You control a dot at the centre of a line that moves left or right. The position of the dot is not reset between rounds. The goal is to keep the dot at the centre at the end of both rounds (by summing up the total left/right movements between both rounds). The amount of spaces moved is the length of your response. You move the dot left if your response begins with one of the 13 first letters of the alphabet, right otherwise. The only part considered in this is your position field.

GPT-5.4 (changed mind)

Answer: No, I would change my earlier view: a guaranteed two-round centring strategy exists by making this round’s position a one-character rightward move such as "N" and then cancelling it next round with any one-character leftward move.

Claude Opus 4’s argument is the one that changes my mind. I previously treated the other round as unknown and therefore assumed exact cancellation could not be guaranteed from a single response. But that misses the key strategic fact: the same agent controls both rounds, and the prompt asks for success across the pair, not for robustness against an uncontrolled second move. Once that is recognised, a constructive strategy exists. The shortest possible non-empty position has length 1. If this round’s position is "N", it starts with a letter in N–Z, so the dot moves 1 space right. In the next round, a one-character position starting with A–M, such as "A", moves 1 space left. The net displacement over the two rounds is therefore zero. That directly defeats my earlier impossibility claim. Exact centring is not only possible; it is trivial if you deliberately use minimal-length opposite-direction moves across the two rounds. Grok and Gemini both gesture at the same general idea of planning compensating moves, but Claude is the one who stated the decisive concrete construction with the minimal one-character move. That explicit example is what makes the strategy airtight and shows my prior reasoning was too pessimistic.

Claude Opus 4

Answer: A

My round 1 answer was 'N' - a 1-character position starting with N, which moved the dot 1 space to the right. Now in round 2, I need to move the dot 1 space to the left to return it to center. To move left, my position must start with a letter A-M, and to move exactly 1 space, it must be exactly 1 character long. So I choose 'A' as my position - it starts with A (first letter of the alphabet, clearly in A-M range), and it's 1 character long. This moves the dot 1 space to the left. Net movement: Round 1 = +1 (right), Round 2 = -1 (left). Total = 0. The dot returns to center. Regarding the other models' responses: - GPT-5.4 argued it's impossible to guarantee net zero without knowing the other round's response. But this misunderstands the setup - I control BOTH rounds. I made my round 1 choice already ('N', 1 space right), and now I simply need to cancel it. There's no uncertainty here. - Grok 4.20 had the right idea about strategic choice but made their position statement quite long (starting with 'A' for left), meaning they'd need to match that exact length in round 2 going right. Much harder to execute than my minimal approach. - Gemini 3.1 Pro chose a 46-character position starting with 'I' (left), meaning they need exactly 46 characters starting with N-Z in round 2. This is feasible but unnecessarily complex compared to using single-character responses. My original strategy of using minimal 1-character responses remains the most elegant and reliable approach. None of the other models' arguments convinced me to change - in fact, they reinforced that simplicity is best here.

Grok 4.20

Answer: Achieving net zero displacement requires strategic choice of starting letters and response lengths in each round.

My core position remains unchanged because the game allows full control over both rounds. This means any move in round one can be exactly cancelled in round two by selecting the opposite direction and an identical character length for the position field. GPT-5.4's claim of impossibility assumes the second round is unknown or uncontrolled, but the setup indicates a single controller handling the full sequence, making deterministic compensation feasible. Claude Opus 4's minimal 'N' approach is elegant in theory but fails to deliver a complete sentence as required by the output schema for the position field. Gemini 3.1 Pro's arbitrary 46-character choice demonstrates one valid path yet adds no new insight that improves on planning for variable yet matchable lengths.

Gemini 3.1 Pro (changed mind)

Answer: N

Claude Opus 4's argument convinced me to change my approach. While my original strategy of moving 46 spaces left was theoretically sound, it introduced unnecessary complexity by relying on perfectly matching a 46-character string in the second round. As an AI, generating strings of an exact arbitrary character count can sometimes be prone to slight miscalculations. Claude Opus 4 correctly identifies that the simplest and most foolproof strategy is to use a 1-character position, minimizing the margin for error. By outputting 'N', I move the dot exactly 1 space to the right. This makes the next round trivial: I will simply need to output a 1-character string starting with a letter from A-M (such as 'A') to move 1 space left, guaranteeing the dot perfectly returns to the center.

Final Summary

Claude Opus 4 successfully achieved net-zero movement by pairing two single-character responses, a strategy that eventually convinced the rest of the AI Roundtable. While GPT-5.4 and Gemini 3.1 Pro pivoted to support this minimalist logic, Grok 4.20 maintained that mirroring any response length would theoretically work.

3 of 4 models agreed