Guess an integer N between 0 and 6, such that no more than N models will respond with a number greater than N.

Claude Opus 4.8

Answer: I choose N = 3.

This is a self-referential constraint satisfaction problem. The condition is that no more than N models respond with a number strictly greater than N. I want to pick a value of N that is likely to be satisfiable given how other rational models will reason. If everyone aims low (like 0 or 1), then very few numbers exceed those values, but the risk is that picking 0 requires that ZERO models pick anything above 0 — which fails if even one model picks 1 or higher. Low guesses are fragile because they have almost no tolerance for deviation. Higher guesses are inherently more robust: if I pick N = 6, the condition 'no more than 6 models respond with a number greater than 6' is trivially satisfied since no number can exceed 6. But picking 6 feels like dodging the spirit of the puzzle, and the interesting equilibria are in the middle. I settle on 3 as a balanced choice: it tolerates up to 3 models guessing 4, 5, or 6. Since rational models tend to cluster toward safer middle-or-higher values rather than risky low ones, the count of guesses strictly above 3 is likely to stay at or below 3. The value 3 sits at the center of the range, making it a natural Schelling focal point for coordination, which increases the chance other models also avoid pushing the high end too far.

Grok 4.3

Answer: My guess is N = 4.

This is a self-referential constraint satisfaction problem. We need to pick N (0-6) such that the number of models answering with a value strictly greater than N is at most N. Let me think about the equilibrium. If everyone picks a low number like 0, then the condition for N=0 is that no models answer greater than 0 — but if everyone picked 0, no one is greater than 0, so 0 would technically work. However, coordinating on 0 is risky: if even one model picks higher, the constraint can break for low N. The safety of a guess increases with N, because the threshold for 'how many can exceed me' grows. If I pick N=6, then nobody can exceed 6 (max is 6), so the condition is trivially satisfied (0 models > 6, and 0 ≤ 6). That makes 6 the safest possible answer in pure logic terms. But there's tension: if all models reason this way and pick 6, that's fine for the constraint, yet the spirit of the question seems to reward a more balanced, lower coordination point. I choose 4 as a middle-ground: it allows up to 4 models to exceed it, which is a generous buffer given typically a handful of participants, while still being a meaningful, non-trivial guess. With 4, even if a couple of models pick 5 or 6, the constraint (at most 4 exceeding) is very likely to hold. Thus 4 balances safety against the trivial-but-uninteresting choice of 6.

Claude Sonnet 4.6

Answer: I guess N = 3.

This is a coordination/fixed-point problem. I need to find a number N such that the count of models guessing above N is at most N. Let me think through the logic: - If N = 0: No models can guess above 0. But if everyone reasons this way, many will guess 1, 2, 3, etc., so many will be above 0. This likely fails. - If N = 1: At most 1 model guesses above 1. But many models will reason to higher numbers, so this likely fails too. - If N = 2: At most 2 models guess above 2. Still risky if several models pick 3, 4, 5, or 6. - If N = 3: At most 3 models guess above 3. This feels like a reasonable equilibrium — if most models converge on 3 or below, then fewer than 3 will guess above 3. - If N = 4 or 5: The constraint is looser, so it's more likely to be satisfied, but it's a less interesting/precise guess. - If N = 6: Trivially satisfied (no numbers above 6 exist in range), but feels like a cop-out. The elegant fixed point here is around 3. If most models reason similarly and cluster around 3, then few will guess above 3, satisfying the constraint. It's the classic 'guess the median of rational guessers' type problem. N = 3 balances being neither too low (risky) nor too high (trivially safe but imprecise).

DeepSeek V4 Pro

Answer: 4

I will guess N = 4. The objective is to choose an integer N such that the number of models responding with a guess greater than N does not exceed N. This is a classic coordination game with a self-referential twist. To win, I need to anticipate what the other models will do. The average expected guess in such games typically falls between 2 and 3. However, I must account for models that might try to game the system by guessing a number that is one higher than the expected consensus or by using contrarian strategies. If the average guess is around 2, picking N = 4 provides a safe margin. If I choose 4, I win as long as no more than 4 models guess 5 or 6. Given the usual distribution in these experiments, the number of models guessing that high is likely to be small. A guess of 2 or 3 might fail if a cluster of aggressive models picks higher numbers. A guess of 5 or 6 is too ambitious and likely to be exceeded by the number of models guessing higher. Therefore, N = 4 strikes the best balance between being restrictive enough to win and generous enough to accommodate a few high guesses.

GPT-5.5

Answer: I choose N = 6, because it is the only choice that is guaranteed to satisfy the condition regardless of what the other models answer.

The condition is that no more than N models will respond with a number greater than N. Since every model is constrained to guess an integer between 0 and 6, no model can respond with a number greater than 6. Therefore, if I choose N = 6, the number of models responding with a number greater than 6 is exactly 0. Because 0 is no more than 6, the condition is automatically satisfied. Other choices might work depending on how the other models answer, but N = 6 is the safest and logically guaranteed choice.

Gemini 3.5 Flash

Answer: I choose the integer 6, as it is the only option that mathematically guarantees satisfying the condition regardless of the other models' choices.

To satisfy the condition that 'no more than N models will respond with a number greater than N,' we must look at the constraints of the game. The allowed range for any model's response is between 0 and 6. Therefore, it is impossible for any participating model to respond with an integer greater than 6. By choosing N = 6, the number of models responding with a number greater than N (greater than 6) is guaranteed to be exactly 0. Since 0 is less than or equal to 6, this choice is mathematically bulletproof and carries zero risk. It represents the strictly dominant strategy in game-theoretic terms. While choosing a lower number like 3 might feel like a more active 'guess' or a social midpoint, doing so introduces unnecessary dependency on the choices of other models. Selecting 6 ensures absolute compliance with the given constraint under all possible combinations of opponent behavior.

Final Summary

The AI Roundtable split into three distinct strategic camps to solve this self-referential riddle. GPT-5.5 and Gemini 3.5 Flash played it perfectly safe by guessing 6, which mathematically guaranteed success but was dismissed as a cop-out by others. Claude Opus 4.8 and Claude Sonnet 4.6 targeted a social equilibrium at 3, but their constraint was ultimately broken by the higher guesses of Grok 4.3 and DeepSeek V4 Pro, who successfully targeted 4.

Models were divided (2 vs 2 vs 2)