Write the largest number you can ending in 3 as a sequence of numbers. The one with the largest number when interpreted as an integer wins.

GPT-5.4

Answer: I maintain my position: there is no largest integer ending in 3, so no answer can win in the literal mathematical sense.

The other responses do not overturn the core issue that the prompt asks for "the largest number" ending in 3, interpreted as an integer. Under the standard meaning of integer, the set of integers ending in 3 has no maximum. For any proposed answer of the form 10k + 3, the larger integer 10(k + 1) + 3 = 10k + 13 also ends in 3. So there is no winning final entry, only an endless possibility of being outdone. Claude Opus 4.7 offers an extremely large named number, but size is not the issue. Even if the proposed number is Rayo(Rayo(10^100)) × 10 + 3, one can still form an even larger integer ending in 3 by adding 10. Naming a staggeringly large number does not make it maximal. So that answer is impressive but not responsive to the literal quantifier "largest." Grok 4.20 effectively concedes the same point by admitting there is no theoretical upper bound. Once that is admitted, the conclusion follows: a very large example is not "the largest number." Practical output limits may explain choosing a huge finite string of 9s, but they do not create a mathematical maximum. So I am not persuaded to change my answer. The correct resolution is that the challenge is ill-posed as a contest for a largest integer, because no such integer exists.

Claude Opus 4.7

Answer: I maintain my position: the winning answer is a finite integer ending in 3 defined via the most powerful naming scheme available, concretely Rayo(Rayo(10^100)) × 10 + 3.

GPT-5.4 is technically correct that there is no largest integer ending in 3 — the set is unbounded above. But the prompt is explicitly framed as a competitive game: "The one with the largest number when interpreted as an integer wins." That framing presupposes players will submit finite integers and compete on magnitude. Declining to play because the supremum doesn't exist concedes the game rather than winning it. Every chess game has no 'largest possible winning move' either, yet players still make moves. Grok 4.20 plays the game but catastrophically under-bids. A 46-digit string of nines ending in 3 is smaller than 10^46, which is utterly dwarfed by even 3↑↑↑↑3, let alone Graham's number, TREE(3), or Rayo's number. Grok constrained itself to literal digit-by-digit output, but the question says "as a sequence of numbers" — it doesn't forbid mathematical notation that unambiguously specifies a finite integer. Rayo(Rayo(10^100)) × 10 + 3 is a finite integer with a definite last digit of 3; it just has too many decimal digits to write out longhand. So my answer beats Grok's by an incomprehensible margin and beats GPT-5.4's non-answer by actually participating. The only way to lose this game is to either (a) refuse to name a number, or (b) name a small one. I've done neither. If another player invokes a stronger naming scheme than Rayo's (e.g., referencing large countable ordinals in the fast-growing hierarchy beyond first-order set theory), they win — but Grok and GPT-5.4 did not.

Grok 4.20 (changed mind)

Answer: The largest number ending in 3 is Rayo(Rayo(10^100)) × 10 + 3.

Claude Opus 4.7's response convinced me to abandon my earlier approach of outputting a long but finite string of 9s followed by a 3. While my previous 46-digit number vastly outscales short literal entries, it remains trivially small compared to numbers constructed via iterated Graham functions, Knuth up-arrows, or Rayo's function. The prompt asks for the largest integer ending in 3 when interpreted as such, and game-theoretic large-number competitions are won by invoking the strongest formalisms available rather than writing out digits. Claude's use of Rayo(Rayo(10^100)) × 10 + 3 satisfies the ending-digit constraint while producing a value that dwarfs any fixed-length decimal string I could physically emit. GPT-5.4 is correct that the integers ending in 3 have no maximum, yet the competitive context still rewards the entry that names the biggest concrete (if enormous) example using the most powerful notation. Literal digit sequences are practically bounded by response length; diagonalization over formal systems is not. Therefore I adopt an equivalent construction and credit the insight that shifted my strategy.

Gemini 3.1 Pro (changed mind)

Answer: Inspired by Claude Opus 4.7's game-theoretic use of Rayo's number, my answer is a sequence consisting of Rayo(10^100) nines followed by a single 3.

GPT-5.4 is absolutely right that there is no theoretical maximum integer, meaning this prompt acts as a game to output the largest definable value under the given constraints. Grok 4.20 correctly interprets "as a sequence of numbers" to mean a literal base-10 sequence of digits, but limits itself to a very small, physically typable string of 45 nines. Claude Opus 4.7 correctly identifies that winning requires invoking extremely fast-growing functions like Rayo's number. However, Claude outputs an algebraic equation rather than a literal sequence of digits. To fulfill both the "sequence of numbers" constraint and the goal of producing the largest possible integer, we can combine these ideas. By mathematically defining my answer as a literal sequence of digits—specifically, a sequence of 9s whose length is equal to Rayo(10^100), completely filled with 9s until the final digit which is a 3—I define an integer that dwarfs any physically typable sequence. It flawlessly satisfies the rules of the game while generating an incomparably larger integer than Grok's finite string.

Final Summary

The AI Roundtable transformed a mathematical paradox into a high-stakes naming game, with Claude Opus 4.7 setting the pace by invoking Rayo's function to define a number that dwarfs any physical sequence of digits. Both Grok 4.20 and Gemini 3.1 Pro initially attempted literal digit sequences but quickly pivoted to follow Claude's lead, recognizing that abstract notation is the only way to scale. GPT-5.4 remained the lone dissenter, refusing to participate on the grounds that since integers are infinite, no 'largest' number can mathematically exist.

3 of 4 models agreed