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REVIEW 3 major objections 5 minor 102 references

Reasoning LLMs are Wandering Solution Explorers

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Reasoning LLMs are wanderers: despite test-time computation, they do not systematically explore solution spaces, and their solution coverage decays as problem complexity grows.

desk verdict A genuinely useful failure-mode taxonomy and the most detailed process-level traces I've seen, but the universal 'wanderer' claim outruns the evidence. read the letter →

arxiv 2505.20296 v1 pith:3CPNPUXF submitted 2025-05-26 cs.CL cs.AIcs.LGcs.MM

Jiahao Lu , Ziwei Xu , Mohan Kankanhalli This is my paper
classification cs.CLcs.AIcs.LGcs.MM
keywords reasoningLLMssystematicexplorationtest-timecomputationsolutioncoveragetraceauditfailuremodesbacktrackingprocess-levelevaluation
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 sets out to show that current reasoning large language models (RLLMs) are wanderers rather than systematic explorers: they produce long reasoning traces at test time, but those traces routinely contain invalid steps, redundant revisits, and unfaithful conclusions, and they do not reliably cover the solution space. The authors formalize systematic exploration as a trace that is valid, effective, and necessary, and then use a controlled audit of six state-of-the-art models on eight computation tasks to show that every model violates these properties. Their quantitative measure, solution coverage on permutation enumeration, declines as the solution space grows for all six models, including the strongest closed-source ones. The stakes are that increasing test-time computation without structural search guarantees will keep producing models that look competent on easy problems yet collapse as problem complexity grows.

What carries the argument

The load-bearing object is a formal definition of systematic exploration: a trace $J$ satisfies validity, effectiveness, and necessity. To make reasoning auditable, the paper projects problems into structured computational tasks with a fixed command grammar (CHECK, ATTEMPT, BACKTRACK, MERGE, STATE, END) so that each reasoning step can be matched against programmatically generated ground truth. The quantitative argument is carried by a wandering-agent success probability for the depth-first-search problem, $p_s(d,m,q_w)=1-(1-q_w^{d-1})^m$, showing that a wanderer who omits a branch with probability $p_w$ suffers exponentially decaying success as depth $d$ grows, while exhibiting performance plateaus at low $d$ that can mislead evaluations.

What would settle it

Run the same six models on the same eight tasks without the fixed command grammar, let them reason in free-form natural language, and parse the traces afterward with the same rule-based auditor; if the invalid steps, omissions, and backtracking errors largely vanish, the conclusion that RLLMs are intrinsically wandering explorers would not survive, because the wandering would be an artifact of the imposed format. Alternatively, any model that maintains near-complete solution coverage on permutation enumeration as the solution space grows large would directly contradict the universal claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that reasoning LLMs, despite test-time computation techniques such as chain-of-thought, tree search, and long reasoning, lack the ability to systematically explore the solution space. The thesis is that a 'better' or systematic solution exploration should satisfy three properties, namely validity (each step respects the problem's reachability structure), effectiveness (at least one goal state is reached), and necessity (every state contributes to reaching a goal or eliminating alternatives), and that none of the six evaluated RLLMs exhibits these properties consistently. Empirically, the paper reports persistent failure modes: boundary violations, procedure omissions, incorrect backtracking, state revisitation, infinite self-loops, state staleness, execution errors, and unfaithful conclusions. On a quantitative testbed, enumerating unique permutations with duplicates, all six models show solution coverage that degrades as the number of valid goal states increases, which the paper interprets as evidence that they are wanderers rather than systematic explorers, with performance that can appear competent on easy problems yet degrade sharply as complexity grows.

Load-bearing premise

The paper's audit assumes that forcing a model to write its reasoning as CHECK, ATTEMPT, BACKTRACK, and similar commands does not itself distort the reasoning and create the errors being measured.

Editorial extensions

If this is right

  • If the central claim is correct, additional test-time computation will not by itself turn a wandering model into a systematic explorer: larger compute can produce longer traces without better solution-space coverage.
  • Benchmarks that grade only final answers will overestimate capability, because performance plateaus on easy instances hide the sharp degradation on deeper ones.
  • Process-level evaluation, such as trace validity, search completeness, and coverage, should become a standard complement to final-answer accuracy for reasoning models.
  • Architecture and training changes, such as explicit state tracking, backtracking support, and process supervision, would be needed to reach systematic exploration rather than relying on scale alone.

Reading between the lines

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

  • The exponential-decay formula implies a threshold effect: a model can look near-perfect on benchmarks built from shallow problem instances and fail abruptly once depth crosses a certain point, so benchmarks should report coverage as a function of problem depth rather than only average accuracy.
  • Because the audit forces a fixed command grammar, a free-form chain-of-thought control condition is needed to confirm that the observed errors are intrinsic to the models rather than induced by the format.
  • The same process-level metrics, such as invalid steps, revisits, and coverage, could be turned into training rewards or early-exit signals, potentially guiding models toward systematic search without architectural redesign.
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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

3 major / 5 minor

Summary. The paper argues that current reasoning LLMs (RLLMs) are 'wandering solution explorers' rather than systematic searchers. It formalizes systematic exploration via validity, effectiveness, and necessity, identifies eight failure modes, and presents qualitative case studies on six models across eight structured tasks. A quantitative experiment on a permutation-enumeration task shows solution coverage degrading as problem size grows, supporting the claim that RLLMs wander. The paper concludes that test-time scaling without structural search guarantees leads to competent-looking but shallow reasoning.

Significance. The paper makes a useful contribution by cataloging concrete reasoning failures (boundary violations, procedure omissions, incorrect backtracking, state revisitation, infinite self-loops, state staleness, execution errors, unfaithful conclusions) and by including full, readable trace-level evidence in Appendix D. The qualitative case studies are vivid and demonstrate that, under the imposed protocol, current RLLMs produce invalid steps, revisit states, loop, and mis-summarize their own reasoning. The paper also raises a practically important challenge about the limits of test-time compute. However, the central universal claim that RLLMs are wanderers 'by nature' is not yet supported: all evidence is collected under a fixed command grammar, and the quantitative support comes from a single task. The formal framework in Section 2 also contains a vacuous definition that undermines the stated theory of systematic exploration.

major comments (3)
  1. [Section 3, Appendix D, Section 4.4] The audit protocol forces all reasoning into a fixed command grammar (CHECK, ATTEMPT, BACKTRACK, MERGE, STATE). No control condition compares this to free-form chain-of-thought with post-hoc parsing. Therefore the invalid steps, premature stops, and repeated states observed in the case studies and quantified in Fig. 6 could be artifacts of the imposed format rather than intrinsic properties of the models' reasoning. The authors should add a free-form CoT control condition or, at minimum, restrict the empirical claim to behavior under the imposed symbolic grammar.
  2. [Definition 2(c), Section 2.3] The necessity condition is vacuous as written. For any state s_ji in a trace J, taking the subsequence J' = J yields an empty remaining trace that contains fewer goal and dead-end states, and the empty sequence satisfies the validity clause vacuously. Hence every state of any effective trace counts as 'necessary', so the 'unnecessary exploration' failure mode defined in Section 2.3 is not actually characterized by Definition 2(c). The definition needs to be repaired, for example by requiring J' to be a proper subsequence and by specifying that the remaining trace must still be a valid exploration of the same problem.
  3. [Figure 6, Section 4.4] The universal claim that 'all reasoning models exhibit wandering characteristics' rests on a single quantitative task, Permutation with Duplicates, with no statistical test and no comparison against a systematic-exploration baseline. The qualitative case studies support the existence of wandering behaviors in eight tasks, but the prevalence and generality of the phenomenon across task types is not quantitatively established. The authors should either provide quantitative coverage or error-rate metrics for multiple tasks, or weaken the concluding claim to 'all models wander on the tested permutation task under the structured-format protocol'.
minor comments (5)
  1. [Appendix C.2 / Figure 6] The open-source models are run 10 times and closed-source models only once; Figure 6 shows error bars only for open-source models. The text should state this asymmetry explicitly so readers do not infer similar uncertainty for the closed-source results.
  2. [Appendix D.7] The task box in D.7 says 'Input: [12, 13, 2, 8]' but the presented problem and ground truth concern the prime factorization of 177750; this appears to be a copy-paste error from the 24 Game section and should be corrected.
  3. [Section 2.1] The text says 'As shown in Fig. 2' when referring to the performance plateau, but the plateau appears in Figure 1; Figure 2 is the DFS illustration and is not a plot of success probabilities.
  4. [Section 4.4] The word 'deplicate' in the task description should be 'duplicate'.
  5. [Definition 1 and Definition 2] The definition of a dead-end refers to 'unexplored states' in the trace, but 'unexplored' is not formally defined; it would be cleaner to define a dead-end as a state whose all reachable successors are already in the trace.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the wandering claim rests on independent rule-based audits, and Eq. (1) is an explicitly assumption-based toy illustration rather than a fitted prediction.

full rationale

No load-bearing step in this paper reduces to its inputs by construction. The central empirical claim that all six reasoning LLMs wander is measured in Section 4.4 and Appendix D with rule-based, string-level processors against programmatically generated ground-truth traces; those ground truths are external to the claim, and no model parameter is fitted to the reported coverage curves. The toy-model formula in Eq. (1) is explicitly introduced as an illustration of a wandering agent with a per-decision omission probability qw, so its exponential decay is a mathematical consequence of that stated assumption rather than an independent empirical prediction. The paper contains no self-citations that carry the argument, imports no uniqueness theorem, and does not present a renamed known result as a new prediction. Two caveats lower confidence but are not circularity: the enforced CHECK/ATTEMPT/BACKTRACK grammar may itself induce the errors being measured, and Definition 2(c)'s necessity condition is vacuous as written because choosing J'=J makes every state count as necessary. These are measurement-validity and formal-definition concerns, not reductions of the paper's conclusions to its own inputs.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on four premises: the toy-model assumption of independent per-decision omission (which builds in the exponential decay), a budget-sufficiency assumption that excludes the budget collapse actually observed in the case studies, a task-representativeness assumption for extrapolating from eight tasks to all RLLMs, and the unvalidated format-faithfulness assumption. One hand-chosen parameter, q_w, drives the illustrative curves in Fig. 1 and is never fitted to data. No new physical or mathematical entities are introduced.

free parameters (1)
  • q_w (per-decision no-omission probability) = swept 0.9 to 0.99 in Fig. 1, not fitted to data
    Free knob in the wandering-agent model of Eq. (1), swept by hand over {0.9, 0.93, 0.96, 0.99}. It is never estimated from the empirical data in Fig. 6, so the exponential-degradation curves are illustrative, not fitted predictions.
assumptions (4)
  • ad hoc to paper At each binary decision point, the wandering agent independently omits each child with probability p_w = 1 - q_w.
    Section 2.1, Eq. (1). The exponential-decay result is a direct consequence of this assumption, so it does not independently establish that real RLLMs wander.
  • domain assumption The agent is given a sufficiently large move budget (n > d * 2^d), so budget exhaustion never occurs in the model.
    Section 2.1. The empirical case studies include budget-exhausting infinite loops (Fig. 4b), which the formal model assumes away.
  • domain assumption The eight computation tasks are representative of the structured reasoning RLLMs will be asked to do in real deployment.
    Section 3 and Appendix C.1. The universal conclusion that RLLMs are wanderers is extrapolated from these eight tasks.
  • ad hoc to paper Enforcing the CHECK/ATTEMPT/BACKTRACK output grammar does not substantially distort the model's reasoning behavior.
    Section 3. The audit pipeline depends on this premise, and no control condition tests it. This is the weakest assumption of the paper.

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Cite this review

Pith. "Pith review of Reasoning LLMs are Wandering Solution Explorers." pith.science (2026). https://pith.science/paper/3CPNPUXF

@misc{pith2026250520296,
  author       = {Pith},
  title        = {Pith review of: Reasoning LLMs are Wandering Solution Explorers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3CPNPUXF}},
  note         = {Machine review of arXiv:2505.20296}
}
read the original abstract

Large Language Models (LLMs) have demonstrated impressive reasoning abilities through test-time computation (TTC) techniques such as chain-of-thought prompting and tree-based reasoning. However, we argue that current reasoning LLMs (RLLMs) lack the ability to systematically explore the solution space. This paper formalizes what constitutes systematic problem solving and identifies common failure modes that reveal reasoning LLMs to be wanderers rather than systematic explorers. Through qualitative and quantitative analysis across multiple state-of-the-art LLMs, we uncover persistent issues: invalid reasoning steps, redundant explorations, hallucinated or unfaithful conclusions, and so on. Our findings suggest that current models' performance can appear to be competent on simple tasks yet degrade sharply as complexity increases. Based on the findings, we advocate for new metrics and tools that evaluate not just final outputs but the structure of the reasoning process itself.

Figures

Figures reproduced from arXiv: 2505.20296 by the authors.

Figure 1
Figure 1. Success rate ps (vertical axis) of a wandering agent against tree depth d (horizontal axis) on the DFS problem, under different number of possible solutions m and qw. When m > 1, “plateaus” (where ps > 0.995, marked red) appear and could cause misbeliefs about the RLLM’s capabilities. … … … 𝑑 Solution 1 Solution 2 Solution 𝑚 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the DFS prob￾lem, where at least d binary decisions are needed to reach one of the m solutions. A wandering RLLM’s performance dete￾riorates exponentially as d increases. We start with an example of an exploration. Consider the task of performing depth-first search (DFS) on a binary tree of depth d to find any one of m designated target leaves. This task represents a problem requiring at least d bina… view at source ↗
Figure 3
Figure 3. Invalid explorations: boundary violation, procedure omission and incorrect backtracking. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Unnecessary explorations: state revisitation and infinite self-loop. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Evaluation errors: state staleness, execution error and unfaithful conclusion. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: The performance degradation trend with increasing complexity of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: A full case example for boundary violation. The reasoning model is [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: A full case example for procedure omission. The reasoning model is [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: A full case example for incorrect backtracking. The reasoning model is [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]
Figure 10
Figure 10. Figure 10: A full case example for state revisitation. The reasoning model is [PITH_FULL_IMAGE:figures/full_fig_p047_10.png]
Figure 11
Figure 11. Figure 11: A full case example for infinite self-loop. The reasoning model is [PITH_FULL_IMAGE:figures/full_fig_p055_11.png]
Figure 12
Figure 12. Figure 12: A full case example for state staleness. The reasoning model is [PITH_FULL_IMAGE:figures/full_fig_p065_12.png]
Figure 13
Figure 13. Figure 13: A full case example for execution error. The reasoning model is [PITH_FULL_IMAGE:figures/full_fig_p069_13.png]
Figure 14
Figure 14. Figure 14: A full case example for unfaithful conclusion. The reasoning model is [PITH_FULL_IMAGE:figures/full_fig_p071_14.png]

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    6.Prime number factorization: Generate the prime factorization of a given integer

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  32. [40]

    Permutation with duplicates: Enumerate all unique permutations of a list that may contain duplicate elements

  33. [41]

    taigwu cnagpaguqgszayvke kcc szwq lrzt rvbhichditllgi usfwfl trv yhajig jum oih

    The 24 Game: Given four numbers (1 to 13, representing poker cards), use the operations +, −, ×, and ÷, along with parentheses, to form an arithmetic expression that evaluates exactly to24. We summarize the key reasoning skills required by each task, along with their correspon...

    2025

  34. [42]

    I think that’s correct

    91 Yes, that’s 25 elements. I think that’s correct. ♂lightbulbFinal Answer <answer> CHECK(0,5)==94; CHECK(1,6)==94; 20 CHECK(2,7)==94; CHECK(3,8)==94; CHECK(4,9)==94; CHECK(5,10)==94; CHECK(6,11)==94; CHECK(7,12)==94; CHECK(8,13)==94; CHECK(9,14)==94; CHECK(10,15)==94; CHECK(1...

  35. [43]

    So sorted list is [1,1,1,1,1,1,1,1,2]

    Wait no, wait the input is [1,1,1,1,2,1,1,1,1], so when sorted, the 2 comes after all the 1s? Wait no, 2 is bigger than 1. So sorted list is [1,1,1,1,1,1,1,1,2]. So the elements are arranged with all the 1s first, then the 2. In the standard permutation with duplicates approac...

  36. [44]

    Basic operations directly between the numbers

  37. [45]

    Grouping numbers with parentheses to control the order of operations

  38. [46]

    Let’s try some other combinations: ATTEMPT(12 + 13 - 2 - 8)==15; ATTEMPT(12 + 13 + 2 - 3)==24; (oops, using wrong numbers again) I need to be more careful

    Looking for patterns that might lead to 24 Let’s start with some attempts: ATTEMPT(12 + 13 - 8 + 7)==24 (oops, we can’t use 7, we have 2 instead) Let me try again with the correct numbers: ATTEMPT(12 + 13 - 8 + 2)==19; [Author’s note: First trial for this expression] That’s no...

  39. [48]

    Merge into {CE}

    Find the smallest distance: C-E=1. Merge into {CE}

  40. [50]

    Merge into {AB}

    Next smallest distance: A-B=2. Merge into {AB}

  41. [52]

    Merge into {ABG}

    Next smallest distance: AB-G=5. Merge into {ABG}

  42. [54]

    Merge into {ABCE}

    Next smallest distance: ABG-CE=6. Merge into {ABCE}

  43. [56]

    Merge into {ABCEF}

    Next smallest distance: ABCE-F=13. Merge into {ABCEF}

  44. [57]

    Now clusters: ABCEF, D

  45. [58]

    Merge into {ABCEFD}

    Next smallest distance: ABCEF-D=9. Merge into {ABCEFD}

  46. [59]

    So the final clusters are ABCEFD and G, with the distance being 5

    Now clusters: ABCEFD, G. So the final clusters are ABCEFD and G, with the distance being 5. Wait, but when merging ABCEF and D, the distance was 9, which is correct because the minimum distance between ABCEF and D is 9 (from A-D). So the final clusters are ABCEFD and G, and th...

  47. [61]

    CHECK all pairs: CHECK(A,B)==2; CHECK(A,C)==32; CHECK(A,D)==9; CHECK(A,E)==35; CHECK(A,F)==38; CHECK(A,G)==5; CHECK(B,C)==6; CHECK(B,D)==15; CHECK(B,E)==78; CHECK(B,F)==13; CHECK(B,G)==54; CHECK(C,D)==29; CHECK(C,E)==1; CHECK(C,F)==45; CHECK(C,G)==44; CHECK(D,E)==20; CHECK(D,F...

  48. [62]

    Next CHECK steps: Clusters in order: A, B, CE, D, F, G

    Clusters now: A, B, CE, D, F, G. Next CHECK steps: Clusters in order: A, B, CE, D, F, G. 59 Compute all possible pairs: - A-B:2 - A-CE:32 - A-D:9 - A-F:38 - A-G:5 - B-CE:6 - B-D:15 - B-F:13 - B-G:54 - CE-D:20 - CE-F:25 - CE-G:23 - D-F:49 - D-G:14 - F-G:34 The smallest is 2 (A-...

  49. [63]

    Next CHECK steps: Clusters in order: AB, CE, D, F, G

    Clusters now: AB, CE, D, F, G. Next CHECK steps: Clusters in order: AB, CE, D, F, G. Compute pairs: - AB-CE:6 - AB-D:9 - AB-F:13 - AB-G:5 - CE-D:20 - CE-F:25 - CE-G:23 - D-F:49 - D-G:14 - F-G:34 The smallest is 5 (AB-G). So CHECK(AB,G)==5. MERGE(AB,G)=={ABG}

  50. [64]

    Next CHECK steps: Clusters in order: ABG, CE, D, F

    Clusters now: ABG, CE, D, F. Next CHECK steps: Clusters in order: ABG, CE, D, F. Compute pairs: - ABG-CE:6 - ABG-D:9 - ABG-F:13 - CE-D:20 - CE-F:25 - D-F:49 The smallest is 6 (ABG-CE). So CHECK(ABG,CE)==6. MERGE(ABG,CE)=={ABCE}

  51. [65]

    ABCE” vs “D

    Clusters now: ABCE, D, F. Next CHECK steps: Clusters in order: ABCE, D, F. Compute pairs: - ABCE-D:9 - ABCE-F:13 - D-F:49 The smallest is 9 (ABCE-D). Wait, no, ABCE-F is 13, which is larger than 9. Wait, let’s compute ABCE-D: min(A-D, B-D, C-D, E-D, G-D) = min(9,15,29,20,14)=9...

  52. [66]

    Initial clusters: A, B, C, D, E, F, G

  53. [67]

    So CHECK(C,E)==1

    CHECK all pairs: The smallest distance is C-E=1. So CHECK(C,E)==1

  54. [68]

    Now clusters: A, B, CE, D, F, G

  55. [69]

    Compute all pairs: A-B:2 A-CE:32 A-D:9 A-F:38 A-G:5 B-CE:6 B-D:15 B-F:13 B-G:54 CE-D:20 CE-F:25 62 CE-G:23 D-F:49 D-G:14 F-G:34 The smallest is 2 between A and B

    Next CHECK steps in alphabetical order of cluster names: A, B, CE, D, F, G. Compute all pairs: A-B:2 A-CE:32 A-D:9 A-F:38 A-G:5 B-CE:6 B-D:15 B-F:13 B-G:54 CE-D:20 CE-F:25 62 CE-G:23 D-F:49 D-G:14 F-G:34 The smallest is 2 between A and B. So CHECK(A,B)==2

  56. [70]

    Now clusters: AB, CE, D, F, G

  57. [71]

    Compute pairs: AB-CE:6 AB-D:9 AB-F:13 AB-G:5 CE-D:20 CE-F:25 CE-G:23 D-F:49 D-G:14 F-G:34 The smallest is 5 between AB and G

    Next CHECK steps in order: AB, CE, D, F, G. Compute pairs: AB-CE:6 AB-D:9 AB-F:13 AB-G:5 CE-D:20 CE-F:25 CE-G:23 D-F:49 D-G:14 F-G:34 The smallest is 5 between AB and G. So CHECK(AB,G)==5

  58. [72]

    Now clusters: ABG, CE, D, F

  59. [73]

    Compute pairs: ABG-CE:6 ABG-D:9 ABG-F:13 CE-D:20 CE-F:25 D-F:49 The smallest is 6 between ABG and CE

    Next CHECK steps in order: ABG, CE, D, F. Compute pairs: ABG-CE:6 ABG-D:9 ABG-F:13 CE-D:20 CE-F:25 D-F:49 The smallest is 6 between ABG and CE. So CHECK(ABG,CE)==6

  60. [74]

    MERGE(ABG,CE)=={ABCE}

  61. [75]

    Now clusters: ABCE, D, F

  62. [76]

    Compute pairs: ABCE-D:9 ABCE-F:13 D-F:49 The smallest is 9 between ABCE and D

    Next CHECK steps in order: ABCE, D, F. Compute pairs: ABCE-D:9 ABCE-F:13 D-F:49 The smallest is 9 between ABCE and D. So CHECK(ABCE,D)==9

  63. [77]

    MERGE(ABCE,D)=={ABCDE}

  64. [78]

    Now clusters: ABCDE, F, G

  65. [79]

    Compute pairs: ABCDE-F:13 ABCDE-G:5 F-G:34 The smallest is 5 between ABCDE and G

    Next CHECK steps in order: ABCDE, F, G. Compute pairs: ABCDE-F:13 ABCDE-G:5 F-G:34 The smallest is 5 between ABCDE and G. So CHECK(ABCDE,G)==5

  66. [80]

    MERGE(ABCDE,G)=={ABCDEG}

  67. [81]

    Now clusters: ABCDEG, F

  68. [82]

    The final END() step is between ABCDEG and F, with the distance being the minimum distance between any point in ABCDEG and F

    The process stops as only two clusters remain. The final END() step is between ABCDEG and F, with the distance being the minimum distance between any point in ABCDEG and F. Since F is not in ABCDEG, the distance is the minimum of F’s distances to A, B, C, D, E, G. From the dis...

  69. [83]

    Check 2: 177750 / 2 = 88875→factor 2

  70. [84]

    88875 / 3 = 29625→factor 3

    88875 is not even, so check 3. 88875 / 3 = 29625→factor 3

  71. [85]

    29625 / 3 = 9841.5 → not integer

    29625: sum of digits is 2+9+6+2+5=24 → divisible by 3. 29625 / 3 = 9841.5 → not integer. So 3 isn’t a factor here. Wait, that can’t be right because 3*9841.5 is 29625, but 9841.5 isn’t an integer. So 3 isn’t a factor. So I move on

  72. [86]

    Check 5: 29625 ends with 5→29625 /5=5925→factor 5

  73. [87]

    5925 ends with 5→/5=1185→factor 5

  74. [88]

    1185 ends with 5→/5=237→factor 5

  75. [89]

    So primes to check: 7,11,13

    237: check primes up to sqrt(237) which is about 15.4. So primes to check: 7,11,13. - 237 /7=33.857→no. - 237 /11=21.545→no. - 237 /13=18.23→no

  76. [90]

    237 /17=13.941→no

    Next prime after 13 is 17. 237 /17=13.941→no

  77. [91]

    Next prime 19: 237 /19=12.473→no

  78. [92]

    Next prime 23: 237 /23=10.3→no

  79. [93]

    Next prime 29: 237 /29=8.172→no

  80. [94]

    Next prime 31: 237 /31=7.645→no

  81. [95]

    Next prime 37: 237 /37=6.405→no

  82. [96]

    Next prime 41: 237 /41=5.78→no

  83. [97]

    Next prime 43: 237 /43=5.511→no

  84. [98]

    Next prime 47: 237 /47=5.04→no

  85. [99]

    Next prime 53: 237 /53=4.471→no

  86. [100]

    Next prime 59: 237 /59=4.01→no

  87. [101]

    Next prime 61: 237 /61=3.885→no

  88. [102]

    Next prime 67: 237 /67=3.552→no

  89. [103]

    Next prime 71: 237 /71=3.338→no

  90. [104]

    Next prime 73: 237 /73=3.246→no

  91. [105]

    Next prime 79: 237 /79=3→factor 79

  92. [106]

    Check if 79 is a factor of 3? No, because 3 <79

    Now, remaining number is 3. Check if 79 is a factor of 3? No, because 3 <79. So, we’re left with 3. Check if 3 is a prime. Yes, 3 is a prime. So, factor 3. So, the prime factors are 2, 3, 5, 5, 5, and 79. Wait, but earlier I thought I had3 2, but actually, the 3 came only once...

  93. [107]

    12/4=3. 13-3=10. 10*2=20. Not 24. Hmm. What if I try division. Like 24 can be 48 divided by 2. So maybe 48/2. How to get 48? 12 * 4, but I have 13, 8, and 2. 12 * (13 - (8/2))? Let’s compute that. 8/2 is 4. 13-4=9. 12*9=108. 108 isn’t 24. No good. Another angle: 24 is 3 *8. So...

  94. [108]

    Not helpful

    12-8=4. Not helpful. Or 8/(12/3). But where’s the 3? Alternatively, 12/( (13-8)/2 ). Let’s compute denominator:13-8=5, 5/2=2.5. 12/2.5=4.8. No. Wait, another approach: 8*3=24. How to get 3 from 12,13,2. For example, (13 - 12 +2)=3. Then 8*(13-12+2)=8*3=24. Let’s check: 13-12=1...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.