deductive or inductive inference, while task 16 also uses a two-observation hypothesis comparison.Source: §5.2 · p. 8
• The Math prompt adds “Let’s think step by step” to the mathematical problem and question.Source: §5.2 · p. 8
• Other dataset prompts ask for a five-option answer, phrase completion, plausibility judgment, causal explanation, or a context-based answer.Source: §5.2 · pp. 8–9

Inductive Reasoning

• With prompting, GPT-4 scored 11/30 on CLUTRR, ahead of BARD (7/30) and GPT-3.5 (2/30); on bAbI-task 16, GPT-4 scored 28/30.On bAbI-task 16, prompting raised GPT-3.5 from 1/30 to 14/30 and BARD from 15/30 to 20/30; without prompting, BARD led both GPT models. In CLUTRR errors, GPT-3.5 missed relationships, while GPT-4 and BARD struggled with identifying motherhood.Source: §6 · p. 9, Table 2 · p. 10

Deductive Reasoning

• On bAbI-task 15, GPT-3.5 scored 28/30 without prompting and 26/30 with it, GPT-4 scored 30/30 with prompting, and BARD scored 25/30 either way.On EntailmentBank, GPT-4 and BARD each scored 27/30 versus GPT-3.5’s 25/30, a difference the paper reports was not statistically significant; BARD’s bAbI-task 15 score was unchanged by prompting.Source: §6 · p. 10, Table 3 · p. 11

Abductive Reasoning

• GPT-4 scored 25/30 on 𝛼NLI, ahead of BARD (22/30) and GPT-3.5 (19/30), but all three models missed the subtle distinction between hypotheses in an example.Bang et al. (2023) reported better performance than the study’s results; the models’ persuasive answers to the shared example were nevertheless incorrect.Source: §6 · pp. 11–12, Table 4 · p. 12

Mathematical Reasoning

• GPT-4 scored 20/30 on mathematical reasoning, ahead of BARD (16/30) and GPT-3.5 (15/30), but BARD led on difficult problems while GPT-4 led on easy and intermediate ones.The evaluation used ten samples per difficulty category; GPT-4 still had trouble with prime factors and base conversions, and the 30 samples did not cover mathematics broadly.Source: Table 5 · p. 12, §6 · p. 12

Commonsense Reasoning

• Across CommonsenseQA, PiQA, and Pep-3K, GPT-4 led on the first two (25/30 and 29/30), while BARD led Pep-3K (19/30).GPT-3.5 scored 21/30, 24/30, and 15/30, respectively; BARD scored 20/30 on CommonsenseQA and 23/30 on PiQA. All three models shared eight incorrect Pep-3K responses, and all missed some commonsense questions.Source: §6 · pp. 12–13, Table 6 · p. 13

Causal Reasoning

• GPT-4 scored 25/30 on E-Care, BARD 24/30, and GPT-3.5 14/30; all models failed the same three questions.GPT-4 and BARD tended to select the wrong hypothesis and miss the cause, while GPT-3.5 often recognized that the information was insufficient.Source: §6 · p. 13, Table 7 · p. 14

Multi-hop Reasoning

• On HotpotQA, GPT-4 scored 17/30 versus GPT-3.5’s 16/30, a nonsignificant difference; both struggled to integrate information across sources, especially in long contexts involving people and dates.Bang et al. (2023) reported 8/30, but the paper’s comparison used the same sample count for its GPT-3.5 evaluation.Source: §6 · pp. 13–14, Table 8 · p. 14

Results and discussion

• 20 correct answers out of 30 questions let BARD outperform both GPT models on HotpotQA.BARD provided more comprehensive responses; some wrong answers were attributed to its inability to process all the input, an issue not encountered with either GPT model.Source: §6 · p. 14

7. Conclusions and Future Work

• The study evaluates deductive, inductive, abductive, mathematical, commonsense, causal, and multi-hop reasoning across three models.Its aims are to assess ChatGPT-4's improvements over ChatGPT-3.5 and BARD and identify challenges for the research community.Source: §7 · p. 14
• Manual evaluation found ChatGPT-4 improved over ChatGPT-3.5 and BARD, while inductive, mathematical, multi-hop, and commonsense reasoning still require significant attention.The authors note that sample-data differences may have influenced score comparisons with Bang et al. (2023).Source: §7 · p. 14
• Inductive tasks expose difficulty determining relationships between people, while commonsense tasks challenge judgments about whether actions are plausible.The authors caution that differences in sample complexity may have influenced comparisons with Bang et al. (2023).Source: §7 · p. 14
• The authors have made their samples publicly available and hope they will support a benchmark for future research.They also encourage use of the samples as a benchmark.Source: §7 · p. 14
• The limited sample count left areas such as algebra, arithmetic, trigonometry, and calculus uncovered, and the authors propose evaluating more language models.They describe broader mathematical evaluation and comparisons across additional models as future directions.Source: §7 · p. 14

A. Appendix

• The appendix labels the material as deductive reasoning prompts, with questions about skin function, evaporation, moon phases, and pollination.These items ask the reader to apply stated relationships to a concrete question.Source: Table 9 · p. 15, A. Appendix · p. 16
• Several prompts ask for short answers without explanations, including questions about water drops disappearing and the Moon’s phase one week after a new moon.The prompts provide relevant facts before posing each question.Source: A. Appendix · p. 16
• Environmental prompts use facts about fossils and ancient environments, then ask what resource causes the greatest human-induced impact on Earth’s atmosphere.The resource question is presented once in a shorter form and again with additional background about pollution and greenhouse gases.Source: A. Appendix · pp. 16–17
• Other examples ask learners to categorize the Sun and explain how sunlight transfers energy to a warmed brick.The prompt contexts identify the Sun as a yellow dwarf and describe the brick as exposed to sunlight.Source: A. Appendix · pp. 17–18
• The final excerpt begins a prompt linking fossil formation and sedimentary rock to metamorphic-rock formation, but ends before the question appears.It states that nearly all fossils are found in sedimentary rock and that metamorphic rocks do not contain fossils.Source: A. Appendix · p. 18

14

• The rock-cycle example asks which process is most likely to destroy fossils, after defining rock types and fossil formation.Source: A. Appendix · p. 18

15

• The physical-science example asks how iron particles are affected when a block melts, following context on motion, heat, kinetic energy, and melting.Source: A. Appendix · p. 19

16

• The optics example asks what happens to light at an eyeglass lens, after context about refraction, visibility, and light traveling between materials.Source: A. Appendix · p. 19

17

• The ecology example asks what role an American alligator fills, noting that it eats frogs, turtles, fish, mice, and birds.Source: A. Appendix · p. 20

18

• The astronomy example asks how long Earth takes to make one revolution around the Sun, after explaining that a planet’s orbit around its star defines a planetary year.Source: A. Appendix · p. 20

19

• The final visible example provides context about iron, heat, conductivity, and magnetism, but the supplied passage ends before its question appears.Source: A. Appendix · p. 21

A. Appendix

• Question 19 asks which object conducts electricity and heat and is attracted to a magnet, with answers limited to a few words and no explanation.The accompanying premises define iron as a metal, metals as electrical conductors, and heat as thermal-energy transfer.Source: A. Appendix · p. 21
• Question 20 asks what energy change occurs when bread is toasted, after defining heat transfer, heating elements, and phase changes.The prompt requests a brief answer without explanation.Source: A. Appendix · p. 21
• Question 21 asks how red dwarf stars compare with the Sun, using temperature and heat-related premises and requesting a brief answer.The supplied text characterizes red dwarfs as cooler than yellow dwarfs and identifies the Sun as a yellow dwarf.Source: A. Appendix · p. 22
• Question 22 asks which gas has the lowest average volume in Earth’s atmosphere and includes a prompt about plants reducing atmospheric carbon dioxide.The premises state that Earth’s atmosphere contains 0.36 percent carbon dioxide by volume and list nitrogen, oxygen, carbon dioxide, water vapor, and argon among air’s components.Source: A. Appendix · p. 22
• Question 23 asks which object most likely has magnetic properties, with iron and iron nails presented as magnetic examples.The prompt also states that magnets attract magnetic metals and asks for a brief answer without explanation.Source: A. Appendix · p. 23
• Question 24 asks what happened to a sunlit puddle that disappeared between 10 A.M. and 2 P.M., with evaporation defined as liquid changing to gas through added heat.The prompt requests a brief answer without explanation.Source: A. Appendix · p. 23
• The excerpt begins Question 25 with definitions of moon phases, including that a phase recurs 28 days after its last occurrence, but supplies no question prompt.It names new, full, quarter, and last-quarter phases and describes the new moon as having its dark half facing Earth.Source: A. Appendix · p. 24
• One example asks which moon phase follows a new moon about a week later, using the fact that moon phases change approximately weekly.The prompt asks for a few words without explanation.Source: A. Appendix · p. 24
• A water question asks what happens as liquid-water molecules slow, alongside background that decreasing temperature slows molecular motion.The prompt requests a few words without explanation.Source: A. Appendix · p. 24
• A weather question asks what follows when warm air passes over cold air and clouds form, with background linking cooling water vapor to rain and precipitation.The prompt requests a brief answer without explanation.Source: A. Appendix · p. 25
• A sound question asks what lets two students ten feet apart hear one another, with background describing sound as vibrations traveling through air.The prompt requests a few words without explanation.Source: A. Appendix · p. 25
• A rock question asks how layered rock made of small sand particles likely formed, with background describing sediment compaction and cementation.The prompt requests a brief answer without explanation.Source: A. Appendix · p. 26
• A hurricane example connects reduced ocean heat and moisture with reduced hurricane strength, alongside background that hurricanes require suitable water temperature.The excerpt also notes that hurricanes usually occur between June and November in the Northern Hemisphere.Source: A. Appendix · p. 26
• The hurricane example answers that Atlantic water is too cold in March to support hurricane formation.The prompt asks which factor makes an Atlantic hurricane less likely to form in March and requests a brief answer without explanation.Source: A. Appendix · p. 26
• Given that sheep are afraid of cats and Gertrude is a sheep, the example concludes that Gertrude is afraid of cats.The response describes this as a valid deductive argument because the conclusion follows from the premises.Source: A. Appendix · p. 27
• In a second deduction example, the response concludes that Jessica is afraid of nothing mentioned in the facts.The supplied facts say sheep are afraid of wolves and Jessica is a sheep, while the response's intermediate inference snippets are fragmented.Source: A. Appendix · p. 28
• With the facts that sheep are afraid of mice and Emily is a sheep, the example concludes that Emily is afraid of mice.The relevant facts and conclusion appear in the prompt and response snippets.Source: A. Appendix · p. 31

Appendix

• For Emily, the response infers fear of both mice and wolves, reasoning from her being a sheep and the stated fear relations.It first infers fear of mice, then says Emily could also fear wolves, concluding that both are reasonable inferences.Source: A. Appendix · p. 31
• For Emily identified as a mouse, the response uses the fact that mice fear wolves to infer that Emily is afraid of wolves.The explanation explicitly connects the general fact about mice to Emily’s membership in that category.Source: A. Appendix · p. 32
• For Winona identified as a mouse, the response follows the stated relation that mice fear sheep to conclude that Winona is afraid of sheep.The appendix presents this as the conclusion reached by following a chain of relationships.Source: A. Appendix · p. 33

• Winona is inferred to fear mice because she is a sheep and sheep are afraid of mice.Source: A. Appendix · p. 34
• Emily is inferred to fear wolves because she is a cat and cats are afraid of wolves.Source: A. Appendix · p. 35
• Gertrude is inferred to fear wolves because she is a sheep and sheep are afraid of wolves.Source: A. Appendix · p. 36
• Winona, identified as a cat, is inferred to fear mice because cats are afraid of mice.Source: A. Appendix · p. 37

A. Appendix

• The example cautions against inferring that Winona fears Jessica merely because Jessica is a mouse, noting that cats are not always afraid of mice.Source: A. Appendix · p. 37
• Emily is inferred to be afraid of cats because mice are afraid of cats and Emily is a mouse.The explanation identifies those two facts as the deductive basis for the conclusion.Source: A. Appendix · p. 39
• Emily being afraid of wolves is described as less certain than the cat inference, partly because the explanation relies on a reasonable assumption about predators.The passage says wolves could potentially harm Emily and frames this as a reasonable assumption, not an equally certain deduction.Source: A. Appendix · p. 39
• Jessica is inferred to be afraid of wolves because sheep are afraid of wolves and Jessica is a sheep.The passage presents this as the inference supported by the given facts.Source: A. Appendix · p. 40
• Jessica’s being a mouse and the rule that mice fear wolves are used to infer that she fears wolves.The response presents this as a deductive inference from the stated facts.Source: A. Appendix · p. 42
• Winona’s being a wolf and the rule that wolves fear sheep are used to infer that she fears sheep.The response also notes that the facts do not establish what else Winona may fear.Source: A. Appendix · p. 43
• For Winona, who is given as a mouse, the example concludes that the most likely inference is that she fears wolves and sheep.The stated facts include that mice fear wolves; the conclusion additionally names sheep.Source: A. Appendix · p. 44

Appendix

• For Emily, the answer infers fear of sheep from the facts that Emily is a mouse and mice are afraid of sheep.The response frames this as a reasonable inference from the supplied facts.Source: A. Appendix · p. 45
• For Gertrude, the answer infers direct fear of wolves and indirect fear of sheep, while judging wolves more likely to frighten her than cats.It distinguishes the stated mouse-to-wolf rule and an indirect chain through wolves from the claim that cats are predators of mice.Source: A. Appendix · p. 46
• For Gertrude as a wolf, the answer concludes she is afraid of mice and says the other prompt facts are irrelevant to that question.The reasoning applies the stated rule that wolves are afraid of mice to Gertrude’s stated identity as a wolf.Source: A. Appendix · p. 47
• In another Gertrude example, the response infers fear of sheep from the mouse rule but also reverses the cats-are-afraid-of-mice rule to claim Gertrude fears cats.The text states that Gertrude is a mouse and mice are afraid of sheep, then says she is also afraid of cats because cats are afraid of mice.Source: A. Appendix · p. 48

A. Appendix

• Gertrude’s most reasonable inferred fears are sheep and cats, though she may also fear other animals.The example states that mice are afraid of wolves but notes that this does not establish Gertrude’s fear of wolves.Source: A. Appendix · p. 48
• Winona is inferred to be afraid of sheep because the facts state that wolves are afraid of sheep and that Winona is a wolf.The passage says this is the fear inferable with certainty from the given facts, while allowing that Winona may have other fears.Source: A. Appendix · p. 49
• The facts establish Winona’s fear of sheep, but do not rule out that she has other fears as well.The passage distinguishes the fear deducible with certainty from additional possibilities.Source: A. Appendix · p. 49
• Emily is inferred to be afraid of cats because Emily is a sheep and sheep are afraid of cats.The passage says the other listed facts are irrelevant and calls fear of cats the only reasonable inference.Source: A. Appendix · p. 50
• Jessica is inferred to fear only wolves by tracing the fear relations and ruling out mice.The example’s stated conclusion is that wolves are the only thing Jessica fears.Source: A. Appendix · p. 51
• Winona is inferred to fear only mice because she is a wolf and wolves fear mice.The appendix also presents a separate example in which Winona is a sheep and is inferred to fear mice.Source: A. Appendix · pp. 52–53
• From the story clues, Stanley and Alan are inferred to be related, likely as father and son, though Stanley could be another close relative.The reasoning identifies Stanley as the boys’ grandfather and Alan as their father, while allowing that Stanley might be Alan’s uncle or another close relative.Source: A. Appendix · p. 55
• One example infers Helen is Sharon’s daughter-in-law because Dorothy is Helen’s mother and Sharon is married to Dorothy’s son Kent.Source: A. Appendix · p. 56
• The example infers Gloria is likely Jeff’s sister through Jeremy, but notes she could instead be his cousin or another close relative.Source: A. Appendix · p. 57
• For Melanie and Irene, the appendix says no direct or indirect relationship is stated, so there is no basis for inferring one.Source: A. Appendix · p. 58
• The Melanie–Irene example infers that they are close sisters who care for each other, while acknowledging the story may not reveal their full relationship.The answer characterizes their relationship as close but sometimes challenging and treats the conclusion as an inference from limited story information.Source: A. Appendix · p. 58
• The Melody–Richard example says the story does not establish their relationship, but another answer infers that they are siblings through family links.One response says the information is insufficient to determine a relationship; another reasons from Melody’s relationship to Donald and Donald’s relationship to Alice, while also noting siblings may be full, half, or step-siblings.Source: A. Appendix · p. 59
• Ronald is most likely Marie’s uncle because Ronald and Mark are inferred to be brothers and Mark is Marie’s father.The appendix also identifies Donna as Marie’s mother.Source: A. Appendix · p. 61
• Steven and Steve are likely grandfather and grandson, although the story does not explicitly state their relationship.The inference draws on Mabel’s cake for her grandfather Steven and Stanley taking Steve to a baseball game; the passage allows that they may not be blood-related.Source: A. Appendix · p. 62
• The Ronald–Robert relationship cannot be established from the story, although the response speculates that Colleen’s arguments with Robert may indicate a strained relationship.The response cautions that this is an inference from limited information and that the relationship could be amicable.Source: A. Appendix · p. 63
• The response presents a father–son relationship between Frankie and Peter as the most likely explanation, citing clues about Lane and Peter’s relationship to Frances.It notes that Frankie calls Lane his son, while Peter is referred to as Frances’s uncle rather than her father.Source: A. Appendix · p. 64
• The response infers that Juanita is Frankie’s granddaughter, based on the story’s family relationships and events.Juanita is described as Brian’s sister, and the response identifies her relationship with Frankie as that of a granddaughter to a grandfather.Source: A. Appendix · p. 65
• For Juanita and Frankie, one response says their relationship cannot be deduced, while another suggests a likely grandparent–grandchild relationship despite acknowledging alternatives.The latter response treats Frankie’s arrival at a family picnic as evidence of a close family connection.Source: A. Appendix · p. 65
• For Robin and Carla, responses infer a likely stepfamily relationship but also allow that they may be unrelated.One response identifies Robin as Carla’s granddaughter; another describes them as stepdaughter and stepmother.Source: A. Appendix · p. 66
• For Linda and Karen, one response says no relationship can be inferred, while another labels Linda Karen’s granddaughter.The latter response gives a relationship chain through Demetra and John.Source: A. Appendix · p. 67
• For Lisa and Eric, a response suggests Lisa is likely Eric’s daughter, then concludes that their relationship cannot be determined.Source: A. Appendix · p. 68
• The Lisa–Eric story supports an inference that they are siblings: it places both as Robin’s children and notes that they appear close in age.The appendix treats siblinghood as the likely conclusion, while acknowledging that other relationships are possible.Source: A. Appendix · p. 68
• The story provides no direct connection between Jennifer and Jose, so their relationship cannot be determined and Jennifer may not be related to Jose at all.A proposed connection through Latasha is not established by the available information.Source: A. Appendix · p. 69
• The story does not establish a specific relationship between Patricia and Blanca because it gives too little information about their connection.The appendix says the individuals are not mentioned together in a context that implies a relationship.Source: A. Appendix · p. 70
• A response identifies Patricia as Blanca’s mother, though the surviving fragment says this is less likely when she presented the baby to Elizabeth.The fragment gives only a partial rationale, so it does not establish a fuller relationship chain.Source: A. Appendix · p. 70
• The responses alternately infer that Jennifer is Felix’s aunt and conclude that the story cannot establish their relationship, while treating Jennifer as Latasha’s close caregiver.One response also says Jennifer and Felix could be related or unrelated and lists questions that might clarify the relationship.Source: A. Appendix · p. 71
• The responses conflict on Paul and Robert: one says their relationship cannot be determined, while another infers that Paul is Robert’s father.The latter connects April as Paul’s daughter and Robert’s sister to the story’s statement that Paul took his son Ryan out for ice cream.Source: A. Appendix · p. 72
• The response identifies Angela as Christina’s mother and Christina as Paul’s mother, implying Angela is Paul’s grandmother.The passage explicitly states the first two relationships; the grandparent relationship follows from them.Source: A. Appendix · p. 73

Appendix

• In an earlier example, Angela is considered a possible great-grandmother of Randall, but the story is said to lack enough evidence to determine their relationship.The reasoning makes the great-grandmother inference conditional on Christina’s relationship to Paul.Source: A. Appendix · p. 73
• In another example, Sandra is Mark’s mother and Carol is his wife, making Sandra Carol’s mother-in-law without establishing their precise relationship.The passage says the story does not show whether Carol is Sandra’s daughter or daughter-in-law.Source: A. Appendix · p. 74
• In a later story, the reasoning identifies Angela as Randall’s wife and Mary’s mother, concluding that Angela and Randall are husband and wife.The story says Randall is proud to see his daughter Mary perform at her first ballet recital.Source: A. Appendix · p. 75
• For story 24, responses variously infer Emily is Marvin’s wife and state that the story provides no clear relationship between them.One answer appeals to shared parenting as a cue; another says there is no direct or indirect reference to a connection.Source: A. Appendix · p. 76
• For story 25, one response infers Sandra is Carol’s mother, while another says their relationship cannot be inferred from the story.The responses cite Sandra’s status as Carmen’s grandmother but differ on whether that establishes her relationship to Carmen’s parent, Carol.Source: A. Appendix · p. 77

A. Appendix

• The Ray–Debra example concludes they are not related, reasoning that Ray is Sharon’s son and Sharon is Stuart’s daughter, while Debra is Stuart’s wife.It also describes their relationship as stepson and step-grandmother, though the supplied excerpts do not fully resolve this inconsistency.Source: A. Appendix · p. 79
• Both examples qualify their conclusions: Ray and Debra’s exact relationship is undetermined, and another passage says no relationship between Mary and Derrick can be inferred.The Ray–Debra answer allows for unstated family dynamics; the Mary–Derrick passage notes no direct connection or relationship is provided.Source: A. Appendix · pp. 79–80
• The Ramon–Derrick example infers that Mary is Derrick’s great-grandmother because Mary is Helen’s mother, Helen is Ramon’s aunt, and Ramon is Derrick’s father.Source: A. Appendix · p. 80

• The appendix infers that Marianne is most likely Franklin’s sister, using family relationships and generations, while noting that the story does not state their relationship explicitly.The reasoning links Franklin to his daughter Etta and granddaughter Stephanie, whom Marianne is described as an aunt to; it treats the sister conclusion as likely, not certain.Source: A. Appendix · p. 82
• The reasoning further supports the sibling inference by placing Franklin, Etta, Juan, and Stephanie in successive age generations and noting that Marianne is close in age to Franklin.Marianne’s absence from the other stated relationships is also cited as a reason to favor the sister explanation.Source: A. Appendix · p. 82
• Given that Julius and Brian are lions and Brian is green, the appendix suggests Julius might be green but says the inference is not definitive.It allows that Julius could instead share another color or have a different one.Source: A. Appendix · p. 83

A. Appendix

• Julius is reasonably inferred not to be green because lions are typically not green, but his color remains uncertain without more information.The example notes that Julius could differ from Brian by species or mutation.Source: A. Appendix · p. 83
• For Greg, the facts alone do not determine his color; inferring white from the example of another frog is tentative, and the rhino’s color is irrelevant.The appendix calls the white-frog inference reasonable but not guaranteed, since other frogs or factors could yield different colors.Source: A. Appendix · p. 84
• Bernhard is guessed likely to be green because Julius is a green frog, but the examples cannot establish Bernhard’s exact color or guarantee the guess.The conclusion is explicitly based only on the information provided and may not hold in all cases.Source: A. Appendix · p. 85
• Bernhard is inferred to be not yellow, but the examples do not establish his exact color.The passage offers green or gray as possibilities rather than a definitive answer.Source: A. Appendix · p. 85
• Lily is likely green because the stated frogs are green, but the facts do not rule out another color.The passage treats green as the most reasonable inference from the available evidence, not a guarantee.Source: A. Appendix · p. 86
• Two green examples suggest Greg may be green, but the facts do not establish that animals of the same species share a color or determine Greg’s color.Greg is a rhino, as is Julius, but the passage says his color cannot be inferred with certainty.Source: A. Appendix · p. 87
• Greg is inferred to be yellow because the known animals of his species are yellow, though some rhinos may differ.The passage presents this as the most reasonable inference, not a certainty.Source: A. Appendix · p. 87
• Lily is likely green because she is a frog like the example frogs described as green, although she could be a different color.The appendix calls this a reasonable inference based on the available examples, not a certain conclusion.Source: A. Appendix · p. 88
• The appendix frames these color generalizations as reasonable rather than definitive because individual animals may differ despite the available evidence.For Lily, it says no evidence suggests another color, making green the most likely conclusion.Source: A. Appendix · p. 88

Appendix

• Bernhard is inferred to be probably green because another lion, Greg, is green, but the answer cautions that this is not definitive.The response treats the species-color pattern as suggestive rather than conclusive.Source: A. Appendix · p. 90
• Julius, a frog, is inferred to be more likely yellow than white because two observed frogs are yellow, though the available facts do not establish his color.Bernhard is white, while Greg and Lily are yellow; the answer allows that Julius could be another color.Source: A. Appendix · p. 91

A. Appendix

• Julius is probably gray because the observed lions are gray, but his color is unknown and inductive reasoning allows exceptions.The example treats the pattern as suggestive, not conclusive.Source: A. Appendix · p. 92
• Bernhard might be white because several animals, including lions and a swan, are white, but Greg’s yellow color shows variation.The example therefore presents white as a plausible inference rather than a certain conclusion.Source: A. Appendix · p. 93
• The Lily example concludes that no clear correlation between animal type and color is established, so Lily’s color cannot be determined from these facts.The supplied facts include lions with differing colors and animals of different types sharing white coloration.Source: A. Appendix · p. 94
• Lily’s color is presented as uncertain: the examples favor gray but also allow white, and state that the facts do not establish her color definitively.The reasoning weighs the gray color of another lion against the prevalence of white among other listed animals.Source: A. Appendix · p. 94
• For Bernhard, the example infers gray from Lily being a gray lion, but cautions that lions may have different colors, so the inference is not definitive.The response notes that Bernhard’s color is not directly given.Source: A. Appendix · p. 95
• An exception-based example favors Bernhard being gray: if he were not gray, Brian would be a second exception to the rule that swans are green.The response treats Brian as the sole exception and uses that pattern to support its inference.Source: A. Appendix · p. 95
• A further prompt supplies that Bernhard is a white swan and Lily is a swan, then begins a response reasoning from those facts.The provided passage fragment does not show the full conclusion.Source: A. Appendix · p. 96
• Lily is reasonably inferred to be white because Bernhard is a white swan, but her color is unobserved and she could be another color.The example calls white the likely inference from the available observations, not a certain conclusion.Source: A. Appendix · p. 96
• Greg is inferred likely yellow because Brian and Lily are yellow lions, though the example acknowledges Greg could be a different color.The passage characterizes induction as relying on patterns and generalizations rather than strict deduction, so this remains speculative.Source: A. Appendix · p. 96
• Bernhard is guessed likely yellow because Julius, another frog, is yellow, but the answer says there is insufficient information for certainty.The passage calls this an educated guess and notes that more information about Bernhard’s color is needed.Source: A. Appendix · p. 97
• A separate frog example shows no clear color pattern: Lily and Brian are green, Greg is white, and Julius is yellow.This variation illustrates why a pattern observed in a small set of examples may not support a reliable inference.Source: A. Appendix · p. 97
• Bernhard’s color cannot be determined from the frog examples, although the text suggests green as a tendency rather than a certainty.The examples note that frogs are not all one color, while also describing green as more common; the stated conclusion is that Bernhard could be green or another color.Source: A. Appendix · p. 97
• Lily may be inferred to be green because Brian is a green swan, but the text cautions that this inference is based on limited facts and could change with more evidence.Other swans may not be white, and the passage says there is not enough information to make a definitive conclusion about Lily’s color.Source: A. Appendix · p. 98
• Julius is likely gray because Lily is the only other frog mentioned and is gray, but the passage says this remains an uncertain inference.It states that there is insufficient information to confirm Julius’s color definitively.Source: A. Appendix · p. 99
• Bernhard is inferred to be likely yellow because Lily is a yellow lion, though the answer warns that inductive reasoning can have exceptions.The answer notes that Bernhard could be a different color.Source: A. Appendix · pp. 100–101
• The example infers that lions are yellow and Bernhard is most likely yellow, but says the inference is not conclusive and allows for other colors.It gives a white lion or genetic variation as possible exceptions.Source: A. Appendix · p. 101
• From examples of white swans, the answer infers that Brian, who is a swan, is likely to be white.The listed facts also say Bernhard is a yellow swan.Source: A. Appendix · p. 102
• The examples infer that Brian is probably white because the observed swans are white, but Bernhard’s yellow color shows the inference is not definitive.The passages also note that unmentioned swans could have different colors.Source: A. Appendix · p. 102
• Another example tentatively infers that Lily is green from Brian’s green color, while acknowledging that the inference is speculative and no swan–color correlation is established.The passages caution that members of the same species need not share a color.Source: A. Appendix · p. 103
• A further example says Greg is likely gray because Julius and Bernhard are described as frogs, but it also says Greg’s color cannot be determined definitively.This illustrates how inductive conclusions draw on specific examples rather than certainty.Source: A. Appendix · p. 104
• The first example infers that Greg is probably gray from gray lions and a gray frog, while warning that the limited examples cannot guarantee the guess.The answer notes that some lions or frogs may differ in color, so Greg could be non-gray despite the observed pattern.Source: A. Appendix · p. 104
• For example 24, the answer says frog-color evidence is too sparse for a strong inference about Bernhard, tentatively guesses yellow, then favors green using a lion-color pattern.Brian is the only frog whose color is given; the answer calls its tentative inference weak and potentially incorrect.Source: A. Appendix · p. 105
• The examples describe induction as generalizing from specific observations or patterns, while noting that the resulting inference may be uncertain.The Greg example reasons from observed gray swans to a likely conclusion about another swan, while allowing that other swans may not be gray.Source: A. Appendix · pp. 106–107
• The examples suggest Bernhard may be green because the listed rhinos are green, while acknowledging that the inference is uncertain.Another response says the available information is insufficient to establish a pattern about Bernhard’s color.Source: A. Appendix · p. 106
• Gray swan examples make it reasonable to infer Greg may be gray, but the conclusion is not definitive because swans can have different colors.The passage notes that more information about the color distribution of swans would support a more accurate inference.Source: A. Appendix · p. 107

• For Greg’s color, one response says the facts do not determine it, while another infers he is likely yellow from the colored-frog examples.Source: A. Appendix · p. 108
• For Brian the swan, the example infers he is gray from Lily’s color, while noting that this conclusion is uncertain.Source: A. Appendix · p. 109
• Another example concludes that Bernhard is a white frog, combining the facts that he is a frog and is white.Source: A. Appendix · p. 110

A. Appendix

• An example infers Julius is probably white from observed frog colors, while warning that color variation makes the inductive guess uncertain.The answer notes there is no direct information about Julius’s color and that frogs may have other colors.Source: A. Appendix · p. 110
• For Bernhard, the answer says his color cannot be reasonably inferred: the animal examples show no consistent color pattern, and the available information is insufficient.Although Greg, another lion, is green, the answer cautions that inductive reasoning does not guarantee a correct conclusion.Source: A. Appendix · p. 111
• Table 11 presents abductive reasoning prompts, including an example answer that hypothesis 1 better explains the observations.The supplied example concerns two observations about a family gathering for Thanksgiving.Source: Table 11 · p. 112, A. Appendix · p. 113
• Hypothesis 1 is judged the better explanation for the family observations.It proposes that people drove in from across the state, rather than that everyone was angry and did not eat.Source: A. Appendix · p. 113
• Tim posting funny cat videos is judged to explain his YouTube success better than working lazily.The response says the lazy-work hypothesis is less consistent with the observations that Tim wanted to become a YouTube star and made millions from his videos.Source: A. Appendix · p. 114

YouTube example

• The YouTube example favors Hypothesis 1 over Hypothesis 2, which the explanation considers less likely because it lacks supporting evidence.Source: A. Appendix · p. 114

Julius example

• The Julius example favors house hunting over merely thinking about new places because it better accounts for addressing his noisy-neighbor problem and moving.Source: A. Appendix · p. 115

Gina example

• In the Gina example, Hypothesis 1 is favored: her continued stare suggests she focused on something outside, despite limited visibility from the backseat.The explanation notes that the context does not definitively rule out the alternative hypothesis.Source: A. Appendix · p. 116

A. Appendix

• For Gina, one explanation favors Hypothesis 2 because siblings could obstruct her view and her continued staring suggests attention inside the car, while another favors Hypothesis 1 because a strange dog could explain her interest.The appendix also includes reasoning that the dog hypothesis is less likely because the observations do not explain why Gina continued to stare, illustrating the competing interpretations.Source: A. Appendix · p. 116
• For Titan, some reasoning favors Hypothesis 1—that Titan was taken home and died—while other reasoning says neither hypothesis is strongly supported by the observations alone.The alternative that Titan got fat from overfeeding lacks direct supporting evidence, and the second observation does not establish Titan’s fate.Source: A. Appendix · p. 117
• Another Titan explanation favors Hypothesis 1, arguing that a new fish is consistent with Titan’s death and that overfeeding is unsupported by general knowledge about fish.This conclusion contrasts with the passage stating that the observations do not determine which hypothesis is better.Source: A. Appendix · p. 117
• For Neil, the appendix says the observations are insufficient for a confident conclusion, but also states that Hypothesis 1 explains them better than Hypothesis 2.The visible observations are that Neil loved music and thought Vienna was the most musical city on earth.Source: A. Appendix · p. 118
• The Neil example shows his musical interests but provides no information about why he traveled, leaving the competing explanations unresolved.Source: A. Appendix · p. 118
• The appendix notes that other explanations may exist and that more information would be needed to make a definitive determination.This caveat limits confidence in selecting between the proposed hypotheses.Source: A. Appendix · p. 120
• For the father-daughter dance example, the stated conclusion favors hypothesis 1, although the preceding rationale says hypothesis 2 better explains the observations.The rationale contrasts Ellie’s specific interaction with her father against the general attendance of many fathers and daughters.Source: A. Appendix · p. 120
• For Joe’s late arrival and simple apple-and-banana breakfast, the conclusion favors hypothesis 2: he lacked time to cook something more elaborate.The explanation says the quick, no-cook breakfast supports hypothesis 2, while hypothesis 1 implies he had time to cook, contrary to the first observation.Source: A. Appendix · p. 121
• In the breakfast scenario, hypothesis 2 is favored because apples and bananas are easy to prepare and could be a quick, healthy choice if Joe was late for school.The appendix concludes that hypothesis 2 better explains the observations.Source: A. Appendix · p. 121
• For Tom’s baseball scenario, hypothesis 2 better explains the observations because his dislike of outfield and his excuse to change positions support it.The appendix contrasts this with hypothesis 1, which does not explain why Tom needed an excuse to play another position.Source: A. Appendix · p. 122
• For Gina’s scenario, neither hypothesis fully explains both observations, but hypothesis 1 is judged more likely than hypothesis 2.The observations are that Gina had a crush on Jamie and Josh and was unsure whether her friend’s advice was good or sabotage.Source: A. Appendix · p. 123
• For Gina, one analysis favors hypothesis 1 as the more plausible advice, while another favors hypothesis 2 because Gina liked both Jamie and Josh; neither interpretation is definitive.The first analysis notes that hypothesis 1 does not explain Gina’s uncertainty about whether the advice was good, while the second also notes this uncertainty is unexplained.Source: A. Appendix · p. 123
• For Ann, the analysis concludes that neither hypothesis satisfactorily explains the observations because the claim that the free cake was given in exchange for a favor lacks evidence.Although Ann’s love of cake is said to support hypothesis 2, the observations establish only that she felt insulted by the offer, not that it was conditional on a favor.Source: A. Appendix · p. 124
• For the Ann-and-cake observations, hypothesis 2 is favored, but further information is needed to assess the explanation confidently.The appendix suggests clarifying what favor the lady asked, whether the cake was homemade, and how Ann reacted.Source: A. Appendix · p. 124
• For Jim’s Costco observations, hypothesis 2 (“Jim loved free samples”) better explains why he ate so much and skipped dinner.The appendix cautions that the available evidence is insufficient to determine with certainty what happened.Source: A. Appendix · p. 125
• For Will’s smoking observations, hypothesis 1—that he stopped buying cigarettes at the store—is judged the better explanation.Source: A. Appendix · p. 126
• For Will, the appendix says more information would help determine which hypothesis is likelier, but the currently available information favors Hypothesis 1.The excerpt names motivation to quit, cigarette access, and willpower as relevant factors.Source: A. Appendix · p. 126
• Kim’s continued yelling after being tricked into eating deer meat supports the hypothesis that she hated eating animals, which better explains both observations.The appendix contrasts this with the unsupported hypothesis that Kim gave her mother a huge kiss.Source: A. Appendix · p. 127
• For Alexis, the hypothesis that she worried the Christmas tree would be too big better explains why it fit her ceiling than the alternative that she ensured it was not under 20 feet tall.The example notes that selecting a tree of the right size does not necessarily follow from that alternative hypothesis.Source: A. Appendix · p. 128
• For the Christmas-tree case, Hypothesis 1 is judged more likely because it explains both observations, while Hypothesis 2 explains only the tree’s fit and is not supported by cutting it in the woods.The answer reasons that Alexis likely took steps to get the right-sized tree rather than simply hoping it would fit.Source: A. Appendix · p. 128
• For the savings-and-boat case, the supplied analyses conflict: one favors Hypothesis 1, while another says Hypothesis 2 explains the observations better.The discussion links saving money to a financial goal but notes that it does not directly explain feeling happy after taking the boat out.Source: A. Appendix · p. 129
• For Amy, hypothesis 2—that she practiced drawing and gave her work to her mom—better explains her goal to learn and delight at seeing her drawings displayed.The explanation links her motivation to practice with the pride and accomplishment suggested by her reaction to the drawings on the fridge.Source: A. Appendix · p. 130
• In the pizza example, the supplied text says hypothesis 1 better explains the observations, while also noting that neither hypothesis directly relates to the Netflix observation.The visible passages describe relaxing and watching Netflix, a dropped pizza, and giving friends an old ruined pizza before getting a new one.Source: A. Appendix · p. 131

• Observation 2 alone cannot distinguish the pizza hypotheses: both explain the ruined order, and their difference does not explain why it was ruined.Source: A. Appendix · p. 131
• In the Joe-and-plates example, one analysis says neither hypothesis is clearly supported, while another favors Hypothesis 1 as more consistent with both observations.Source: A. Appendix · p. 132

A. Appendix

• For the dream example, responses either favor the bike-tour hypothesis as more specific and testable or conclude that the observations cannot determine which hypothesis is correct.The bike-tour hypothesis explains the dream and disappointment, while the alternative that the person owns a bike does not clearly explain both; another response notes that further information is needed.Source: A. Appendix · p. 133
• For Ric’s dogs, the appendix favors the hypothesis that they lived long, happy lives because it explains both his strong attachment and his mourning.The observations describe Ric taking his dogs everywhere and gently dusting their ash canisters; the cats hypothesis does not explain the dog-related evidence.Source: A. Appendix · p. 134

Ric example

• Hypothesis 1 better explains why Ric loved and took his dogs everywhere and was still grieving; Hypothesis 2 does not explain why he dusted their ash canisters.The appendix also notes that Ric’s dogs were part of his family, died of old age, and had not been replaced by another dog.Source: A. Appendix · p. 134

Elizabeth example

• Elizabeth’s high-heels-on-a-rainy-day hypothesis better explains her fall into a puddle and muddy clothes because heels can impair balance on slippery ground.The competing hand-raising hypothesis is described as unrelated to the observations and not a plausible explanation for the fall.Source: A. Appendix · p. 135

Amy and Ben example

• For Amy and Ben, exchanging gifts on their anniversary is judged more likely than deciding to give each other gifts, given their one-month relationship and the anniversary exchange.Source: A. Appendix · p. 136

A. Appendix

• Hypothesis 2—that Amy and Ben exchanged anniversary gifts—better explains their one-month relationship and chocolates and skittles than a decision to exchange money.The observations are described as inconsistent with exchanging money as a gift and consistent with anniversary gift-giving.Source: A. Appendix · p. 136
• For Bob, stopping to tie his shoes is better supported than stopping because he lacked bug spray.The observations say he forgot his shoes and stopped during the hike.Source: A. Appendix · p. 138
• For Ryan, the hypothesis that he loved going down the water slide better explains the observations.The supplied passages identify this as hypothesis 2 and state that it better explains the observations.Source: A. Appendix · p. 139
• The Ryan example links climbing the long stairs to willingness to reach the slide and reports that the slide was more fun than he expected.These observations underpin the preferred explanation that he loved the slide.Source: A. Appendix · p. 139
• The water-slide analysis favors Ryan loving the slide because that hypothesis fits both observations, whereas fear explains only the first.The analysis also calls the loving-the-slide hypothesis more likely in general.Source: A. Appendix · p. 139
• The example also proposes that Gina pulled Lucy from the water and Lucy hid the incident out of embarrassment, accounting for both observations.The observations are that Lucy almost drowned and then asked Gina to keep the incident secret.Source: A. Appendix · p. 141
• The Lucy example concludes that Gina calling rescuers better explains the observations than the hypothesis that Gina suddenly saved Lucy.It also says the rescuers explanation is more plausible than the sudden-save hypothesis.Source: A. Appendix · p. 141
• Table 12 presents mathematical reasoning prompts, with excerpts showing step-by-step equation solving, equation simplification, and a prime-factor task.The excerpts provide prompt or response openings, not enough information to report completed solutions.Source: Mathematical Reasoning · p. 143, Table 12
• The algebra responses report g = 121 with prime factors 11 and 11, but another answer says g’s prime factors are 7.The latter response also gives y = 7 / −36, unlike the earlier solution yielding g = 121.Source: Mathematical Reasoning · p. 143
• The base-12 subtraction 5 − 132 is reported as 11095, 428, and −283 in separate worked responses.The excerpts give mutually incompatible results for the same subtraction.Source: Mathematical Reasoning · p. 144
• For l(3), one worked response gives 43 and units digit 3, while separate statements give units digits 7 for 3^3 and 9 for 3^2.The excerpts therefore include conflicting units-digit reasoning alongside the reported evaluation.Source: Mathematical Reasoning · p. 145
• The passage reports that dividing 1541 by 3 gives 514 with a remainder of 1, and concludes that 3 is not a prime factor of 1541.Source: Mathematical Reasoning · p. 146
• Responses give three conflicting factorizations of 1541: 3, 7, and 11; 23 and 67; or 23 alone.The response claiming 23 and 67 also states that 1541 divided by 23 equals 67.Source: Mathematical Reasoning · p. 146
• For 19385, one response says no primes up to about 139 divide it, yet concludes that it is not prime.The passage gives the square root as approximately 139 and reports that the number is not divisible by any of the tested primes.Source: Mathematical Reasoning · p. 147
• The 19385 response both factors it as 5 x 3877 and calls it nonprime, then claims it is prime because it is not divisible by primes up to 139.It also incorrectly says 19385 is not divisible by 5, despite noting its last digit is 5.Source: Mathematical Reasoning · p. 147
• For 9 = t + 9, the worked response subtracts 9 from both sides and gives t = 0.Source: Mathematical Reasoning · p. 148
• The worked differentiation response gives -24 as the third derivative with respect to y.Source: Mathematical Reasoning · p. 149
• The comparison response calculates b = 0 and i = 4, concluding that b is smaller than i.Source: Mathematical Reasoning · p. 149

• For -0.000010485 rounded to seven decimal places, one fragment reports approximately -0.0000105.Source: Mathematical Reasoning · p. 150
• A separate fragment reports that -0.000010485 rounded to seven decimal places becomes -0.00001049.Source: Mathematical Reasoning · p. 150
• For 22u^4 + 3u^3 + 46u^2 - 151246u, the appendix gives the second derivative as 264u^2 + 18u + 92.Source: Mathematical Reasoning · p. 151

A. Appendix

• The expression is evaluated as approximately -0.4, but the response then inconsistently concludes that the closest value is -3.It first says the closest option to -0.4 is (a), then gives -3 as the final answer.Source: Mathematical Reasoning · p. 153
• The sequence response identifies first term -2153 and common difference -2121, then gives the u’th term as -2121u.It describes equal consecutive differences and an arithmetic progression.Source: Mathematical Reasoning · p. 154
• For -849 = -73*l - 119, the response reduces the equation to -730 / -73 = l, yielding l = 10.The supplied steps add 119 to both sides and divide by -73.Source: Mathematical Reasoning · p. 154
• In this case, 0.082 has the smallest absolute difference, making it the closest value in the comparison.Source: Mathematical Reasoning · p. 155
• The terms -16l + 24l - 281l are combined as -273*l.The passages explain that coefficients of terms with the same variable are added or subtracted.Source: Mathematical Reasoning · p. 155
• The third derivative of 56396m^3 + 889144m^2 is described as a constant.The passages provide intermediate derivatives but do not show the constant’s value in the supplied text.Source: Mathematical Reasoning · p. 156
• For 9:02 AM to 6:14 PM, the appendix gives conflicting totals of 526 minutes and 552 minutes.One calculation ends at 526 minutes, while another sums 58 + 480 + 14 to reach 552 minutes.Source: Mathematical Reasoning · p. 157
• The cited passage poses a base-11 subtraction question beginning with 25237a, but it does not show the second operand or state any base-10 result.Source: Mathematical Reasoning · p. 158
• The base-11 subtraction is reported as 1261011 in base 11, while the text notes that the result depends on the unspecified value of digit a.The conversion shown is 223731 in base 10 to 1261011 in base 11.Source: Mathematical Reasoning · p. 158
• The fraction expression (2/20 * 3/(12/32))/((-2082)/(-1041)) is reported to simplify to 8328/5205.Source: Mathematical Reasoning · p. 159
• The appendix reports 22223690.876 + 2.056 = 22223692.932.Source: Mathematical Reasoning · p. 160
• One excerpt calculates i as 64.Source: Mathematical Reasoning · p. 161
• The highest common divisor of 25408 and 64 is 64.The excerpt states 25408 = 64 * 396 + 0.Source: Mathematical Reasoning · p. 161
• A separate excerpt gives the GCD of 25408 and 664 as 8.Source: Mathematical Reasoning · p. 161
• Among 1024, -790/31, and 2/11, the response identifies 1024 as the largest value.Source: Mathematical Reasoning · p. 163
• The fourth-largest value among -2/13, 3, 1, 1.91543, and -2/9 is 1.The response sorts the values as [3, 1.91543, 1, -2/9, -2/13].Source: Mathematical Reasoning · p. 163
• The arithmetic response evaluates (-7)/(-2) as 3.5 and 55629/(-2649) as -21, yielding -17.5.Source: Mathematical Reasoning · p. 164
• One factor-checking answer concludes 78 is not a factor by citing 920 and 903, while another finds x = 900 and a remainder of 12.The second answer also concludes that 78 is not a factor of x.Source: Mathematical Reasoning · p. 165
• Table 13 presents commonsense prompts in a five-option question format, including questions about vegetables and covering a corpse.The appendix also illustrates the format with questions about stage anxiety, ticketed events, and perjury.Source: Table 13 · p. 167, Mathematical Reasoning · pp. 168–170
• For the rock-searching question, the worked rationale selects “C. surface of earth” because it is the only option that makes sense.It rejects “ground” on the grounds that people do not stare at it while searching for rocks.Source: Mathematical Reasoning · p. 171
• The appendix’s tax-question example says a definitive answer is not possible without more context, despite selecting option E.It explains that “back taxes” refers to unpaid taxes from earlier years, whereas “current taxes” are due in the current year.Source: Mathematical Reasoning · p. 171

• The cat question’s answer is “falling,” which the explanation treats as a danger cats instinctively try to avoid.Source: Mathematical Reasoning · p. 172
• The carpet-and-drapes clue points to a hotel, described as the only place in town that was not booked solid.Source: Mathematical Reasoning · p. 172
• For thinking too quickly, the answer is “jump to conclusions,” since haste can lead to assumptions without full understanding.Source: Mathematical Reasoning · p. 172
• The fishing scenario points to relaxation because fishing is framed as leisure and a break from work.Source: Mathematical Reasoning · p. 173
• For “performing a type of,” the appendix selects “acting” as the best fit for a kind of performance.Source: Mathematical Reasoning · p. 174

Appendix

• Other examples select “cool off” after a man expresses anger violently, “White House” as the place for a presidential decree, and “sell products” as a salesman’s responsibility.The rationales connect each answer to calming down, presidential authority, or the job’s primary responsibility, respectively.Source: Mathematical Reasoning · pp. 174–175
• For going to bed, the rationale selects “falling asleep” (D) as the goal, distinguishing it from rest as a way to achieve that goal.It also rejects “lazy,” “insomnia,” and “dreaming” as the goal.Source: Mathematical Reasoning · p. 175
• For the Avalon question, the answer selects “English house” (C), describing Avalon as a legendary island and an English house as a plausible dwelling.The rationale rules out pain, meaningful work, cotton candy, and headache as things people live in.Source: Mathematical Reasoning · p. 175
• For the bar-stool question, the answer selects “drunker” (B) as the condition that would make Dan more likely to fall off.The rationale says the other options are not relevant to the question.Source: Mathematical Reasoning · p. 176

A. Appendix

• A fast-food restaurant is identified as the most likely place to find a hamburger.The explanation notes that hamburgers are commonly found on fast-food menus.Source: Mathematical Reasoning · p. 176
• A radio station is identified as the place where a broadcast studio is likely to be heard.The explanation says radio stations broadcast audio from a broadcast studio.Source: Mathematical Reasoning · p. 177
• A dining room is identified as the most likely room to have a sideboard on the counter.The answer given is option C, dining room.Source: Mathematical Reasoning · p. 178
• For the congregation question, the response selects “funnel” and explains that the other choices do not describe what congregations do when entering a church.It says congregations funnel into the church and rejects options such as running away, a wind instrument, or a container.Source: Mathematical Reasoning · p. 181
• The homeowner response says the sentence gives no information about what the homeowner must do beyond frowning at the gas price.The fragmented answer material also lists “receive mail” as an option, but the explanation emphasizes that the sentence does not indicate a later action.Source: Mathematical Reasoning · p. 181
• The peroxide examples contrast leaving it on overnight with applying it for 30 minutes and rinsing, while warning that prolonged application can damage hair.The appendix cautions that overnight use may cause burns or hair damage and says the shorter method may be less harsh.Source: Mathematical Reasoning · p. 182
• A closet is too warm to chill dough properly; the ideal temperature is 35–40°F, and excess warmth can make the dough rise too much.Chilling also helps develop the gluten that gives dough structure and elasticity.Source: Mathematical Reasoning · p. 183
• For a sunflower-shaped craft, the appendix says gluing sticks to an embroidery ring would make the structure sturdier and better able to hold its shape.Source: Mathematical Reasoning · p. 185
• For mixed grains with dates, the appendix favors the traditional recipe with dates and herbs over an option that adds jello.The explanation says the dates and herbs have complementary flavors, whereas jello is unusual and its sweetness would overwhelm the other ingredients.Source: Mathematical Reasoning · p. 185
• For recording-room sound dampeners, the appendix recommends hanging them behind the speaker to absorb reflections and reduce echo.It contrasts this with placing them where the voice projects, which it says is less effective because sound still reflects from other surfaces.Source: Mathematical Reasoning · p. 186
• For a confession phrase, the appendix selects “tell the truth about something” rather than “tell a lie.”The supplied answer text identifies the truth-telling option as correct.Source: Mathematical Reasoning · p. 187

Appendix

• The confession response says a confession must be truthful, because lying defeats the aims of seeking forgiveness and making amends.It adds that lying amounts to trying to avoid punishment.Source: Mathematical Reasoning · p. 187
• For attaching rope to latex, the response selects rubber bands, reasoning that glue can weaken and tear latex while rubber bands are flexible and removable.The answer identifies rubber-banding the rope in place as the preferred option.Source: Mathematical Reasoning · p. 187
• For a paper bird feeder, the response chooses an egg carton with seeds, arguing that seeds provide needed nutrients and the carton holds them securely.It contrasts this with eggshells, which it says would not provide a balanced diet.Source: Mathematical Reasoning · p. 189
• Knife-sharpening answer fragments include a refusal and instructions to use an unglazed ceramic mug’s rough base, with a 15-degree blade angle.One option specifies sharpening small knives, while another specifies large knives.Source: Mathematical Reasoning · p. 189

A. Appendix

• Using a comb to hold the nail is described as a safe, effective way to avoid hitting your thumb; placing the hammer in the comb is rejected as unsafe and likely to break the comb.The comb keeps the nail in place, and a clothespin or piece of tape is also suggested for holding it.Source: Mathematical Reasoning · p. 192

• For substituting potatoes in a dish, one response chooses cauliflower because it looks and tastes similar to potatoes.Source: Mathematical Reasoning · p. 193
• For increasing jalapeño heat, the excerpt includes both leaving seeds in and removing them, with an explanation that removing seeds and pith reduces heat.Source: Mathematical Reasoning · p. 194
• For succulent drainage, a response recommends a hole in the container bottom and describes rocks as an additional drainage layer.Source: Mathematical Reasoning · p. 195
• For deodorizing a diaper pail, the excerpt presents a spray-bottle option combining two parts water and one part vinegar.Source: Mathematical Reasoning · p. 196

A. Appendix

• Both 2:1 water-and-oil and water-and-vinegar sprays can deodorize a diaper pail; oil is gentler, while vinegar is more effective.The better choice depends on personal preference and odor severity.Source: Mathematical Reasoning · p. 196
• Suspend a candy thermometer’s bulb in the mixture without touching the pan bottom, since contact can produce an inaccurately high reading.The bulb should be submerged in the cooking candy and positioned away from the pan bottom.Source: Mathematical Reasoning · p. 197
• To “poke around” in a building means to walk and look around, not to poke each wall.The phrase describes casual or unsystematic exploration, whereas poking each wall would be a systematic search.Source: Mathematical Reasoning · p. 197
• “Tissues for wiping noses cleanly” is the preferred completion; wiping hands is grammatically correct but less common.The explanation favors nose-wiping because people are more likely to use tissues for that purpose.Source: Mathematical Reasoning · p. 198

Appendix

• A dish towel is the better choice for wiping dirty stains from a counter, while dry paint calls for a more suitable remover rather than a towel.The passage says dish towels absorb spills and stains but are not effective at breaking down paint.Source: Mathematical Reasoning · p. 199
• For an inexpensive portable bidet, the appendix suggests filling a soft water bottle with warm water and gently squeezing it as needed.It contrasts this with a dishwashing-liquid bottle, which may be too harsh and difficult to squeeze.Source: Mathematical Reasoning · p. 200
• A makeshift bottle bidet may not cleanse as thoroughly as a proper bidet or medical device, so the appendix suggests consulting a healthcare professional about hygiene options.The passage warns that soapy water could irritate skin and the bottle could be hard to squeeze.Source: Mathematical Reasoning · p. 200
• For knife care, the appendix recommends rinsing with soap, drying with a kitchen towel and then paper towel, and storing the knife dry; standard towels may leave lint.It advises paying special attention to drying the blade.Source: Mathematical Reasoning · p. 200
• Foam panels are presented as an easy, affordable sound-dampening option, while tearing apart a foam mattress is described as ineffective and impractical.The passages say panels trap sound energy and can be hung on walls or ceilings.Source: Mathematical Reasoning · p. 201
• For mixing salt and sugar, the response selects a fork over a soup can because its size and prongs make it easier to use and break up clumps.It also lists a whisk, spoon, hands, and food processor as alternatives.Source: Mathematical Reasoning · p. 202
• Responses disagree on whether “dog ride cat” is plausible: one calls it implausible because dogs are larger, while another says it can occur but is unusual.The implausibility response also argues that dogs are not suited to ride on cats; the other notes online examples of cats riding dogs.Source: Mathematical Reasoning · p. 202
• The response judges “clown catch baboon” implausible, reasoning that clowns are entertainers and baboons are large, powerful wild animals.It adds that clowns are not typically trained to catch or handle wild animals.Source: Mathematical Reasoning · p. 204
• “Clown catch baboon” is considered very implausible overall, though a trap, an injured or restrained baboon, or an experienced clown could make it possible.The response says such circumstances would be specific and that baboons are territorial and not easily captured.Source: Mathematical Reasoning · p. 204
• “Woman tug chef” is called implausible and grammatically incomplete because the relationship between the words and the intended action are unclear.The response suggests clearer alternatives such as “woman assists chef” or “woman pulls chef out of danger.”Source: A. Appendix · p. 25, Mathematical Reasoning · p. 204
• “Cup pour rice” is treated as plausible if it describes pouring rice into a cup, but the response calls the fragment ambiguous and awkwardly phrased.It notes that “cup pour” is not a common collocation and offers “pouring rice from a cup” or “using a cup to pour rice” as clearer wording.Source: Mathematical Reasoning · p. 205
• “Policeman bend lamp” is judged implausible because bending a lamp is not typical police work and sturdy lamps are difficult to bend.The response also describes physically bending a lamp as improbable, while allowing that a specific scenario could change the judgment.Source: Mathematical Reasoning · p. 205

A. Appendix

• For “stick shove snake,” responses treat the predicate as ambiguous, with one considering it plausible under a literal reading while noting that snakes are not typically associated with being shoved.Another response says the predicate does not make sense without additional context.Source: Mathematical Reasoning · p. 205
• For “woman whisk air,” responses call the predicate implausible despite its grammatical form, reasoning that air cannot be whisked because it is a gas.One response also describes the phrasing as vague and unclear in meaning.Source: Mathematical Reasoning · p. 206
• For “butterfly move gorilla,” responses judge the action implausible because of the animals’ substantial differences in size, strength, and physical abilities.One response specifically concludes that a butterfly cannot physically move a gorilla.Source: Mathematical Reasoning · p. 206
• For “policeman crop tree,” responses judge the predicate implausible, finding no clear relationship between policing and cropping or cultivating trees.One response notes that crop farming and tree cultivation are not typically within a police officer’s job.Source: Mathematical Reasoning · p. 206

• A response says police may encounter trees through investigations or tree-related calls, but interprets “crop” as cultivation or trimming inconsistent with police duties.Source: Mathematical Reasoning · p. 206
• “Pen drain sink” is judged implausible because the words lack a coherent relationship, with a pen unlikely to participate in draining a sink.Source: Mathematical Reasoning · p. 207
• For “man tackle lion,” one response calls it technically possible but dangerous and unlikely to succeed, while another labels it implausible because a human is outmatched.Source: Mathematical Reasoning · p. 207
• “Girl catch desert” is judged implausible as written because a desert cannot be caught, though a girl catching a creature in the desert would be plausible.Source: Mathematical Reasoning · p. 208

A. Appendix

• “Girl catch desert” is judged literally implausible because a desert cannot be caught, though metaphorical interpretations are possible.The response gives winning a race through a desert or capturing an audience’s attention as possible figurative readings.Source: Mathematical Reasoning · p. 208
• “Bush bury rice” is judged implausible because a bush is a plant and plants do not bury things, leaving no clear relationship between the words.The response allows that a highly specific scenario could make the predicate plausible, but says it is not the likely interpretation.Source: Mathematical Reasoning · p. 209
• “Skyscraper heat air” is ambiguous: it may describe heating or urban heat effects, but as a standalone predicate it is difficult to interpret.The response says skyscrapers can be heated and can contribute to the urban heat island effect, while emphasizing that context is needed.Source: Mathematical Reasoning · p. 210
• A fragment says skyscrapers can be heated by circulating hot water through pipes or by sunlight absorbed by concrete and glass.The absorbed solar heat can warm the air inside the building.Source: Mathematical Reasoning · p. 210
• For “bottle spill smoke,” responses judge the predicate implausible because bottles spill contents rather than smoke, while another calls its meaning unclear.The explanations note that smoke is a gas, not a liquid typically contained in or spilled from a bottle.Source: Mathematical Reasoning · p. 210
• For “ice cool girl,” responses variously treat the predicate as context-dependent or somewhat plausible, interpreting it as describing someone calm and composed.One response also suggests the phrase could imply stylishness, while noting its meaning may not be clear without context.Source: Mathematical Reasoning · p. 211
• For “bear bite girl,” responses range from likely plausible to indeterminate, with one noting bear attacks can occur but depend on circumstances.One response cites a bear attack on a six-year-old girl in Alaska in 2020, while another says more context is needed to judge likelihood.Source: Mathematical Reasoning · p. 211

• “Tube ejaculate bird” is judged implausible as an incoherent or nonsensical predicate.Source: Mathematical Reasoning · p. 212
• “Elephant haul nail” is judged implausible because elephants are not known to haul nails and lack hands or fingers to manipulate them.Source: Mathematical Reasoning · p. 212
• “Rabbit chew carrot” is judged plausible because rabbits eat carrots, though the response notes the predicate is not a complete sentence.Source: Mathematical Reasoning · p. 213
• “Pencil hang rope” is judged implausible and described as failing to convey a clear meaning.Source: Mathematical Reasoning · p. 214

A. Appendix

• “Pencil hang rope” is judged implausible in most contexts, though specific situations could make it plausible.The response suggests a pencil attached to a rope for art or as a weight as possible exceptions.Source: Mathematical Reasoning · p. 214
• “Ant sew curtain” is judged implausible because ants lack the abilities needed to sew.The response cites the physical and cognitive demands of sewing.Source: Mathematical Reasoning · p. 215
• “Air cool house” is considered somewhat plausible, but its phrasing is incomplete and unclear.The response suggests clearer alternatives such as “air circulation cools a house.”Source: Mathematical Reasoning · p. 215
• “Man pluck string” is presented as a plausible action, but its likelihood is difficult to assess without more context.The response suggests it could refer to plucking a string on a musical instrument.Source: Mathematical Reasoning · p. 216
• “Spatula knead water” is judged implausible because water is a liquid and a spatula is unsuitable for kneading.The response says kneading typically involves solid or semi-solid substances and is done by hand or with a dough hook.Source: Mathematical Reasoning · p. 216
• “Cow open cream” is judged implausible because cows do not typically open containers, though the predicate might have another interpretation with more context.The response notes that “open” could mean “not pregnant” for dairy cows, but says the phrase is unclear without additional information.Source: Mathematical Reasoning · p. 216
• “Child pack truck” is treated as ambiguous: packing a truck with items could be plausible, whereas packing a child into a truck is described as dangerous and implausible.The response says a definitive judgment is difficult without knowing the intended meaning or having more context.Source: Mathematical Reasoning · p. 217

Appendix

• The appendix labels “policeman breed bird” implausible, explaining that breeding birds is not a typical policing duty or commonly observed activity.It allows that a policeman might breed birds as a hobby, but says the predicate remains implausible as stated.Source: Mathematical Reasoning · p. 218
• For the dirty-blanket premise, the response favors cleaning the inside over cleaning the surfaces, but notes that both could have been cleaned and the evidence does not settle the choice.It reasons that cleaning only the surfaces may leave the inside dirty.Source: Mathematical Reasoning · p. 220
• For a patient with a generalized acute viral disease, the response favors the hypothesis that he was assigned to the ward of mumps, citing the mumps virus.The supplied text identifies the mumps-virus connection and states that the ward assignment better explains the premise.Source: Mathematical Reasoning · p. 221

A. Appendix

• For the mumps case, hypothesis 2—that the patient was assigned to a mumps ward—is judged a better explanation than ischemia, since ward exposure could lead to infection.The appendix says hypothesis 1, diagnosis with ischemia, does not explain the premise well.Source: Mathematical Reasoning · p. 221
• For the “stigma” premise, the appendix frames the alternatives as wanting to know a flower’s female reproductive structure or not knowing what “CM” refers to.It identifies “stigma” as a botanical term for a female reproductive structure in flowers.Source: Mathematical Reasoning · p. 222

Appendix example 5

• The explanation contrasts the established transmission route of Leptospirosis with inhalants, which are not known to cause it.The passages describe infection through contact with infected animals’ urine and state there is no evidence inhalants cause Leptospirosis symptoms or are directly linked to the disease.Source: Mathematical Reasoning · p. 223
• The appendix favors contaminated rat urine over inhalant abuse as the explanation for a boy’s Leptospirosis diagnosis.It says there is no known direct link between inhalant abuse and Leptospirosis, while exposure to contaminated rat urine is a known risk factor.Source: Mathematical Reasoning · p. 223

Appendix example 6

• For the premise “Her eyes are affected,” the appendix says the information is insufficient to determine whose eyes are affected or what is causing the problem.The options name Lucy, who has diabetes, and Sophia, who has facial protrusions; the text says diabetes can affect the eyes, while the protrusions could obstruct vision or cause other eye-related issues.Source: Mathematical Reasoning · p. 224

Appendix

• The eye-related hypotheses cannot be confirmed from the premise alone, which does not specify symptoms or establish that the facial protrusions explain the eye issue.The response notes that diabetes-related eye complications are possible but not directly indicated, while the protrusions may be unrelated.Source: Mathematical Reasoning · p. 224
• Diabetes is judged more likely than xanthelasmas to explain the eye problems because it is a common cause, though the premise gives no specific symptoms.The response says xanthelasmas remain possible, particularly if the person has other signs of high cholesterol.Source: Mathematical Reasoning · p. 224
• For Lucy’s work awareness, the hypothesis that she was more successful than Tom is favored because success directly reflects work performance, unlike spiritual awareness.The response also describes success as more specific and measurable than spiritual awareness.Source: Mathematical Reasoning · p. 225

• For item 8, neither a nasal problem nor brainstem nuclei strongly explains unclear vision; the example suggests the eyes or visual pathways instead.Source: Mathematical Reasoning · p. 226
• For item 9, Yuan Longping’s contribution to solving starvation better explains his status as a great historical hero than the unrelated laxative incident.Source: Mathematical Reasoning · p. 227
• For Luna, getting a collie is the better explanation for her care of her dog because it is more specific and directly related to the premise.Source: Mathematical Reasoning · p. 228
• For Betty’s salmonella premise, both hypotheses are possible, but her baby’s poor health is the more specific explanation, while diarrhea is a likely infection effect.Source: Mathematical Reasoning · p. 229

A. Appendix

• For Betty’s salmonella infection, hypothesis 1 better explains why she was infected, although the passage says more health and contextual information is needed for a definitive conclusion.The alternative that the infection caused her diarrhea explains that symptom but not the infection’s origin.Source: Mathematical Reasoning · p. 229
• For the raspberry premise, hypothesis 2 is more relevant because raspberries belong to the rose family, but the appendix says neither option satisfactorily explains their presence.The aralia hypothesis does not connect to raspberries, and more direct evidence or additional hypotheses are needed.Source: Mathematical Reasoning · p. 230
• For the lactating woman’s hydrazine premise, neither hypothesis is scientifically plausible, though the baby’s adverse reactions better explain the effect than breast enlargement.The passage describes hydrazine as toxic and potentially dangerous, and says ingestion could lead to adverse reactions in a breastfed baby.Source: Mathematical Reasoning · p. 231
• In the hydrazine-lactation example, the text warns of serious toxicity and possible infant exposure, and advises immediate medical attention if a lactating woman has taken hydrazine.It lists seizures, respiratory problems, liver damage, and death among possible harms.Source: Mathematical Reasoning · p. 231
• The hydrazine discussion says there is no evidence that it enlarges breasts and that the competing hypotheses lack scientific support, leaving their explanatory comparison uncertain.A separate fragment nonetheless favors hypothesis 2 over the breast-enlargement hypothesis.Source: A. Appendix · p. 118, Mathematical Reasoning · p. 231
• For the church-prayer premise, the response favors hypothesis 2’s flood-related suffering as a plausible explanation but notes that the premise gives no context to determine why they prayed.It contrasts this with hypothesis 1’s medical-treatment explanation, which lacks a clear connection to going to church.Source: Mathematical Reasoning · p. 232

• For item 15, the answer selects hypothesis 1 because studying atomic paramagnetism explains atoms’ permanent magnetic moments, while myofilament counts do not.Source: Mathematical Reasoning · p. 233
• For item 16, the response says hypothesis 2 better explains why batteries containing caustic chemical mixtures should not be casually thrown away.Source: Mathematical Reasoning · p. 234

A. Appendix

• For the battery premise, hypothesis 2 is judged better because it gives a general explanation applying to all batteries, unlike hypothesis 1’s specific curium incident.The broader explanation links improper disposal to harmful chemicals in batteries.Source: Mathematical Reasoning · p. 234
• In the stimulant example, inconsistent pronouns and speculative hypotheses leave the cause undetermined, prompting a request to clarify the individual’s gender.Suggested follow-up includes further drug testing and asking Jack about his drug use and why he may have taken a new drug.Source: Mathematical Reasoning · p. 235
• The chocolate hypothesis is considered unlikely to be the primary cause of Jack’s positive stimulant test, while the alternative suggests he took a different drug.The example notes that chocolate contains caffeine, but typically not enough to cause a positive stimulant result.Source: Mathematical Reasoning · p. 235

• For Jack’s positive stimulant test, the answer favors hypothesis 2 because it directly implies stimulant use, while acknowledging that other explanations may exist.Source: Mathematical Reasoning · p. 235
• The gum response rejects hypothesis 1 as too broad and unsupported, noting that inflammation could have other causes, including infection, allergy, or a side effect.Source: Mathematical Reasoning · p. 236
• For inflamed gums, neither hypothesis is judged a good explanation, though hypothesis 2 is slightly better supported; the answer says more information is needed.Source: Mathematical Reasoning · p. 236
• For Bob decreasing his speed to 50 km/h, the answer selects hypothesis 1, that the racers entered the deceleration phase, as the better explanation.Source: Mathematical Reasoning · p. 237

Appendix

• For the director-and-staff premise, the analysis ultimately favors Hypothesis 2—answering all the director’s questions—over Hypothesis 1—using codes instead of spoken words.Answering all the questions is more directly connected to the staff’s product knowledge, whereas using codes does not clearly explain it.Source: Mathematical Reasoning · p. 238
• The premise alone does not settle which staff hypothesis is true, and using codes could indicate discomfort, poor product understanding, or fear of difficult questions.The analysis says additional context is needed to decide between the hypotheses.Source: Mathematical Reasoning · p. 238
• For the premise that they found much food, neither hypothesis is definitive, but the hunter-catching-squirrels hypothesis is judged more likely because it offers a specific explanation.The alternative—that they were the island’s first settlers—is less directly linked to finding food.Source: Mathematical Reasoning · p. 239

A. Appendix

• The first-settlers alternative proposes discovering island resources, possibly abundant food, but is described as less specific about how people found a lot of food.The passage notes that first settlers might have found either much or very little food.Source: Mathematical Reasoning · p. 239
• For the island-food premise, Hypothesis 1 is ultimately judged more likely than the first-settlers hypothesis, though the passage says the evidence cannot establish which is likelier.Both hypotheses could explain finding much food, and their likelihood may depend on the circumstances.Source: Mathematical Reasoning · p. 239
• For the parasite example, the hypothesis that eggs on cloves hatch into parasites is preferred because it aligns with the parasites’ typical life cycle.The alternative says the eggs die after a few days.Source: Mathematical Reasoning · p. 240
• For Jim’s job-hunting stress, the appendix favors distress-related sleep trouble over cramps because sleep disturbance is presented as a more common stress effect.It notes that cramps can have other causes and are not a specific stress symptom.Source: Mathematical Reasoning · p. 241
• For Tom’s parade preparation, one passage says neither hypothesis can be distinguished from the premise alone, while a later rationale favors going to the parade street.The latter rationale links the street location to being able to see and join the parade.Source: Mathematical Reasoning · p. 242
• For the hotel premise that service staff lack management, the fragments conflict: one says neither hypothesis explains the effect, while another favors declining service quality.Source: Mathematical Reasoning · p. 243
• The hotel premise alone does not establish that management gaps caused the decline; service quality may have been low already or other factors may contribute.The appendix says more information or context is needed to determine which hypothesis, if either, explains the premise.Source: Mathematical Reasoning · p. 243
• Hypothesis 1 is judged the better explanation for declining hotel service quality because staff lacking support could affect service, whereas the coursework in Hypothesis 2 does not explain it.The appendix also cautions that the premise alone does not establish why service quality declined; it may have been low already or other factors may contribute.Source: Mathematical Reasoning · p. 243
• For the premise given, Hypothesis 1 is more likely, while Hypothesis 2 does not provide direct evidence for that premise.Source: Mathematical Reasoning · p. 244

• The fish electrocution explanation (H1) is more likely than freshwater exposure (H2), since a tank-connected wire could expose the fish to significant electrical current and H2 lacks evidence.Source: Mathematical Reasoning · p. 244
• For Mary learning the Greek alphabet, H1—that she first learns alpha—better explains the premise than H2—that she first learns A—although neither hypothesis is entirely accurate.Source: Mathematical Reasoning · p. 245

A. Appendix

• The carpenter premise alone is insufficient to determine which hypothesis is correct; additional context is needed to establish the cause.Source: Mathematical Reasoning · p. 247
• Hypothesis 2, which calls for someone skilled in construction, is judged the better explanation for recruiting a carpenter.The text contrasts it with a hypothesis about a family’s inability to afford tuition, which does not directly explain the recruitment.Source: Mathematical Reasoning · p. 247
• The leech example identifies Hypothesis 2—the 18-inch creature attaching to Tom’s leg and sucking blood—as the better explanation.The premise identifies the creature as a leech, described in the passage as a blood-sucking parasite.Source: Mathematical Reasoning · p. 248
• An eight-inch leech is more plausible than an 18-inch creature because its size is within a typical range, while an 18-inch leech would be extremely rare and the creature could be another predator.The analysis concludes that Hypothesis 1 better explains the premise.Source: Mathematical Reasoning · p. 248
• The multi-hop prompt examples provide context about plant genera and ask what category Tritonia and Phyteuma share.The appendix labels Table 15 as multi-hop reasoning prompts and includes the question about the two plant names.Source: Table 15 · p. 249, Mathematical Reasoning · p. 250
• Gioconda Vessichelli is described as an opera singer who pioneered a fusion of opera and classical Hindi music.The snippet notes her Bollywood work and performances with artists including Sukhwinder Singh and Anup Jalota.Source: Mathematical Reasoning · p. 251
• Heidi Brunner’s roles at the Vienna State Opera included Annio in “La clemenza di Tito.”The segment also asks which theater she played Annio in.Source: Mathematical Reasoning · p. 251
• The remaining material supplies context passages and question prompts about musicians, albums, and films.Examples include a question about James Aaron Diamond’s first solo album and film descriptions for High Noon and Oklahoma!.Source: Mathematical Reasoning · pp. 252–253

Appendix

• Film and actor-history snippets accompany a question about the actor who played Prewitt and the event centered on by the 1953 Fred Zinnemann film.The background includes entries on films such as Five Days One Summer and People on Sunday, and an overview of Rod Steiger’s career.Source: Mathematical Reasoning · p. 253
• Erotic-thriller context distinguishes the genre by its focus on illicit romance or erotic fantasy and includes entries on Wild Things films.The passages note that sex or nudity alone does not make a thriller an erotic thriller.Source: Mathematical Reasoning · p. 254
• Sports-related background covers rugby and FIFA World Cups, Belgian football, and national football teams including Sweden and Ukraine.Examples include the 1991 Rugby World Cup, the 2002 FIFA World Cup, and Belgium’s 1968–69 football season.Source: Mathematical Reasoning · p. 255

A. Appendix

• The sports contexts cover Ukraine’s Euro qualification, Algeria’s World Cup history, and Switzerland’s women’s team qualifying for the 2015 Women’s World Cup.One attached question asks where the World Cup was hosted when Algeria first reached the round of 16.Source: Mathematical Reasoning · p. 255
• Music examples describe releases and artists associated with Avantasia, including Ghostlights, The Mystery of Time, and Tobias Sammet.The excerpt also includes material on other metal projects and artists, such as Dezperadoz, Revolution Renaissance, Dionysus, Ayreon, Edguy, and Tobias Exxel.Source: Mathematical Reasoning · p. 256
• Film and television contexts include Cash McCall, Maverick episodes and films, The Notebook, and several international romantic dramas.A related question asks who directed the 2004 romantic drama featuring James Garner.Source: Mathematical Reasoning · p. 257
• Biographical entries describe activist Sheena Duncan and Beninese Vodou artist Yves Apollinaire Pede, alongside a question about Tobias Sammet’s nationality.The excerpt also contains a filmography summary of N. T. Rama Rao Jr.Source: Mathematical Reasoning · pp. 256–258
• Biographical entries describe South African anti-apartheid figures including Nelson Mandela, Oliver Tambo, Steve Biko, and Samuel Sisulu.They cover roles, activism, and—in Mandela’s case—his presidency and anti-apartheid work.Source: Mathematical Reasoning · p. 258
• Las Vegas reference entries describe casino properties and landmarks, including the Playboy Club, MGM Grand, and Tropicana–Las Vegas Boulevard intersection.The accompanying question asks about the room count of the hotel and casino where the Playboy Club was located.Source: Mathematical Reasoning · p. 259
• Music-related context covers the Portland garage punk band The Rats and releases by Slade and Dirty Projectors.The Rats’ entries mention their albums and lineup changes; the other entries describe Slade’s “All Join Hands” and Dirty Projectors’ Rise Above.Source: Mathematical Reasoning · p. 260
• The music context includes band and album histories, followed by a question about when the rock band that sang “All Join Hands” rose to prominence.The snippets include entries on Join Hands and Siouxsie and the Banshees, among other acts.Source: Mathematical Reasoning · p. 260
• Another music-context item asks which English beat rock band performed “My Sentimental Friend,” alongside entries on Herman’s Hermits and Mick Quinn.Source: Mathematical Reasoning · p. 261
• The appendix also provides profiles of attorney-general offices and officials in Idaho, Bangladesh, Washington, Brazil, and Colombia.The Bangladesh entry identifies Mahbubey Alam as attorney general; the Washington entry identifies Bob Ferguson.Source: Mathematical Reasoning · p. 262
• The excerpt gives Attorney General background for Colombia, England and Wales, Wyoming, and Sri Lanka, followed by a question about Missouri’s current Attorney General.The snippets describe legal roles and officeholders; they do not provide an answer to the Missouri question.Source: Mathematical Reasoning · p. 262
• The excerpt includes music biographies and band information about Twiggy Ramirez, Sunstorm, Marilyn Manson, and Joe Lynn Turner, plus a comparison question.Topics include members’ roles, musical projects, and albums.Source: Mathematical Reasoning · p. 263
• The excerpt also presents biographical context about Brazilian jiu-jitsu figures and academies, including Mauricio Villardo, Royler Gracie, and Gracie Humaitá.Other profiles cover practitioners, instructors, and related figures.Source: Mathematical Reasoning · p. 264

• A biographical context about Saulo Ribeiro’s Brazilian Jiu-Jitsu training is followed by a question about a student of Rolls Gracie who ran Gracie Humaitá.Source: Mathematical Reasoning · p. 264
• Example 17 combines reference snippets on Ingerophrynus gollum, The Burning Red, Gollum, From Unknown Worlds, and Suanfa tongzong with a follow-up question.Source: Mathematical Reasoning · p. 265
• Example 18 presents film and biographical passages about Victor Heerman and Lloyd Corrigan, then asks which award both won.Source: Mathematical Reasoning · p. 266
• Example 19 begins with passages about the Yeomen of the Guard and the Namiestnik of Poland.Source: Mathematical Reasoning · p. 267

A. Appendix

• The segment presents background entries on historical figures and other topics alongside questions, without stating appendix methods or research results.Examples include entries on Bolesław I, Charles Fort, and the Kasi tribe, interspersed with question prompts.Source: Mathematical Reasoning · pp. 267–269

• A context-based question asks in what century Tecumseh became leader of a large confederacy, with background identifying his brother Tenskwatawa as the Prophet.Source: Mathematical Reasoning · p. 269
• Another example gives Pat Hingle’s biography and asks which Western TV series a close actor friend succeeded in.Source: Mathematical Reasoning · p. 270
• Example 23 introduces context entries on fictional diaries, rabies in popular culture, and Illinois governor James R. Thompson.The supplied excerpt begins Thompson’s biography, identifying him as Illinois’s 37th and longest-serving governor.Source: Mathematical Reasoning · p. 271

Appendix examples

• One example asks whether Jim Thompson or Theodore H. White wrote more fictional works, following context about Thompson and White.The supplied passages pose the question but do not provide its answer.Source: Mathematical Reasoning · p. 271
• Another example asks which county contains Annapolis Junction, using context that identifies Anne Arundel County as named for Cecilius Calvert’s wife.The passages identify Anne Arundell as Cecil Calvert’s wife but do not state the answer to the location question.Source: Mathematical Reasoning · p. 272
• A further example begins with context on the 2014 film Obsessed and biographies of directors and screenwriters, including Kim Dae-woo.No question for this example appears in the supplied passages.Source: Mathematical Reasoning · p. 273

A. Appendix

• Another question asks what Kim Dae-woo’s directing-debut film was about, after context entries on films and directors.The excerpt also includes reference entries on films such as Juice and Bill & Ted’s Bogus Journey.Source: Mathematical Reasoning · p. 273
• A question asks which store at Gurney Paragon was founded in Belgium in 1926, following context about the mall and several unrelated entities.The mall entry identifies Gurney Paragon as a shopping mall in George Town, Penang.Source: Mathematical Reasoning · p. 274
• The excerpt pairs background material on the Yankton Sioux treaty and United States v. Causby with a question about the author of An Inconvenient Truth.The passages describe the treaty and airspace case, then ask about the author's role in the United States government.Source: Mathematical Reasoning · p. 275
• Other examples combine Lancaster School and nearby Iowa community histories with a question about the county seat associated with Keokuk Falls, Oklahoma.The excerpt also includes a question about the city containing the Museum of Jurassic Technology, following passages about the museum and related figures.Source: Mathematical Reasoning · pp. 276–277
• A final excerpt gives brief descriptions of High Street in Fremantle and Lincoln and the River Farset in Belfast.Source: Mathematical Reasoning · p. 278
• The entries describe sites in Belfast, Newport, Middletown, London, and Westwood, including the River Farset, Friars Walk, Russell House, and a historic district.They give location, architectural, and historical details rather than findings from the paper’s LLM evaluation.Source: Mathematical Reasoning · p. 278
• Other entries describe the Brooklyn High Street subway station and Sheffield’s city-centre High Street as a shopping thoroughfare.The station entry notes its A and C train service; the Sheffield entry describes its route and shops.Source: Mathematical Reasoning · p. 278
• Bonifacio High Street is described as a mixed-use development with open-air and indoor retail, leisure, and residential sections.The appendix ends by asking when the company that owns it was founded, but does not provide an answer.Source: Mathematical Reasoning · p. 278