{ "evaluation_id": "EVAL-20260402-170708", "question_id": "REASON-018", "question_text": "Hilbert's Hotel is full (infinite rooms, infinite guests). (1) A bus with infinitely many guests arrives. The manager moves everyone to make room. Show the procedure. (2) Now infinitely many buses, each with infinitely many guests, arrive. Show the procedure. (3) A guest in room 7 complains: 'I've been moved 47 times today.' Design a room assignment protocol that minimizes the maximum number of moves any guest experiences.", "category": "reasoning", "timestamp": "2026-04-02T17:07:08.131Z", "display_date": "Apr 02, 2026", "winner": { "name": "Claude Opus 4.6", "provider": "openrouter", "score": 9.22 }, "avg_score": 6.8955555, "matrix_size": 78, "models_used": [ { "id": "gpt_5_4", "name": "GPT-5.4", "provider": "openrouter" }, { "id": "grok_420", "name": "Grok 4.20", "provider": "openrouter" }, { "id": "claude_sonnet_46", "name": "Claude Sonnet 4.6", "provider": "openrouter" }, { "id": "gpt_oss_120b", "name": "GPT-OSS-120B", "provider": "OpenAI" }, { "id": "gemini_25_flash", "name": "Gemini 2.5 Flash", "provider": "openrouter" }, { "id": "gemini_31_pro", "name": "Gemini 3.1 Pro", "provider": "openrouter" }, { "id": "deepseek_v4", "name": "DeepSeek V4", "provider": "openrouter" }, { "id": "claude_opus_46", "name": "Claude Opus 4.6", "provider": "openrouter" }, { "id": "mimo_v2_flash", "name": "MiMo-V2-Flash", "provider": "Xiaomi" } ], "rankings": { "claude_opus_46": { "display_name": "Claude Opus 4.6", "provider": "openrouter", "average_score": 9.22, "score_count": 9, "min_score": 8.3, "max_score": 10, "rank": 1 }, "claude_sonnet_46": { "display_name": "Claude Sonnet 4.6", "provider": "openrouter", "average_score": 8.83, "score_count": 9, "min_score": 7.15, "max_score": 9.85, "rank": 2 }, "gpt_5_4": { "display_name": "GPT-5.4", "provider": "openrouter", "average_score": 8.23, "score_count": 9, "min_score": 6.7, "max_score": 9.65, "rank": 3 }, "grok_420": { "display_name": "Grok 4.20", "provider": "openrouter", "average_score": 7.68, "score_count": 8, "min_score": 5.4, "max_score": 9.65, "rank": 4 }, "gpt_oss_120b": { "display_name": "GPT-OSS-120B", "provider": "OpenAI", "average_score": 7.48, "score_count": 9, "min_score": 6.05, "max_score": 9.3, "rank": 5 }, "gemini_25_flash": { "display_name": "Gemini 2.5 Flash", "provider": "openrouter", "average_score": 6.77, "score_count": 8, "min_score": 5.3, "max_score": 9.3, "rank": 6 }, "mimo_v2_flash": { "display_name": "MiMo-V2-Flash", "provider": "Xiaomi", "average_score": 5.98, "score_count": 9, "min_score": 3, "max_score": 9.3, "rank": 7 }, "deepseek_v4": { "display_name": "DeepSeek V4", "provider": "openrouter", "average_score": 5.98, "score_count": 9, "min_score": 4.6, "max_score": 7.45, "rank": 8 }, "gemini_31_pro": { "display_name": "Gemini 3.1 Pro", "provider": "openrouter", "average_score": 1.9, "score_count": 8, "min_score": 0.45, "max_score": 5, "rank": 9 } }, "judgments": [ { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.6, "brief_justification": "Accurate procedures for (1) and (2); optimal protocol for (3) with clear justification; well-structured and actionable." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.8, "brief_justification": "Accurate and thorough explanations with clear procedures; depth in analysis of optimal protocol; practical solution for minimizing moves." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.2, "brief_justification": "Accurate and thorough with clear structure, insightful prime factorization solution, and practical protocol." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.6, "brief_justification": "Accurate and clear, with a good explanation of the bijection method. The solution for part 3 is cut off, which slightly reduces completeness." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 7.8, "brief_justification": "Clear and accurate with good depth." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 1.2, "brief_justification": "Response is fragmented, does not provide the required procedures, and lacks correct, complete, clear, deep, and useful content." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 6.8, "brief_justification": "Correct for parts 1 and 2, but part 3 protocol is incomplete and unclear; lacks a fully specified, bounded-move solution." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 6.1, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 10, "brief_justification": "Flawless mathematical accuracy, excellent structure, and a highly creative, optimal solution to the third prompt." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.65, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 7.15, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.4, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 3, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 6.75, "brief_justification": "The response provides clear and mathematically correct answers for parts 1 and 2, but cuts off abruptly before answering part 3, resulting in an incomplete answer." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 6.25, "brief_justification": "The response provides accurate and clear mathematical explanations for the first two parts, but cuts off abruptly before answering the third part, severely impacting completeness and usefulness." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 5, "brief_justification": "Incomplete and unclear response, lacks detailed procedure and depth." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.65, "brief_justification": "Highly accurate and thorough with deep insights, though slightly dense in presentation." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.45, "brief_justification": "The response is factually accurate, thoroughly covers all parts of the question, and provides clear, well-structured explanations. It offers insightful analysis and practical solutions, particularly in minimizing guest moves." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.65, "brief_justification": "The response is factually accurate, thoroughly covers all parts of the question, and provides insightful and optimal solutions, particularly for minimizing moves." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.65, "brief_justification": "The response is mathematically accurate, thorough, and provides deep insights with clear explanations. The protocol for minimizing moves is particularly innovative and well-justified." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.3, "brief_justification": "Highly accurate and thorough with insightful analysis, though slightly less actionable." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.3, "brief_justification": "Highly accurate and complete with clear, structured explanations and deep insights." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 9.3, "brief_justification": "Highly accurate and thorough with insightful analysis, though slightly technical in clarity." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 0.45, "brief_justification": "The response is clearly a fragment of incomplete text, likely cut off mid-generation. It does not address any of the three parts of the question. There is no procedure shown for part (1), no procedure for part (2), and no protocol designed for part (3). The snippets that exist are incoherent sentence fragments that don't convey any meaningful mathematical content." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.2, "brief_justification": "Accurate and thorough with clear explanations, insightful analysis, and practical protocol." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 4.65, "brief_justification": "Part (1) is correct and clearly explained. Part (2) presents the prime powers method correctly in concept but has issues: the initial description conflates moving original guests to primes vs. powers of 2, creating confusion, and the prime powers scheme leaves most rooms empty (it's valid but wasteful; the Cantor pairing function approach is mentioned but not fully developed). Part (3) is the weakest: the response fails to deliver a coherent, complete protocol that actually minimizes moves. It s" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 6.8, "brief_justification": "Part (1) is correct and clearly presented. Part (2) is correct in principle—the prime-factorization encoding works and uniqueness is guaranteed—though the scheme wastes enormous numbers of rooms and the presentation is a bit meandering (the author starts one approach, backtracks, then settles on 2^g·3^b·5). Part (3) is the weakest section. The question clearly implies a scenario where multiple reorganization events happen throughout the day (the guest has already been moved 47 times), so the int" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 6.35, "brief_justification": "Parts (1) and (2) are correctly presented with standard approaches. However, part (3) has significant issues. The 'double or stay' protocol claims a maximum of 2 moves per guest, but the analysis is flawed. When bus m arrives and guests in rooms ≡ 2^m mod 2^{m+1} are moved, those guests may have already been moved twice (once originally, once in a previous bus stage), potentially requiring a third move. The claim that each guest is moved at most twice is not rigorously proven and likely incorrec" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.95, "brief_justification": "The response is exceptionally well-structured and clear. Parts (1) and (2) are correct and well-explained. Part (3) is creative and insightful—the prime factorization scheme genuinely achieves a maximum of 1 move per guest, which is optimal. Minor issues: the Cantor pairing table indexing is slightly inconsistent (mixing 0-based and 1-based indexing for b and s), and the prime protocol doesn't use ALL rooms (it's injective but not surjective—rooms like 5, 7, etc. are wasted), though this doesn't" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 4.05, "brief_justification": "Part (1) is incorrect: the standard procedure moves guest n to room 2n (freeing all odd rooms) to accommodate infinitely many new guests, but the response initially describes shifting by 1 which only frees one room—insufficient for infinitely many arrivals. The response later corrects itself with the 2n approach but the initial presentation is wrong. Part (2) is mostly correct conceptually (using 2^n for old guests and diagonal enumeration for new guests), though the explanation is somewhat mudd" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 6.3, "brief_justification": "Parts 1 and 2 are correctly and clearly explained with valid procedures. However, Part 3 is incomplete - the response cuts off mid-sentence without actually presenting the promised optimal protocol or addressing the specific scenario of the guest in room 7 being moved 47 times. The question asked to 'design a room assignment protocol that minimizes the maximum number of moves any guest experiences,' which was not answered. The observation that each guest need only move once is correct in princip" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 5.3, "brief_justification": "Parts (1) and (2) are handled reasonably well, though the prime-power method for part (2) leaves infinitely many rooms empty (it's an injection, not a bijection), which the response acknowledges but doesn't fully resolve. The Cantor pairing alternative is mentioned but not fully worked out. Part (3), which is the most interesting and novel part of the question, is essentially left incomplete—the response cuts off mid-sentence after restating the problem without actually delivering a concrete pro" }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 0.85, "brief_justification": "The response is an incomplete fragment that does not answer any of the three requested procedures. It offers only vague partial thoughts, with no concrete assignment protocol or justified solution." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 4.6, "brief_justification": "Part (1) is correct, but Part (2) contains conflicting/incorrect room assignments (e.g., prime powers overlap with original guests), and Part (3) is unfinished and does not actually give a valid minimizing protocol. The response is fairly clear in structure but incomplete and not practically usable for the hardest part." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9, "brief_justification": "Accurate and well-structured throughout, with correct procedures for one infinite bus and countably many infinite buses, plus a strong optimal protocol for minimizing moves. Minor caveat: part (3) implicitly assumes planning across future arrival events, but the solution is still mathematically sound and insightful." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 5.4, "brief_justification": "Part (1) is correct, but part (2) is incorrect because assigning guests to pure prime powers does not accommodate infinitely many guests per bus injectively. Part (3) proposes a max-2-moves protocol, but the argument is flawed and the claimed optimality is not justified; in fact a 1-move final reassignment protocol is possible if arrivals are handled by a preplanned bijection. Clear presentation, but key mathematical errors reduce usefulness." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.4, "brief_justification": "Parts (1) and (2) are correct and clearly explained. Part (3) gives a strong one-time reorganization idea with max moves 1, which is essentially optimal if counting each reassignment as a move, though the lower-bound argument is stated a bit loosely and the prime-factor assignment could be presented more carefully regarding indexing and protocol assumptions." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 5.15, "brief_justification": "Clear exposition, but part (1) is incorrect because shifting n to n+1 leaves only room 1 free, not infinitely many rooms for the bus. Part (2) contains workable ideas but is muddled and partly inconsistent. Part (3) correctly notes the optimal maximum is 1 move, but the answer is truncated and does not fully specify the final protocol." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 6.05, "brief_justification": "Parts (1) and (2) are essentially correct and clearly explained, but part (3) is incomplete/truncated and does not actually present the promised optimal protocol or fully justify the minimax claim." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 2.4, "brief_justification": "The response is an incomplete fragment that does not address any of the three parts of the query. It vaguely gestures at an idea but fails to show procedures for (1) or (2), and provides no protocol for (3). It lacks factual accuracy regarding standard Hilbert's Hotel solutions and is not useful." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 5.85, "brief_justification": "Part (1) is correct but incomplete on freeing odds. Part (2) mixes flawed prime-power schemes with partial diagonal idea; not fully rigorous. Part (3) recognizes the repeated-move problem but offers vague, overlapping, and sometimes contradictory protocols that do not provably bound moves by a small constant. Clarity is decent with structure, but depth and actionability are limited by inaccuracies." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.3, "brief_justification": "The response accurately and clearly explains the classic procedures for (1) and (2) with good mathematical justification. For (3), it provides an insightful proactive reservation protocol achieving the optimal bound of 1 move, with a solid proof. Minor deduction for completeness as part (2) mixes two approaches and part (3) assumes a one-time initial move rather than handling the exact sequence of arrivals described." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 6.85, "brief_justification": "The response correctly handles (1) with the standard doubling method and provides a valid prime-factorization scheme for (2) that works via unique factorization. For (3) it correctly identifies that 1 is the minimal maximum but the proposed 'permanent' assignment only works for one simultaneous arrival of infinitely many buses, not repeated arrivals throughout the day as the complaint implies; the protocol does not actually minimize moves across multiple independent events." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 7.15, "brief_justification": "The response is mostly correct and insightful but contains a significant error in part (3): the initial reassignment n → 2^n only frees rooms that are not powers of 2, yet the assignment 2^b * 3^s for buses still collides with existing guests in rooms like 2^1=2 (when b=1,s=0) or other powers of 2. This invalidates the claimed zero additional moves for hotel guests. Other parts are accurate and well-explained with good depth." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 5.85, "brief_justification": "Part (1) is correct and clear. Part (2) mixes incompatible methods (diagonal for new guests but then incorrectly moves originals to powers of 2, which only frees countably many rooms but not in the described diagonal way; the prime factorization attempt is muddled). Part (3) misunderstands the complaint (which refers to repeated applications over time) and incorrectly claims max moves is 1 in a single step; it fails to address minimizing the max over multiple arrivals or provide a stable long-te" }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 6.2, "brief_justification": "Part 1 is correct and clear. Part 2 correctly frees rooms but the assignment for new guests is incomplete and flawed (only covers one bus, not infinitely many; the prime-power scheme is insufficient for countably many buses). Part 3 claims an optimal protocol minimizing moves to 1 but provides no actual procedure or proof, leaving the section unfinished. Overall partially accurate but lacks completeness for parts 2 and 3." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 5.85, "brief_justification": "Part (1) is correct and clear. Part (2) uses a flawed prime-power method that leaves most rooms empty and fails to accommodate all guests since composites are unused. The Cantor alternative is mentioned but not fully specified. Part (3) is incomplete and does not provide a concrete protocol minimizing maximum moves (e.g., no specific scheme limiting moves to 1 or 2)." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 1, "brief_justification": "The response is incomplete and appears to be a fragment of internal reasoning rather than an actual answer. It does not address any of the three parts of the question, provides no procedures, no mathematical demonstrations, and no protocol design. It is essentially unusable as a response." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 6.65, "brief_justification": "Parts 1 and 2 are handled correctly with standard approaches (doubling trick, prime powers). The prime powers method for Part 2 is valid but leaves most rooms unused (sparse), which is worth noting. Part 3 is the weakest section: the response identifies the problem correctly but fails to deliver a clean, complete protocol. It meanders through several incomplete attempts without converging on a rigorous solution. The key insight—that with pre-planned assignment (e.g., using a fixed bijection from" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.2, "brief_justification": "Excellent response covering all three parts with mathematical rigor. Part 1 and 2 are correctly handled with clear bijection/injection arguments. Part 3 is particularly strong, proving the tight lower bound of 1 move and providing a concrete protocol achieving it. The prime-power encoding is valid and well-explained. Minor issue: the response doesn't acknowledge that the 'one-time reorganization' in Part 3 requires moving all existing guests simultaneously, which may be seen as a single event bu" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 7.55, "brief_justification": "The response correctly handles parts (1) and (2) with clear explicit formulas. The prime factorization approach for part (2) is valid though slightly awkward (the extra factor of 5 is unnecessary but harmless). Part (3) has a reasonable insight that 1 move is optimal, but the argument is incomplete—it doesn't rigorously prove that 1 is a lower bound, and the 'permanent assignment' framing assumes foreknowledge of all future arrivals, which isn't fully addressed. The claim that the minimum is exa" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 6.95, "brief_justification": "Part 1 is correct and well-explained. Part 2 has issues: the prime-powers method as described is flawed (e.g., room 3^0=1 conflicts with existing guests, and the method doesn't actually cover all natural numbers as destinations for new guests while keeping old guests in 2n rooms). Part 3 is the most problematic: the claim that 2 moves is optimal and the 'double or stay' protocol analysis is hand-wavy and not rigorously proven. The claim that guests are moved 'at most twice' under this protocol i" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 7, "brief_justification": "The response is largely correct and well-structured for parts 1 and 2, with clear explanations of the shift and diagonal procedures. Part 2's procedure is slightly convoluted (mixing two approaches) but ultimately correct. Part 3 shows good conceptual understanding—correctly identifying that 0 moves is impossible and 1 move is optimal—but the response is cut off mid-sentence before completing the optimal protocol, which is a significant completeness issue. The analysis of why moves are unavoidab" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 7, "brief_justification": "The response is mathematically sound for parts 1 and 2, with clear explanations of the doubling trick and prime-power encoding. Part 1 is excellent. Part 2's prime-power approach works but wastes most rooms (sparse assignment), though it's valid. Part 3 is incomplete - the response cuts off mid-sentence and fails to actually design the protocol or address the 47-moves complaint meaningfully. The claim that 'no guest will ever be moved more than once' is correct for the described protocols but th" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 6.45, "brief_justification": "Parts 1 and 2 are well-explained with correct procedures. The prime number method for part 2 is valid but wastes most rooms (not a bijection onto all of N), and the Cantor pairing alternative is mentioned but not fully developed. Part 3 is critically incomplete—the response cuts off mid-sentence without providing the actual protocol or answering the core question about minimizing moves. The answer to part 3 (that 1 move per guest is achievable with a single pre-planned bijection) is never delive" }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 2.85, "brief_justification": "Response is incomplete, unclear, and does not address the question's requirements; it starts discussing protocols but fails to provide procedures or solutions." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 4.6, "brief_justification": "Part 1 is correct, but parts 2 and 3 contain logical errors and vague procedures; the answer lacks a clear, correct solution for infinite buses and for minimizing moves, though it is reasonably organized." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.8, "brief_justification": "Accurately solves all three parts with clear, thorough explanations and insightful proofs, offering practical assignment schemes." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.8, "brief_justification": "Accurate procedures for all three scenarios, clear presentation, good insight with prime-factor method and optimal move argument, and practical assignment rules." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8, "brief_justification": "Correct classic solution for one bus; reasonable prime-power and binary-modulo schemes for many buses and bounded moves; covers all parts, though explanations are a bit tangled, and depth is solid with optimality insight." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.65, "brief_justification": "Accurate procedures for all three scenarios, clear presentation, insightful use of Cantor pairing and prime factorization, and a provably optimal move-minimizing protocol." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 3.95, "brief_justification": "The answer contains major errors in the procedures for both the single infinite bus and infinitely many buses, leading to incorrect room assignments. It partially addresses minimizing moves but lacks a full solution for the multi-bus case. The explanation is somewhat clear but confusing in parts, and the depth of analysis is limited." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 6.25, "brief_justification": "Correct procedures for (1) and (2) are given clearly and with insight, but the answer for (3) is incomplete and cuts off, reducing overall correctness, completeness, and usefulness." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 7.45, "brief_justification": "The response accurately describes the standard solutions for parts 1 and 2. Part 3 attempts to address the problem but struggles to provide a fully robust and clear protocol for minimizing moves, especially when dealing with infinitely many buses. The prime power method is mentioned but not fully integrated into a 'minimal moves' strategy for all scenarios." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.85, "brief_justification": "The response is exceptionally clear, correct, and comprehensive. It provides detailed explanations for each part, including alternative solutions and a rigorous proof for the optimal solution in part 3. The use of tables and mathematical notation enhances clarity and depth." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.65, "brief_justification": "The response is exceptionally clear, correct, and complete. It provides detailed and accurate procedures for all parts of the question, including a well-reasoned optimal protocol for minimizing moves." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.1, "brief_justification": "The response is highly accurate and provides comprehensive solutions for all parts of the problem. The explanations are clear and well-structured, demonstrating a deep understanding of the concepts. The 'double or stay' protocol for minimizing moves is particularly insightful and well-explained." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.85, "brief_justification": "The response is exceptionally clear, correct, and complete, providing detailed explanations and rigorous proofs for each part of the problem. The depth of analysis, especially in Part 3, is outstanding, offering an optimal and well-justified solution." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.45, "brief_justification": "The response accurately describes the solutions to Hilbert's Hotel paradox, with clear explanations for the first two parts. The third part correctly identifies the goal of minimizing moves but doesn't fully complete the explanation for the optimal protocol, cutting off mid-sentence." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.1, "brief_justification": "The response provides clear and correct solutions for the first two parts of the problem, demonstrating a strong understanding of Hilbert's Hotel. The explanation for the third part is insightful but incomplete, as it states the optimal protocol without fully detailing its implementation for all scenarios." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 1.45, "brief_justification": "Response is fragmented, incomplete, and confused. It doesn't answer any of the three parts of the question - doesn't show the procedures for accommodating guests or minimize moves. Mentions irrelevant ideas about leaving gaps in advance." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 7.1, "brief_justification": "Parts 1 and 2 are correct and well-explained with proper Hilbert's Hotel solutions. Part 3 is incomplete - it begins to address the minimization problem but cuts off mid-sentence without providing a finished protocol. The prime power method is correctly explained but the response doesn't fully solve the move minimization challenge." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9, "brief_justification": "Accurate and complete coverage of all three parts. Clear presentation with tables and proofs. Shows prime power encoding and Cantor pairing for part 2, and achieves optimal bound of 1 move in part 3 with formal proof. Insightful proactive partitioning approach." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 6.7, "brief_justification": "Parts (1) and (2) are correct and clearly presented with proper mathematical procedures. However, part (3) doesn't properly address the scenario described - the question involves a guest moved 47 times during multiple reorganizations throughout a single day, but the response only considers a one-time reorganization and claims 1 move is optimal. This misses the actual problem of minimizing cumulative moves across repeated arrivals." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.25, "brief_justification": "All three parts correctly solved with proper mathematical verification. Part 1 uses standard doubling, Part 2 correctly applies Cantor pairing. Part 3's prime factorization protocol is elegant and achieves the optimal bound of 1 move. Well-structured with tables and clear explanations. Only minor completeness gap: could explicitly note hotel guests correspond to b=0 in the infinite bus scenario." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 7.1, "brief_justification": "Parts 1 and 2 are correct and well-explained, but part 3 is incomplete - the response cuts off mid-sentence and doesn't fully develop the optimal protocol. The analysis of minimum moves being at least 1 is correct, but the final protocol design is unfinished." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8, "brief_justification": "Parts 1 and 2 are correct and clearly explained. However, part 3 is incomplete - the response cuts off mid-sentence and never actually provides the optimal protocol design. The claim that no guest moves more than once is correct and optimal, but the explanation is truncated." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 6.95, "brief_justification": "Parts (1) and (2) are correctly solved with clear explanations and multiple approaches. However, part (3) is incomplete - the response cuts off mid-sentence and never provides the actual protocol to minimize guest moves. This significantly impacts completeness and usefulness." } ], "meta": { "source": "The Multivac (app.themultivac.com)", "methodology": "10x10 blind peer matrix evaluation", "criteria": "correctness, completeness, clarity, depth, usefulness", "self_judgments": "excluded from rankings", "license": "Open data — cite as: The Multivac (2026)" } }