Problem 986
Another Infinite Game
Peter is playing another game on an infinite row of squares, each square of which can hold an unlimited number of tokens.
Initially, every square contains a token.
Given positive integers $c$ and $d$, each move of the game consists of the following steps:
- Choose two tokens $X$ and $Y$ such that $Y$ is $c$ squares to the right of $X$.
- Move both $X$ and $Y$ to the square that is $d$ squares to the right of $Y$.
Peter’s goal is to move as many tokens as possible into one square. For example, with $c = 2$ and $d = 1$, it is possible to move $7$ tokens into one square, following these steps (where red color marks the chosen tokens):
... 1 1 1 1 0 1 0 3 ...
... 1 1 1 0 0 0 2 3 ...
... 0 1 0 2 0 0 2 3 ...
... 0 0 0 1 2 0 2 3 ...
... 0 0 0 1 1 0 1 5 ...
... 0 0 0 1 0 0 0 7 ...
However, it is not possible to move $8$ tokens into one square.
Let $G(c, d)$ be the maximum number of tokens Peter can move into one square. For example, $G(2, 1) = 7$. You are also given that $G(1, 2) = 7$, $G(3, 1) = 11$, $G(2, 2) = 3$ and $G(1, 3) = 15$.
Find the sum of $G(c, d)$ for all pairs of $c, d$ with $1 \leq c, d \leq 160$.
又一个无限游戏
彼得又在玩一个游戏，游戏需要用到一行无限延伸的方格，每个方格上可以放置无限数量的代币。
游戏开始时，每个方格上都有一枚代币。
给定正整数$c$和$d$，游戏的每一步操作如下：
- 选择两枚代币$X$和$Y$，要求$Y$在$X$右侧第$c$格。
- 将$X$和$Y$都移动到$Y$右侧第$d$格。
彼得的目标是将尽可能多的代币移动到同一个方格中。例如，当$c = 2$、$d = 1$时，他可以将$7$枚代币移动到同一个方格中，步骤如下（红色标记所选的代币）：
... 1 1 1 1 0 1 0 3 ...
... 1 1 1 0 0 0 2 3 ...
... 0 1 0 2 0 0 2 3 ...
... 0 0 0 1 2 0 2 3 ...
... 0 0 0 1 1 0 1 5 ...
... 0 0 0 1 0 0 0 7 ...
然而，彼得永远无法将$8$枚代币移动到同一个方格中。
记$G(c, d)$为彼得能移动到同一个方格中的最大代币数。例如，$G(2, 1) = 7$；还已知$G(1, 2) = 7$，$G(3, 1) = 11$，$G(2, 2) = 3$，$G(1, 3) = 15$。
求所有满足$1 \leq c, d \leq 160$的数对$c, d$所对应的$G(c, d)$之和。
译注：彼得玩的上一个无限游戏在第664题。