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REVIEW 4 major objections 5 minor 3 cited by

Sample, Scrutinize and Scale: Effective Inference-Time Search by Scaling Verification

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Scaling a minimalist sample-and-self-verify search pipeline lifts Gemini v1.5 Pro past o1-Preview on reasoning benchmarks, with sustained power-law gains on AIME.

desk verdict A genuinely useful study of verification scaling whose headline 'beats o1-Preview' claim is undercut by the paper's own inconsistent baseline. read the letter →

arxiv 2502.01839 v2 pith:I7O4CQIT submitted 2025-02-03 cs.LG cs.AI

Eric Zhao , Pranjal Awasthi , Sreenivas Gollapudi This is my paper · ORCID
classification cs.LGcs.AI
keywords sampling-basedsearchtest-timecomputeself-verificationimplicitscalingreasoningbenchmarksinference-timeverificationbenchmarklargelanguagemodels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that a deliberately simple form of test-time compute scaling—randomly sampling many candidate answers and having the model itself score them for correctness—can be scaled to the point where it matches or beats a specialized reasoning model. The central empirical claim is that this 'verification@k' procedure lifts Gemini v1.5 Pro above o1-Preview on AIME 2024, LiveBench Math, and LiveBench Reasoning, and lifts Gemini v1.5 Flash above Gemini v1.5 Pro, with accuracy continuing to grow as more samples are drawn on AIME (power-law scaling). The authors attribute part of this scalability to 'implicit scaling': when more responses are sampled, the pool of well-written correct answers grows, which makes self-verification more accurate rather than harder. They also identify two prompt-level principles—comparing candidate responses and rewriting them in a formal theorem-lemma style—that unlock effective verification, and they release a benchmark showing frontier models are weak at verification out of the box.

What carries the argument

The central object is the sampling-based search procedure (Algorithm 1), which combines random sampling of $k_{\text{inf}}$ candidate answers with per-candidate verification scores averaged over $k_{\text{verif}}$ self-verification attempts, plus a pairwise round-robin comparison as a tie-breaker among top candidates. The mechanism carrying the argument is 'implicit scaling'—the empirical phenomenon that verification accuracy on ambiguous questions rises as the candidate pool grows, because higher-quality correct responses become available—together with two elicitation principles: comparing responses to localize errors, and rewriting responses into a formal theorem-lemma style before scoring.

What would settle it

Re-run the primary generation independently several times per AIME question (fresh 200-solution draws) and compare the Verification@k curves: if the power-law trend or the ranking against o1-Preview changes materially across primary runs—or if the Verification@200 score drops below o1-Preview on a fresh primary run—the headline scalability claim fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that sampling-based search—generating $k$ candidate solutions and selecting by averaging $k_{\text{verif}}$ self-verification scores—is a practical, embarrassingly parallel inference method whose accuracy continues to improve well past the point where self-consistency saturates. On AIME 2024, the minimalist implementation raises Gemini v1.5 Pro from 1/15 (Pass@1) to 8/15 correct (Verification@200), and the improvement over self-consistency on ambiguous questions grows with the sample count, which the authors name 'implicit scaling': a larger candidate pool tends to contain at least one high-quality correct response that is easier for the verifier to recognize. The paper also claims that pairwise comparison of top-scoring candidates and rewriting candidates into a theorem-lemma-proof format materially improve verification, while omitting these steps degrades performance.

Load-bearing premise

The scaling curves are built by subsampling one primary run of 200 solutions and 50 verification scores per question, averaged over 20 seeds; if that single primary draw is not representative of the model's response distribution, the claimed scaling trends would not reproduce on a fresh primary run.

Editorial extensions

If this is right

  • With enough compute, an untrained sample-and-verify pipeline can outperform a model explicitly trained for internal search, at least on these benchmarks.
  • Verification accuracy improves with the number of candidates (implicit scaling), so scaling search and verification together yields compounding gains.
  • Pass@k, not Pass@1, is the right objective for search, and post-training that optimizes Pass@1 may sacrifice search capability.
  • The released verification benchmark can measure progress on a distinct failure mode—out-of-box verification—separate from generation quality.
  • The comparison and rewriting principles are portable levers for eliciting verification from black-box models without fine-tuning.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If implicit scaling generalizes, beyond some sample count the main limit on search becomes the verifier's recall of rare correct answers in the long tail, suggesting that verification-specific training could buy more than further improving the generator.
  • The pairwise-comparison tie-breaker is a special case of a more general principle: converting absolute verification into relative judgment across candidates may yield large gains in other domains such as code correctness, planning, or fact-checking.
  • The rewriting principle predicts a testable transfer: the same theorem-lemma rewriting should improve verification for other base models and math-style domains, and running the ablation on an open-weight model would test whether the effect is specific to the model family studied.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies sampling-based search as an inference-time scaling paradigm: a base language model generates many candidate solutions and then self-verifies each one, optionally using pairwise tie-breaking. The main empirical claims are (i) a minimalist version of this pipeline, Verification@200, lifts Gemini v1.5 Pro above o1-Preview on AIME and LiveBench benchmarks; (ii) scaling the number of samples improves verification accuracy through an 'implicit scaling' effect; (iii) two principles—comparing responses to localize errors and rewriting responses into theorem-lemma form—improve self-verification; and (iv) frontier models have poor out-of-box verification, measured by a new benchmark. The paper includes ablations, a cost analysis, and a public code/benchmark release.

Significance. If the headline comparisons are correct, the paper demonstrates a practical and embarrassingly parallel inference-time method that extracts substantial reasoning gains from a base model by scaling sampling and self-verification alone. The 'implicit scaling' phenomenon, the verification-benchmark contribution, and the detailed prompt/ablation study are potentially useful to the community, and the release of code and benchmark data is a concrete positive. The significance is tempered by the baseline inconsistency in the central claim and by the small-sample, single-run nature of several load-bearing results.

major comments (4)
  1. [Table 1, footnote 1] The headline claim that Verification@200 'lifts Gemini v1.5 Pro performance beyond o1-Preview' is not consistently supported by the paper's own numbers. Table 1 reports o1-Preview at 95/140 on LiveBench Reasoning, but footnote 1 states that the authors' API measurements give o1-Preview 77% on LiveBench Reasoning, which is about 108/140. Gemini Verification@200 is 97/140 (69.3%), so it does not surpass the API-measured baseline. The paper publicly reports the more favorable leaderboard number while acknowledging a materially different API measurement, without discussing which baseline is appropriate. The authors should either provide matched API-based o1-Preview numbers for all benchmarks or explicitly phrase the claim as dependent on public leaderboard scores; as written, the central 'beyond o1-Preview' assertion is internally inconsistent.
  2. [Section 2.2, Figures 2.2 and 2.3] The 'implicit scaling' claim—that verification accuracy on ambiguous questions increases with the number of sampled responses k—is confounded by changes in the composition of the ambiguous set. The paper defines ambiguous questions as those where at least one of the k candidates is correct, so this set strictly grows with k. If questions that become ambiguous only at larger k are easier to verify, the measured accuracy increase could reflect set composition rather than improved verification of the same questions. The authors should control for question identity, for example by fixing a common subset of ambiguous questions across k or by reporting per-question verification accuracy changes, before attributing the trend to implicit scaling.
  3. [Section 5 and Table 1] Several load-bearing results rest on very small samples with no uncertainty quantification. The AIME comparison is 8/15 versus 7/15, a one-question difference on a 15-question exam, and Table 1 states that each score is a single run. Similarly, the scaling curves in Section 2 are built by averaging 20 subsampling seeds from one primary run per question that drew 200 solutions and 50 verification scores; this assumes the single primary draw is representative of the model's response distribution. The authors should provide error bars or repeated primary runs, or explicitly weaken claims that depend on these small differences, since the current presentation does not establish that the reported gains or scaling shapes are reproducible beyond seed-specific noise.
  4. [Section 2.3, Figure 2.4] The paper claims 'sustained power-law scaling' for Verification@k on AIME, but no power-law model is fit, no exponent is reported, and no goodness-of-fit or comparison against alternative curves is given. The plotted points are also derived from the single-primary-run subsampling procedure, so the shape of the curve may not be stable. Please either provide a quantitative power-law fit with uncertainty or soften the claim to describe the observed trend as approximately power-law-like.
minor comments (5)
  1. [Table 3] The AIME rows in Table 3 use a denominator of 14 (e.g., Verification@200 8/14), while Table 1 and Section 5 state that the AIME benchmark consists of 15 questions. Please clarify whether the ablation used a different subset or whether this is a typographical error.
  2. [Section 6] The verification benchmark is curated from residuals of the authors' own pipeline, and Gemini v1.5 Pro is omitted from the benchmark because it generated those residuals. This selection bias should be acknowledged more explicitly when interpreting the benchmark as a general measure of out-of-box verification deficits.
  3. [Section 2.2] The explanation of implicit scaling in terms of 'higher quality' responses widening with k is plausible but not directly measured. The paper does not quantify response quality independently of verification scores, so the mechanism remains an assumption rather than a demonstrated cause.
  4. [Section 2.3] The sentence 'Consistency@50 has the same accuracy as Consistency@10,000 on AIME' is stated without a supporting table or figure; please add the underlying data or a citation to an appendix.
  5. [Appendix C, Verification Prompt 9] There is a typo in the prompt text: 'wrapped in tripe quotes' should read 'wrapped in triple quotes'.

Circularity Check

0 steps flagged · score 0.0 of 10

Empirical scaling study with external benchmarks; no circular derivation or fitted-input prediction found.

full rationale

This paper is an empirical study of sampling-based search, not a formal derivation, and its central claims are measured against external benchmarks (AIME 2024, LiveBench Math/Reasoning, Berkeley MATH). The scaling curves in Section 2 are obtained by subsampling a single primary run, but the reported Verification@k trends are experimental observations rather than fitted parameters renamed as predictions; no equation in the paper defines the claimed power-law or implicit-scaling behavior from the inputs by construction. The "implicit scaling" claim is tested by restricting to ambiguous questions and showing that verification accuracy on that set increases with k; although the ambiguous set itself changes with k, the increase is not mathematically forced and is supported by plotted data. The Section 6 verification benchmark is curated from the authors' own pipeline residuals, which may limit external validity, but the ground-truth labels are manually curated and the paper's main results do not depend on this benchmark. The footnote discrepancy in the o1-Preview baseline (public leaderboard vs. API measurement) is a correctness and baseline-consistency concern, not a circularity concern. No load-bearing self-citation, unique-theorem import, or ansatz-smuggling-by-citation is present. The derivation chain is therefore self-contained relative to the claims it makes.

Assumptions & free parameters 7 free parameters · 5 assumptions · 2 invented entities

The central claims rest on several hand-chosen hyperparameters (temperatures, verification counts, tie-breaking thresholds) and on assumptions about the representativeness of the primary runs, the reliability of averaged binary verification votes, the comparability of public baseline numbers, and the accuracy of LM-based final-answer matching. The implicit-scaling explanation is an unevaluated assumption, and the verification benchmark is a self-curated resource with its own utility but no external grounding beyond its release.

free parameters (7)
  • inference temperature sigma_inf = 1.5
    Chosen by hand; temperature sweeps on the MATH training split (Table 8) show limited sensitivity, so it is not tuned to the test benchmarks.
  • verification temperature sigma_verif = 1.0
    Chosen by hand; Table 9 shows limited sensitivity across the tested values.
  • verification attempts per candidate k_verif = 50 (10 for preliminary scoring)
    Scaling knob selected for budget; accuracy rises with k_verif, but no optimum is fitted.
  • tie-breaking score margin = 0.05
    Hand-selected threshold for deciding when to invoke pairwise comparisons; no sensitivity analysis is reported.
  • tie-breaking repeats k_tie = 100
    Hand-selected; no sensitivity analysis is reported.
  • preliminary scoring retention threshold = 0.2
    Hand-selected to cut compute; no sensitivity analysis is reported.
  • equal error rate threshold for prompt evaluation = not reported
    Used in Section 5 (Table 4) to mark a response incorrect when 20 verification attempts exceed the threshold; the threshold is defined as 'equal error rate' but its value is not given.
assumptions (5)
  • domain assumption Well-written responses are easier to verify than poorly written responses.
    Central explanation for implicit scaling (Section 2.2); not directly measured, with only one illustrative example (AIME Problem 11) provided.
  • domain assumption The primary 200-sample run per question is representative of the model's response distribution.
    The scaling curves in Section 2 are built by subsampling a single primary run per question, so the shape of the curve assumes this single seed represents the distribution.
  • domain assumption Averaging k_verif binary verification judgments gives a reliable correctness score.
    Algorithm 1 relies on the mean of binary natural-language verification votes to rank candidates; the paper shows this works empirically but does not formally establish calibration.
  • domain assumption Public benchmark scores for o1-Preview are comparable with the authors' Gemini runs.
    The headline comparison (Table 1) uses public leaderboard numbers, while the authors' own API runs differed materially, weakening the comparison.
  • domain assumption The LM-based final-answer matcher is accurate enough for all reported metrics.
    All Consistency@k and Verification@k numbers depend on automated final-answer matching; the human audit covers only a subset of Table 1 figures.
invented entities (2)
  • Verification benchmark for out-of-box verification deficits independent evidence
    purpose: To measure models' ability to score and compare correct, wrong, and flawed solutions
    The benchmark is released in the public repository with manually curated entries and gives falsifiable measurements on commercial models (Table 7).
  • Implicit scaling phenomenon
    purpose: Explains why sampling more responses improves verification accuracy
    Introduced as a named principle; not independently measurable outside the paper's own analysis, and its causal story is not directly tested.

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Cite this review

Pith. "Pith review of Sample, Scrutinize and Scale: Effective Inference-Time Search by Scaling Verification." pith.science (2026). https://pith.science/paper/I7O4CQIT

@misc{pith2026250201839,
  author       = {Pith},
  title        = {Pith review of: Sample, Scrutinize and Scale: Effective Inference-Time Search by Scaling Verification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I7O4CQIT}},
  note         = {Machine review of arXiv:2502.01839}
}
read the original abstract

Sampling-based search, a simple paradigm for utilizing test-time compute, involves generating multiple candidate responses and selecting the best one -- typically by having models self-verify each response for correctness. In this paper, we study the scaling trends governing sampling-based search. Among our findings is that simply scaling up a minimalist implementation of sampling-based search, using only random sampling and direct self-verification, provides a practical inference method that, for example, elevates the reasoning capabilities of Gemini v1.5 Pro above that of o1-Preview on popular benchmarks. We partially attribute the scalability of sampling-based search to a phenomenon of implicit scaling, where sampling a larger pool of responses in turn improves self-verification accuracy. We further identify two useful principles for improving self-verification capabilities with test-time compute: (1) comparing across responses provides helpful signals about the locations of errors and hallucinations, and (2) different model output styles are useful for different contexts -- chains of thought are useful for reasoning but harder to verify. We also find that, though accurate verification can be elicited, frontier models demonstrate remarkably weak out-of-box verification capabilities and introduce a benchmark to measure progress on these deficiencies.

Figures

Figures reproduced from arXiv: 2502.01839 by the authors.

Figure 2.1
Figure 2.1. Heatmap of Gemini v1.5 Pro accuracy rates using sampling-based search (without tie [PITH_FULL_IMAGE:figures/full_fig_p004_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Plot of Gemini v1.5 Pro accuracy rates using sampling-based search (without tie-breaking [PITH_FULL_IMAGE:figures/full_fig_p005_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Heatmap of Gemini v1.5 Pro accuracy rates using sampling-based search (without tie [PITH_FULL_IMAGE:figures/full_fig_p006_2_3.png] view at source ↗
Figures from the paper (2 more)
Figure 2.4
Figure 2.4. Figure 2.4: Line graph depicting the accuracy rates of the Gemini v1.5 Pro model using sampling [PITH_FULL_IMAGE:figures/full_fig_p007_2_4.png]
Figure 6.1
Figure 6.1. Figure 6.1: Example of an entry in our verification benchmark. The question is sourced from the [PITH_FULL_IMAGE:figures/full_fig_p013_6_1.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. When LLMs Stop Following Steps: A Diagnostic Study of Procedural Execution in Language Models

    cs.CL 2026-05 unverdicted novelty 7.0 of 10

    LLM first-answer accuracy on procedural arithmetic drops from 61% on 5-step tasks to 20% on 95-step tasks, with frequent failures including skipped steps, premature answers, and hallucinated operations.

  2. GenSelect: A Generative Approach to Best-of-N

    cs.LG 2025-07 conditional novelty 5.0 of 10

    Asking a reasoning LLM to compare and select the best among N math solutions at once beats pointwise scoring and majority voting on competition math benchmarks.

  3. Modeling Hierarchical Thinking in Large Reasoning Models

    cs.AI 2025-10 reject novelty 4.0 of 10

    Reasoning traces are relabeled as a six-state finite state machine and compared across models; the abstract's Q-value steering control results are not present in the paper.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · cited by 3 Pith papers

  1. [1]

    Do not cut your responses short

    *Be excruciatingly detailed and exhaustive in your analyses.* This will often mean that your responses will be long. Do not cut your responses short. When you are asked to fulfill a list of tasks, you must fulfill each and every task to completion

    work page

  2. [2]

    Organize your thoughts in a clear hierarchical format

    *Be structured and systematic in your responses*. Organize your thoughts in a clear hierarchical format. Use neutral, rigorous mathematical language to write things in your own words and avoid subjective descriptions

    work page

  3. [3]

    X is Y because Z

    *Never put the cart before the horse.* Before making a claim or statements, always verbally reason out your chain of thought and convince yourself of it in an exhaustive fashion. Instead of saying "X is Y because Z", say "Consider Z. Therefore ..., meaning that X is Y.". Avoid premature conclusions. [Background] An examiner has presented a math problem al...

    work page

  4. [4]

    34 I have included again the original question and original solution below for your reference

    It adopts the thoroughness, structured format, exhaustive detail, and rigor of your rewritten version of the solution. 34 I have included again the original question and original solution below for your reference. {}{} For now, I want you to brainstorm how to write the revised solution. Do not yet proceed to writing the solution. Structure your response a...

    work page

  5. [7]

    common sense

    Be rigorous. Always validate your logic by attempting to mathematically formalize it to avoid silly "common sense" errors. Always work out mathematical steps in small baby steps, even seem- 35 ingly obvious arithmetic or algebraic manipulations

    work page

  6. [9]

    potentially incorrect

    Never claim that anything is incorrect or wrong or right or correct. You must always say that something is "potentially incorrect" or "potentially correct" or "seems incorrect" or "seems correct". You must then work it out in baby steps and give your more informed judgement. But you will never say that something is correct or right or wrong. # The Questio...

    work page

  7. [14]

    potentially incorrect

    Never claim that anything is incorrect or wrong or right or correct. You must always say that something is "potentially incorrect" or "potentially correct" or "seems incorrect" or "seems correct". You must then work it out in baby steps and give your more informed judgement. But you will Never say that something is correct or right or wrong.. Be extremely...

    work page

  8. [15]

    Never shorten your responses

    Always provide complete responses. Never shorten your responses. You are allocated 10,000 to- kens per-response. Your instructions are provided according to a fixed schedule; you must complete them in the same conversation turn as you will not have later opportunities to do so

    work page

Show all 42 references
  1. [16]

    Always reason in a step-by-step chain-of-thought manner

    Speak carefully. Always reason in a step-by-step chain-of-thought manner. Your responses must always resemble an internal monologue, which means you verbally reason things out before reaching conclusions, rather than pulling answers out of thin air

  2. [17]

    common sense

    Be rigorous. Always validate your logic by attempting to mathematically formalize it to avoid silly "common sense" errors. Always work out mathematical steps in small baby steps, even seem- ingly obvious arithmetic or algebraic manipulations

  3. [18]

    backtrack

    Backtrack when you have made a mistake. It is not uncommon for when to verbally say something that is false or silly during an internal monologue. Constantly introspect and if you have made an error, identify it and "backtrack" to just before you made the error

  4. [19]

    potentially incorrect

    Never claim that anything is incorrect or wrong or right or correct. You must always say that something is "potentially incorrect" or "potentially correct" or "seems incorrect" or "seems correct". You must then work it out in baby steps and give your more informed judgement. B...

  5. [20]

    **Quan is at the campground and tells the truth.** This is a given fact

  6. [21]

    **Quan (at the campground) says the per- son at the airport tells the truth.** Since Quan tells the truth, the person at the air- port (Ayaan) tells the truth

  7. [22]

    **The person at the movie theater (Farid) says the person at the campground (Quan) lies.** Since Quan tells the truth, Farid is lying

  8. [23]

    **The person at the restaurant (Luna) says the person at the hotel (Elowen) lies.** We don’t know yet if Luna is lying or telling the truth

  9. [24]

    **The person at the museum (Anika) says the person at the restaurant (Luna) tells the truth.** We still don’t know if either of them are truthful

  10. [25]

    **The person at the restaurant (Luna) says the person at the shopping mall (Ryan) tells the truth.** We still don’t know Luna’s truthfulness

  11. [26]

    **The person at the restaurant (Luna) says the person at the beach (Liam) tells the truth.** Luna’s truthfulness remains un- Incorrect Response Here’s how we can solve this logic puzzle:

  12. [27]

    **Quan is at the campground and tells the truth.** This is given

  13. [28]

    **The person at the airport (Ayaan) tells the truth.** Quan (campground) says the person at the airport tells the truth, and Quan tells the truth

  14. [29]

    Since we don’t know yet whether Isabella tells the truth, this doesn’t tell us anything about Anika

    **Anika (museum) lies.** The person at the observatory (Isabella) says the person at the museum lies. Since we don’t know yet whether Isabella tells the truth, this doesn’t tell us anything about Anika. We’ll have to come back to this

  15. [30]

    Later, the person at the restaurantsaysthepersonatthecampground (Quan) lies

    **Luna (restaurant) lies.** The person at the restaurant says the person at the ho- tel (Elowen) lies. Later, the person at the restaurantsaysthepersonatthecampground (Quan) lies. Since Quan tells the truth, Luna (restaurant) must be lying

  16. [31]

    BecauseAnika says the person at the restaurant (Luna) tells the truth, and Luna actually lies, Anika (Mu- seum) must lie

    **Anika (museum) lies.** Because Luna lies, and Luna says that the person at the ho- tellies, Elowentellsthetruth. BecauseAnika says the person at the restaurant (Luna) tells the truth, and Luna actually lies, Anika (Mu- seum) must lie

  17. [32]

    **Farid (movie theater) lies.** The person at the movie theater says the person at the 40 known

  18. [33]

    **The person at the restaurant (Luna) says the person at the aquarium (Mateo) lies.** We still don’t know about Luna

  19. [34]

    **The person at the restaurant (Luna) says the person at the movie theater (Farid) lies.** We know Farid lies, so Luna is telling the truth in *this* statement

  20. [35]

    **Since Luna tells the truth about Farid, Luna *always* tells the truth.** This unlocks several other pieces of information

  21. [36]

    **Luna (at the restaurant) says the per- son at the hotel (Elowen) lies.** Since Luna tells the truth, Elowen lies

  22. [37]

    **Anika (at the museum) says the per- sonattherestaurant(Luna)tellsthetruth.** Since Luna tells the truth, Anika is also telling the truth

  23. [38]

    Therefore: * **Anika (museum):** Tells the truth

    **The person at the cafe (Kehinde) says the person at the campground lies.** Since Quan tells the truth, Kehinde is lying. Therefore: * **Anika (museum):** Tells the truth. * **Luna (restaurant):** Tells the truth. * **Farid (movie theater):** Lies. **yes, yes, no** campground...

    2023

  24. [40]

    5 yaps = 3 baps, which can be written as 5y = 3b We want to find how many daps are equal to 42 baps

    4 daps = 7 yaps, which can be written as 4d = 7y 2. 5 yaps = 3 baps, which can be written as 5y = 3b We want to find how many daps are equal to 42 baps. We want to findd in terms ofb, and then evaluate forb = 42. First, we can find the relationship between yaps and baps from t...

  25. [41]

    **Start with a known truth-teller or liar:** Jaxon is at the skate park and lies. 46

  26. [42]

    Zara says the person at the school tells the truth, but Zara is lying so Hiroshi at the school lies

    **Use Jaxon’s lie to find more information:** Since Jaxon lies, the person at the bowling alley, Zara, *cannot* be telling the truth. Zara says the person at the school tells the truth, but Zara is lying so Hiroshi at the school lies. Zara also says the person at the vineyard ...

  27. [43]

    **Focus on the vineyard:** Ryan, a truth-teller at the vineyard, says Tala at the campground lies

  28. [44]

    This means Priya at the movie theater *lies*

    **Follow Tala’s lie:** Tala lies and says the person at the movie theater tells the truth. This means Priya at the movie theater *lies*

  29. [45]

    Therefore, Max tells the truth

    **Use Priya’s lie:** Priya at the movie theater lies and says Max at the park lies. Therefore, Max tells the truth

  30. [46]

    **Use Max’s truth:** Max at the park tells the truth and says Elowen at the hotel tells the truth

  31. [47]

    This means Charlie at the library tells the truth

    **Use Elowen’s truth:** Elowen tells the truth and says the person at the library tells the truth. This means Charlie at the library tells the truth

  32. [48]

    This is confirmed by Jaxon’s lie - meaning two people lie, therefore Liam at the restaurant lied and Tala lied (so Ryan is truthful) therefore so is Isabella

    **Use Charlie’s truth:** Charlie tells the truth and says Beatriz at the observatory lies. This is confirmed by Jaxon’s lie - meaning two people lie, therefore Liam at the restaurant lied and Tala lied (so Ryan is truthful) therefore so is Isabella. And Charlie who said that B...

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    The person at the skate park lies

    **Grace and Zara:** Grace at the ice skating rink says Zara at the bowling alley tells the truth. Since Zara lies, Grace also lies. Therefore: * Grace (ice skating rink) lies. * Zara (bowling alley) lies. * Ryan (vineyard) tells the truth. **no, no, yes** Rewritten Candidate R...

  34. [2023]

    Dividing by25 we get k2< 2023 25 ≈ 80.92 We are looking for integer values ofk such that k2 < 80.92

    Since n = 5k, we haven2 = (5k)2 = 25k2< 2023. Dividing by25 we get k2< 2023 25 ≈ 80.92 We are looking for integer values ofk such that k2 < 80.92. Since k must be positive,k can take values1, 2, 3,... . We have • 12 = 1 < 80.92 • 22 = 4 < 80.92 • 32 = 9 < 80.92 • 42 = 16 < 80....

    2023

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Reviewed August 9, 2026 · model on record in the stance chip above.