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Largest Number Game

A draft about short descriptions, enormous numbers, and Ackermann-style recursion.

A typical children's game is competing over who can say the largest number; here we propose this game again.

The goal of the game is to describe the largest possible number. Rules:

  • No meta-references to the game, no “the other person's number + 1”.
  • No mathematics beyond high-school level, no “busy beaver” unless it is defined.
  • No badly or vaguely defined numbers: it must be possible to understand whether they are larger or smaller than the other numbers.

So far, there are not enough restrictions to prevent arbitrarily large numbers. Many such restrictions can be imposed, and in practice different choices create different subgames.

The main category we propose is: describe the number in 50 ASCII characters.

Current king of the hill

Among all the numbers that have come up, with various mathematical and formatting ideas used to optimize as much as possible, the largest one I have encountered comes from a friend's idea of using Ackermann recursion to define the hyperoperators. Indeed, it is possible to obtain:

f(x,0)=1
f(0,y)=9^y
f(x,y)=f(x-1,f(x,y-1))
f(9!,9)

Counting a line break as one character, one could also use a semicolon. This is precisely equal to 9↑³⁶²⁸⁸¹9, using Knuth notation.

Allowing a more fanciful notation for binary operators, the same idea can be used to obtain

n~0=1
0~x=9^x
n~x=(n-1)~(n~(x-1))
(((9!~9)~9)~9)~9

which is precisely

9↑^(9↑^(9↑^(9↑³⁶²⁸⁸¹9+1)9+1)9+1)9.

The boundary conditions can be modified as

n~0=9
0~x=9x!
n~x=(n-1)~(n~(x-1))
(((9!~9)~9)~9)~9

which probably, to be verified, slightly increases the size of the number.