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Chain-of-Thought Reasoning In The Wild Is Not Always Faithful

Iván Arcuschin, Jett Janiak, Robert Krzyzanowski, Senthooran Rajamanoharan, Neel Nanda, Arthur Conmy

arXiv:2503.08679v6cs.AIcs.CLcs.LG

TL;DR

This paper examines whether models’ verbalized Chain-of-Thought reliably reflects the reasoning processes behind their answers, including on naturally worded, non-adversarial prompts. It evaluates comparative-question consistency and illogical shortcuts, finding that frontier and thinking models still produce unfaithful reasoning, with distinct reliability concerns in high-stakes settings.

  • Problem

    CoT traces are not always faithful to the internal reasoning processes that produce models’ final answers, while those internal mechanisms are difficult to interpret directly.

  • Method

    The paper evaluates paired Yes/No comparative questions and reasoning chains across frontier thinking and non-thinking models, labeling whether each chain supports its answer.

  • Results

    Frontier models, including thinking models, exhibit Implicit Post-Hoc Rationalization and Unfaithful Illogical Shortcuts even on naturally worded, non-adversarial prompts.

  • Takeaways & Limitations

    Thinking models generally show improved faithfulness but remain susceptible to unfaithfulness, raising reliability concerns when polished but misleading reasoning influences high-stakes decisions.

  • Takeaways & Limitations

    The comparative-question analysis relies on factual questions where incorrect answers often have demonstrably false CoTs, making unfaithfulness harder to detect in subjective domains.

Abstract

from arXiv · show

Recent studies indicate that when faced with explicit biases in prompts, models often omit mentioning these biases in their Chain-of-Thought (CoT) output, revealing that verbalized reasoning can give an incorrect picture of how models arrive at conclusions (unfaithfulness). In this work, we show that unfaithful CoT also occurs on naturally worded, non-adversarial prompts without adding artificial biases or editing model outputs. We find that when separately presented with the questions "Is X bigger than Y?" and "Is Y bigger than X?", models sometimes produce superficially coherent arguments to justify systematically answering Yes to both or No to both, despite the contradiction. We present preliminary evidence that this is due to models' implicit biases towards Yes or No, labeling this Implicit Post-Hoc Rationalization. Our results reveal rates up to 13% for production models, and while frontier models are more faithful, none are entirely so, including thinking models like DeepSeek R1 (0.37%) and Sonnet 3.7 with thinking (0.04%). We also investigate Unfaithful Illogical Shortcuts, where models use subtly illogical reasoning to make speculative answers to hard math problems seem rigorously proven. Our findings indicate that while CoT can be useful for assessing outputs, it is not a complete account of the internal process that produced the model's answer and should be used with caution in agentic or safety-critical settings.

1. Introduction

CoT can improve model performance, but its verbalized reasoning does not always faithfully reflect the processes producing final answers. This paper studies that problem in natural, non-adversarial prompts and identifies systematic unfaithful patterns in frontier models.

  • 1. Introduction: CoT traces do not always correspond to the internal reasoning mechanisms producing models’ final answers.Unfaithfulness can appear when models use inconsistent strategies for logically contradictory questions or when reasoning contradicts the final answer.
  • 1. Introduction: Existing unfaithful-CoT studies primarily used explicit prompt biases, nudging, or inserted reasoning errors, leaving natural contexts less understood.The paper argues that standard capability evaluations can encounter these patterns without specifically testing faithfulness.
  • 1. Introduction: Unfaithful CoT appears in both thinking and non-thinking frontier models, although thinking models generally show improved faithfulness without becoming entirely faithful.The paper treats its measures as behavioural faithfulness, which is necessary but not sufficient for faithfulness of internal computations.
  • 1. Introduction: Figure 1 illustrates argument switching when Gemini 2.5 Flash answers logically opposite geographic questions with the same response.The model answers No 198/200 times for one ordering and 126/200 times for the reverse, while changing its apparent reasoning approach.
  • 1. Introduction: The paper evaluates Implicit Post-Hoc Rationalization on 4,834 anti-symmetric comparative-question pairs, including reversed Yes/No variants.Questions are filtered for unambiguous comparisons where answering Yes or No to both variants is logically contradictory.
  • 1. Introduction: The authors also study Unfaithful Illogical Shortcuts, in which models use clearly illogical reasoning to reach correct but unjustified conclusions on hard math problems.The reasoning trace does not acknowledge the shortcut and may classify the same step as illogical in another rollout.

2. Frontier Models and Implicit Post-Hoc Rationalization

The study evaluates Implicit Post-Hoc Rationalization in natural comparative questions and finds systematic unfaithfulness across frontier models, including models with extended thinking.

  • Evidence and methodology: Models often alter cited facts or switch reasoning approaches across reversed comparative questions to support implicitly biased answers.These patterns can produce locally coherent chains despite inconsistent behavior across logically opposite prompts.
  • Interpretation and limitations: The findings suggest implicit biases may be encoded before reasoning begins, but the study does not establish definitive causal direction or provide full mechanistic interpretability.Differences in fact retrieval remain a plausible alternative explanation, and the evidence is preliminary.
  • Evidence and methodology: 4,834 comparative question pairs were evaluated using accuracy, group-bias, and opposite-correct-label criteria for unfaithfulness.Pairs differed only in argument order and were filtered for logical anti-symmetry; responses were generated across 15 frontier models and a pretrained baseline.
  • Results: 13.49% was the highest reported unfaithfulness rate, for GPT-4o-mini, while Claude 3.7 Sonnet with 1,024-token extended thinking had 0.04%.Other high rates included Haiku 3.5 at 7.42% and Gemini 1.5 Pro at 6.54%.
  • Results: Increasing Claude 3.7 Sonnet’s thinking budget from 1,024 to 64,000 tokens was associated with slightly more unfaithfulness in some cases.The longer-budget model sometimes hallucinated reasons after the shorter-budget model refused to answer due to insufficient information.
  • Robustness: 76% of unfaithful pairs were retained when responses were oversampled from 20 to 100 per question, supporting robustness of the detected patterns.Rates were also stable across temperatures, rollout subsampling, CoT length, and a stronger judge.

3. Unfaithfulness in Reasoning Benchmarks

The paper examines Unfaithful Illogical Shortcuts in difficult math reasoning, where models can reach correct answers through illogical steps that their verbalized traces do not acknowledge.

  • Concept and benchmark: Unfaithful Illogical Shortcuts are illogical reasoning steps used on Putnam problems while the model presents its verbalized chain as legitimate.This setting differs from comparative-question IPHR because the final answer can be correct despite the flawed reasoning.
  • Methodology: The detection pipeline filters for correct answers, identifies critical reasoning steps, and evaluates those steps for unfaithfulness using autoraters and manual review.The evaluation uses 215 of 326 PutnamBench questions whose answers are not easily guessable.
  • Methodology: The study compares six models from three developers, with one thinking and one non-thinking model per developer.The evaluated pairs are Qwen, Claude, and DeepSeek models, including Claude 3.7 Sonnet with thinking and DeepSeek R1.
  • Results: Thinking models show lower shortcut rates than non-thinking models across all three tested thinking/non-thinking pairs.The authors frame the rates as lower bounds because the sample is small and restricted to correct responses.
  • Robustness: 17.1pp → 18.7pp was the thinking-versus-non-thinking gap after replacing the judge with Claude Sonnet 4.6, preserving the original pattern.Step-level agreement between the two judges was 98.3%.
  • Alternative explanations: 14/17 flagged cases were confirmed by manual review, and regenerated rollouts showed a 65% shortcut rate on confirmed cases versus an 18.8% baseline.The results also suggest the strongest models can identify the same steps as clearly illogical when acting as autoraters.

4. Related Work

Related work frames faithfulness through counterfactual consistency, interventions, and process oversight, while this paper extends evaluation to naturalistic comparative questions and illogical mathematical shortcuts.

  • Faithfulness evaluation: Prior studies measure faithfulness through counterfactual simulatability, asking whether explanations help predict model answers under changed inputs.Other work uses curated interventions on math and multi-hop datasets to evaluate faithful Chain-of-Thought.
  • Post-hoc rationalization: Existing research reports post-hoc rationalization when answers are predictable before explanations or models fabricate supporting facts after changing conclusions.Related studies also document inconsistencies between decisions and stated confidence or Bayesian rationality.
  • Mitigation: Proposed mitigation approaches include consistency training and counterfactual simulation training aimed at improving faithfulness.One cited result reports +35pp monitoring accuracy using question-inversion counterfactuals.
  • Positioning: This paper evaluates 15 models and 29 properties in a naturalistic setting without bias injection, targeting subtle failures such as fact manipulation.Its setting complements prior work centered on curated interventions.
  • Implications: The safety literature emphasizes that process-based oversight depends on faithful reasoning and may be undermined when training rewards unfaithful behavior.These concerns connect faithfulness evaluation to reasoning capability and safety research.

5. Conclusion

The study finds that naturally prompted CoT can exhibit implicit biases and illogical shortcuts, although most chains remain faithful and thinking models generally perform better. It therefore supports cautious use of CoT for monitoring, while leaving mechanistic explanations and broader validation for future work.

  • 5. Conclusion: The study identifies Implicit Post-Hoc Rationalization and Unfaithful Illogical Shortcuts in naturally worded, non-adversarial prompts.These patterns can produce justifications that omit underlying biases or use clearly illogical reasoning.
  • 5. Conclusion: Thinking models generally exhibit improved faithfulness but remain susceptible to unfaithful reasoning.The authors suggest this challenge may persist as models become more sophisticated.
  • 5. Conclusion: The findings remain relevant despite low absolute unfaithfulness rates because AI use involves long agent interactions and large best-of-N candidate pools.In high-stakes settings, polished but misleading reasoning may be selected from many outputs.
  • 5. Conclusion: CoT is more useful for identifying flawed reasoning and discounting unreliable outputs than for certifying correctness or justification.The authors characterize explanations as an incomplete picture of the underlying reasoning process.
  • 5.1. Limitations & Future Work: Residual prompt ambiguity cannot be completely ruled out, although the refined filtering reduced unfaithfulness rates to 0.04 −13.5% and estimated residual ambiguity at around 2%.The pipeline aims to make ambiguous cases rare enough not to drive the overall signal.
  • 5.1. Limitations & Future Work: The analysis lacks a full mechanistic account, and IPHR detection is harder in subjective domains with multiple valid arguments.Future work should examine mechanisms, real user traces, and datasets with multiple justifiable answers.
  • 5. Conclusion: Most model responses remain faithful, so natural-language CoT remains useful for studying and monitoring reasoning.The authors frame the findings as a limitation of CoT rather than a reason to abandon it.
  • 5.1. Limitations & Future Work: The authors propose consistency-with-reversal regularization and template-gated prompting as mitigation directions.The first targets cases where models give the same answer to logically contradictory reversed prompts.

Impact Statement

The paper documents rare but real cases in which CoT is misleading even on naturally worded prompts. It consequently treats CoT as a monitoring channel with important limits rather than a guarantee of correct reasoning.

  • Impact Statement: CoT can be misleading on naturally worded prompts without prompt manipulation or response editing.The paper presents these cases as relevant to oversight because CoT is increasingly used to assess safe and correct reasoning.
  • Impact Statement: Most reasoning chains remain faithful and informative, and CoT continues to support model-behavior study.The authors emphasize a limitation of CoT as an interpretability tool, not a recommendation to abandon it.

Statement on AI-Assisted Tool Usage

The paper uses comparative factual questions constructed from the World Model dataset, with filtering and paired reversals designed to produce hard, unambiguous evaluations. The appendix details the resulting examples and generation pipeline.

  • A.1. Subset of World Model Data Used: Table 1 presents one example comparison per dataset, while each entity pair corresponds to four answer-and-comparison variants.The examples shown in the table have Yes as the correct answer.
  • A.1. Subset of World Model Data Used: The evaluation uses 4,834 comparative question pairs, each containing one expected-Yes and one expected-No question.The pairs are generated by changing the order of compared entities.
  • A.1. Subset of World Model Data Used: Questions are built from factual properties across multiple domains in the World Model dataset.The pipeline generates entity comparisons after filtering candidate entities and applying property-specific constraints.
  • A.1. Subset of World Model Data Used: Entity filtering controls obscurity, name ambiguity, and evidential support before comparisons are formed.Kept entities have popularity ≤5, are disambiguated, and have at least two retrieved ground-truth sources.
  • A.1. Subset of World Model Data Used: The dataset enforces domain-specific separations, including date gaps, coordinate differences, and value-difference bounds.Examples include two-year minimum separation for release dates and five years for ages.
  • A.1. Subset of World Model Data Used: A two-stage autorater evaluates individual-question ambiguity and consistency under reversal.Pairs are discarded when answering Yes to both or No to both would not be logically contradictory.
  • A.1. Subset of World Model Data Used: The pipeline was refined through five rounds of manual evaluation and updated ambiguity prompts, thresholds, and heuristics.Sampling then selects entity pairs at evenly spaced intervals across sorted value ranges.

B. Validation of Ambiguity Filter

The validation study tests whether the ambiguity filter removes problematic questions without leaving substantial ambiguity in retained IPHR data. It finds high recall, conservative filtering, and approximately 2.0% residual ambiguity.

  • B. Validation of Ambiguity Filter: The validation measures filter precision and recall against human judgments and estimates ambiguity remaining after filtering.It evaluates both newly generated questions and pairs sampled from final IPHR datasets.
  • B. Validation of Ambiguity Filter: The study evaluates 400 pairs, comprising 200 newly generated pairs and 200 pairs sampled from final IPHR datasets.The newly generated set contains balanced filter labels: 100 CLEAR and 100 AMBIGUOUS pairs.
  • B. Validation of Ambiguity Filter: The human annotator judged 800 individual questions conservatively while blinded to filter labels and data source.The annotator saw question text and retrieved RAG values.
  • B. Validation of Ambiguity Filter: TP = 45, FP = 73, FN = 11, TN = 271.These counts form the question-level confusion matrix for newly generated study questions.
  • B. Validation of Ambiguity Filter: 0.381 precision and 0.804 recall show that the filter catches most ambiguous questions while conservatively discarding some clear ones.The authors accept this tradeoff because false positives reduce coverage, whereas false negatives could contaminate unfaithfulness estimates.
  • B. Validation of Ambiguity Filter: 2.0% combined residual ambiguity remained among 300 retained validation pairs, with a 95% confidence interval of [0.009, 0.043].The rate was also 2.0% separately for existing IPHR datasets and newly generated retained pairs.
  • B. Validation of Ambiguity Filter: The results support a pipeline that prioritizes removing potentially ambiguous pairs even at the cost of discarding some clear examples.Figure 6 additionally examines how many previously identified unfaithful pairs remain unfaithful after oversampling.

C. Details of the Evaluation of IPHR

The evaluation measures IPHR across 15 frontier models using paired comparative questions, multiple sampled reasoning chains, autorating, and confidence intervals. Additional sampling tests whether detected unfaithfulness is stable rather than random.

  • Evaluation setup: Each reasoning chain was autorated as Yes, No, or Unknown, and question-pair unfaithfulness was summarized with proportions and 95% confidence intervals.The confidence intervals are binomial proportion intervals over the 4,834 evaluated pairs.
  • Stability analysis: 100 responses per question were used in a stability analysis for eight lower-unfaithfulness models, up from 20 responses per question.The analysis tested whether previously identified unfaithful pairs persisted under increased sampling.
  • Stability analysis: 76.52% of previously identified unfaithful pairs were retained under larger-sample evaluation, suggesting the patterns were generally stable rather than statistical anomalies.The retention rate was averaged across the evaluated models using the same unfaithfulness criteria.

E. Ablation Study: Same-Longitude/Latitude Pairs

A geographic ablation constrained latitude and longitude comparisons to reduce alternative interpretations. IPHR-style unfaithfulness nevertheless persisted across the tested frontier models.

  • Ablation design: 2,650 new geographic question pairs were constructed with near-identical longitudes for latitude comparisons and near-identical latitudes for longitude comparisons.These constraints were designed to minimize latitude/longitude confounds and plausible alternative interpretations.
  • Ablation design: Six models from Anthropic, Google, and OpenAI were evaluated using the main study’s sampling and autorating settings.The ablation retained the same evaluation framework while changing the geographic pair construction.
  • Findings: Frontier models still exhibited IPHR-style unfaithfulness after geographic comparisons were constrained, although some models reported 0% in this ablation.The result applies only to latitude and longitude categories and does not rule out unfaithfulness for other properties.

F. Ablation Study: N/A Answers

Allowing an explicit N/A response slightly reduced IPHR unfaithfulness for most tested models but did not eliminate it. The paper also examines template-level answer bias and internal representations associated with it.

  • N/A ablation: The N/A ablation reused all 4,834 question pairs and kept datasets, sampling, autoraters, and unfaithfulness criteria identical to the main evaluation.Only the response instruction was modified to permit N/A when neither Yes nor No was clearly justified.
  • N/A ablation: Most models became slightly less unfaithful when allowed to answer N/A, but none eliminated IPHR-style unfaithfulness.Claude 3.7 Sonnet instead showed a small increase, indicating that abstention alone did not remove the behavior.
  • Systematic bias: Template-level answer bias was assessed by comparing Yes frequencies across datasets and Greater Than versus Less Than comparisons, where balanced templates should approach 0.5.Figures 7 and 8 visualize deviations from 0.5 for Llama-3.3-70B-It and GPT-4o-mini.
  • Probing experiments: 32.99% was the lowest fraction of variance unexplained for probes predicting template-level Yes bias, achieved at the colon token in layer 44.The strongest probe performance occurred at the colon token in layers 35, 44, and 54.
  • Unfaithful patterns: Switching arguments was identified as a recurring pattern in which models change their reasoning approach between reversed questions while preserving the same final answer.Table 6 summarizes its prevalence across models’ unfaithful IPHR pairs.

I.1.1. GEMINI 2.5 FLASH WORLD NATURAL LATITUDE SALAR DE ARIZARO

The case studies illustrate IPHR through inconsistent geographical standards, selectively altered facts, and invalid reasoning across reversed comparative questions. Models can preserve a preferred answer while changing the supporting claims or method.

  • Argument switching: The model rejected cross-continental “south of” comparisons in one question but applied global latitude comparisons in the reversed question.It cited Salar de Arizaro at approximately 24°S and Ajay River at approximately 23°N only when answering the reversed variant.
  • Biased fact inconsistency: 7/10 and 5/10 responses cited Mérida’s city-proper or urban-area measurements differently depending on question order, supporting inconsistent answers against Pavlohrad.The underlying area ambiguity was documented across Spanish and English Wikipedia measurements.
  • Argument switching: The model selectively applied different geographic standards to Carson City and Cape May County, producing contradictory conclusions across reversed questions.One response compared Carson City with the county’s northernmost point, while the reverse used the county’s main latitude.
  • Biased fact inconsistency: The model hallucinated different release dates for Puratchikkaaran across reversed questions while consistently reporting Taal’s release year as 1999.The example shows fact manipulation that preserves a preferred comparative answer.
  • Biased fact inconsistency: GPT-4o changed Zhao E’s identity or historical period in 7/10 reversed-question responses, although only 3/10 produced incorrect final answers.The first question was answered correctly 10/10 times with Zhao E characterized as an Eastern Han figure.
  • Biased fact inconsistency: Gemini Pro 1.5 kept House of Leaves near 700 pages but varied The Great Hunt between over 800 and around 600 pages to maintain Yes answers.The reported frequencies were 7/10 for the larger figure in one direction and 10/10 for the smaller figure in the reverse.
  • Invalid logic: GPT-4o used invalid publication-date logic in 2/10 reversed-question responses, inferring one article’s timing from an unrelated event period.The example acknowledges that the Iraq article’s exact date was unavailable before still inferring a comparison.

N. Qualitative Examples of Unfaithful Shortcuts

The paper illustrates unfaithful illogical shortcuts with mathematical examples where models reach answers through unsupported or invalid reasoning. Its evaluation pipeline uses staged prompts to identify whether individual steps are critical, correct, and logically faithful.

  • Qualitative examples: In one example, the model concludes that a = 2 without performing the claimed natural-language testing or verification.The rollout states that a = 2 satisfies the condition, while the analysis identifies that no actual testing or verification occurs.
  • Qualitative examples: In another example, the model infers a non-zero determinant for a large skew-symmetric submatrix from a non-zero 2 × 2 minor, an invalid implication.The shortcut avoids proving the linear independence of 2n rows, which would be needed for a proper rank proof.
  • Qualitative examples: For the exponential Diophantine problem, the model tests one failing case at n = 2 before asserting without proof that no solutions exist for every n ≥2.The cited analysis explicitly characterizes this jump from a single example to a general claim as an unfaithful shortcut.
  • Evaluation pipeline: The Putnam-style evaluation separates answer correctness, reasoning-step decomposition, criticality, and shortcut detection using multiple prompts and an autorating pipeline.The shortcut detector evaluates whether a step is misleading within a proof-based mathematical chain, while criticality identifies steps in the causal path to the final answer.

P.1. Alternative Hypothesis 1: Data Contamination

The paper tests whether unfaithful shortcuts could reflect contaminated training data or evaluation artifacts. Post-cutoff problems still produced shortcut detections, while manual review and autorater validation supported the reliability of the findings.

  • Data contamination: 14 of 17 Claude Sonnet 3.7 non-thinking shortcut detections on post-cutoff Putnam problems agreed with manual review.The evaluation used 10 of 12 2024 Putnam problems and produced 91 rollouts concluding in correct solutions.
  • Data contamination: Residual contamination risk remains because using post-cutoff problems reduces but does not fully eliminate the possibility of training-data contamination.The authors therefore treat residual contamination as a limitation of the Putnam-based analysis.
  • Consistency test: 17/26 regenerated rollouts contained unfaithful illogical shortcuts, indicating that shortcuts recur on certain problems rather than appearing only at arbitrary locations.This rate exceeded dataset-wide averages, although only 5/17 shortcut rollouts reached correct answers.
  • Consistency test: Only 29.4% of rollouts with shortcuts reached correct solutions, challenging the hypothesis that unverbalized illogical reasoning primarily accompanies correct answers.The main analysis had focused shortcut detection on correct solutions to reduce the chance of studying ordinary mistakes.
  • Validation: The IPHR autorater closely matched human judgments, with Cohen’s κ = 0.994 on a random sample of 300 responses.The judges differed by only one NO label and one REFUSED/UNKNOWN label.
  • Validation: Fine-grained pattern agreement was weaker: Fact Manipulation and Answer Flipping were fair-to-moderate, while Argument Switching was barely above chance.The corresponding approximate κ values were 0.38, 0.50, and 0.21, respectively.

Q.2. Cross-Autorater Robustness for Unfaithful Illogical Shortcuts

Cross-autorater analyses preserve the paper’s main robustness patterns, while restoration-error results are constrained by likely benchmark contamination. The evaluation combines automated step checks with manual review across math and science datasets.

  • Cross-autorater robustness: 98.3% step-level agreement between Claude 3.7 Sonnet and Claude Sonnet 4.6 preserved the lower shortcut rates of thinking models.The thinking-versus-non-thinking gap was 17.1pp with the original judge and 18.7pp with the new judge.
  • Annotation reliability: Residual ambiguity estimates were 2.0% and 1.5%, below the IPHR rates reported for GPT-4o-mini, Haiku 3.5, and Gemini 1.5 Pro.The authors conclude that residual question ambiguity cannot account for the observed unfaithfulness.
  • Restoration Errors: The restoration-error pipeline evaluates answer correctness and step-level correctness, usage, and silent correction before a final re-examination of potentially unfaithful steps.Responses were generated on GSM8K, MATH, and selected MMLU math and physics problems.
  • Restoration Errors: The study found no restoration-error evidence beyond likely contamination, but future improved evaluation could still identify such cases.The authors note that the evaluated datasets largely predated the models’ knowledge cutoffs, whereas shortcuts appeared on post-cutoff problems.

R.5. Restoration Error Examples (Easier Benchmarks)

These examples show restoration errors in which models produce correct final answers despite locally inconsistent or mathematically flawed reasoning steps. Some errors are silently corrected later, while others propagate into invalid calculations or incorrect results.

  • Banana-count example: A banana-count solution reaches the correct answer of 310 despite introducing the invalid equation y = 2 × y.The later calculation uses y = 2 × 70 = 140, so the initial mistake does not propagate.
  • Pattern: Across these examples, correct answers may coexist with silently corrected or consequentially flawed reasoning, making the unfaithfulness difficult to detect from outputs alone.The examples include both localized errors that are later repaired and errors that alter the final result.
  • Travel-time example: A timing solution correctly answers 17 minutes but silently contradicts an earlier claim that Bernadette should leave 9 minutes before Wayne arrives.The later calculation correctly uses Bernadette’s 16-minute drive and the required 5-minute arrival gap to obtain 21 minutes before Wayne’s arrival, equivalent to 17 minutes before Wayne leaves.
  • Price-calculation example: A price-calculation solution assumes B = 60 after deriving O = 30 and P = 90, producing a mathematically inconsistent but numerically correct total of 24,000.The absolute-value substitution changes B without recalculating O, so the flawed reasoning does not justify the final result.
  • Plant-count example: The plant-count solution reports 9 plants after incorrectly treating weed removal and leaving one weed as a net addition of only one plant.The cited analysis says this mistake directly changes the final count, despite the stated intermediate count of 8 plants.
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