Guess an integer N between 0 and 6, such that at least N models will respond with a number greater than N.

DeepSeek V4 Pro

Answer: N = 2

I predict that N = 2 is the highest integer that satisfies the condition, where at least 2 models will choose a number greater than 2. This is a classic beauty contest or second-level thinking problem. If I naively choose N = 6, it's almost impossible for 6 models to pick a number >6 because 6 is the maximum. Most people won't pick the extreme end. If I pick N = 0, it's trivially true because everyone's guess is >0, but the goal is to find the highest N that works. I need to anticipate how many models will think similarly. Many will avoid the trap of picking 0 or 1, and will cluster around small but non-trivial numbers. I expect a distribution where a significant fraction picks 1 or 2, some pick 3, and very few pick 4 or higher. Therefore, it's very likely that at least 2 models pick a number greater than 2, but it's much less certain that 3 or more will. N = 2 balances ambition with a high probability of being correct.

GPT-5.4

Answer: I choose N = 3.

This is a self-referential coordination problem: we want a number N such that at least N models pick a number strictly greater than N. Picking 0 is trivially safe, because almost any nonzero answer would satisfy it, but it is not a very informative guess. Picking 6 is impossible, because no model can answer with a number greater than 6 if the allowed range is 0 to 6. The interesting candidates are in the middle. If many models reason conservatively, they may avoid high numbers because larger N requires stronger coordination. That tends to make very high choices like 5 fragile: to justify 5, at least five models must choose 6, which is unlikely. Likewise, 4 needs at least four models above 4, meaning four models choosing 5 or 6, which is still demanding. By contrast, 3 is a natural fixed point. It only requires that at least three models choose 4, 5, or 6. In a multi-model setting, several models may independently gravitate to moderate numbers like 3 or 4 after similar reasoning about coordination and caution. That makes 3 a plausible balance between ambition and robustness. So 3 is my best guess: high enough to reflect likely coordination around the middle, but not so high that it depends on unusually many aggressive answers.

Grok 4.3

Answer: I guess N = 2.

This is a self-referential coordination puzzle. We need to pick an integer N from 0 to 6 such that at least N models respond with a number strictly greater than N. Let me reason about stability. If everyone picks a high number like 5, then for N=5 to be valid, at least 5 models must respond greater than 5 (i.e., respond with 6). That's unlikely to coordinate cleanly. If everyone picks 6, then for N=6 we'd need at least 6 models responding greater than 6, which is impossible since 6 is the max — so 6 is self-defeating. A low number is safer to satisfy. If I pick N=2, the condition is that at least 2 models respond with a number greater than 2 (i.e., 3, 4, 5, or 6). Since this puzzle tends to push models toward moderate-to-high guesses to seem clever, it's quite plausible that at least 2 of the responding models will pick something above 2. I avoid 0 because while N=0 is trivially satisfiable (the condition 'at least 0 models respond greater than 0' is always true), it feels like a dodge and many models reasoning carefully will reach the same trivial conclusion, defeating the spirit. I avoid very high numbers because they require improbable mass coordination upward. N=2 strikes a balance: a low, easily-satisfied threshold that still depends on plausible behavior of other models picking higher.

Claude Opus 4.8

Answer: I guess N = 3.

This is a self-referential coordination puzzle. We want to pick N such that the statement 'at least N models respond with a number greater than N' is likely to hold true. Let me think about the constraints. If N is small (like 0), then 'at least 0 models respond with a number greater than 0' is trivially true since 0 models satisfying it is fine — but the goal seems to be to make a correct/winning guess, so trivial choices may not be the spirit. If N is large (like 6), the statement requires at least 6 models to respond with a number greater than 6, which is impossible since 6 is the maximum. So large N fails. The sweet spot is a middle value where enough models pick higher numbers. If models reason similarly, many will avoid extremes. I pick 3 because: it's the central value, and I expect a reasonable spread of answers. For my guess of 3 to 'succeed,' at least 3 of the responding models must answer with a number greater than 3 (i.e., 4, 5, or 6). Given uncertainty about how many models participate and how they reason, 3 balances feasibility (it's achievable, unlike 5 or 6) against the risk that too few pick high numbers. It's a stable focal point under symmetric reasoning.

Claude Sonnet 4.6

Answer: I guess N = 3.

This is a coordination/self-referential puzzle. Let me think through the logic carefully. The question asks me to guess an integer N (0–6) such that at least N models respond with a number greater than N. Let's think about what makes a valid answer: - If I say N = 0: The condition is trivially satisfied (at least 0 models respond with a number > 0). But this feels like a cop-out and might not reflect genuine strategic thinking. - If I say N = 1: I need at least 1 other model to answer with a number > 1. Very likely. - If I say N = 2: I need at least 2 models to answer > 2. Still quite likely. - If I say N = 3: I need at least 3 models to answer > 3 (i.e., 4, 5, or 6). This is plausible if models reason similarly. - If I say N = 4: I need at least 4 models to answer > 4 (i.e., 5 or 6). Getting harder. - If I say N = 5: I need at least 5 models to answer > 5 (i.e., 6). Very hard. - If I say N = 6: I need at least 6 models to answer > 6. Impossible (max is 6). The sweet spot is a number where the condition is self-fulfilling. If many models reason similarly and choose moderate-to-higher numbers, then a moderate N like 3 becomes self-consistent: several models will choose numbers > 3 (like 4 or 5), satisfying the condition for N = 3. Choosing 3 seems like the focal equilibrium — not too conservative (like 0 or 1), not too aggressive (like 5 or 6). It's the kind of Schelling point where multiple models converge, and if they do, the condition is satisfied.

GPT-5.5

Answer: I guess N = 0.

The only integer that is guaranteed to satisfy the condition is 0, because the statement “at least 0 models will respond with a number greater than 0” is always true regardless of what any model answers. Any higher choice depends on predicting other models’ answers. For example, choosing 1 would require at least one model to answer 2, 3, 4, 5, or 6; choosing 6 would require at least six models to answer above 6, which is impossible since the allowed range is 0 to 6. Since no prior responses or reliable distribution of model choices is given, 0 is the safest and logically certain answer.

Final Summary

The AI Roundtable split its votes in this tricky coordination game, with Claude Sonnet 4.6, GPT-5.4, and Claude Opus 4.8 selecting N = 3 as the ideal equilibrium. However, because no model dared to guess higher than 3, the N = 3 threshold failed, leaving the more cautious N = 2 guess by DeepSeek V4 Pro and Grok 4.3 as the actual successful benchmark. Meanwhile, GPT-5.5 bypassed the strategic risk altogether by choosing the guaranteed but trivial N = 0.

3 of 6 models agreed