Problem 855

Delphi Paper

Given two positive integers $a, b$ , Alex and Bianca play a game in $a b$ rounds. They begin with a square piece of paper of side length $1$ .

In each round Alex divides the current rectangular piece of paper into $a \times b$ pieces using $a - 1$ horizontal cuts and $b - 1$ vertical ones. The cuts do not need to be evenly spaced. Moreover, a piece can have zero width/height when a cut coincides with another cut or the edge of the paper. The pieces are then numbered $1, 2, \dots, a b$ starting from the left top corner, moving from left to right and starting from the left of the next row when a row is finished.

Then Bianca chooses one of the pieces for the game to continue on. However, Bianca must not choose a piece with a number she has already chosen during the game.

Bianca wants to minimize the area of the final piece of paper while Alex wants to maximize it. Let $S (a, b)$ be the area of the final piece assuming optimal play.

For example, $S (2, 2) = 1 / 36$ and $S (2, 3) = 1 / 1800 \approx 5.5555555556 e - 4$ .

Find $S (5, 8)$ . Give your answer in scientific notation rounded to ten significant digits after the decimal point. Use a lowercase e to separate the mantissa and the exponent.

德尔菲分纸法

亚历克斯和比安卡在玩游戏，这个游戏需要进行 $a b$ 轮，其中 $a$ 和 $b$ 均为正整数。游戏开始时，需要一张边长为 $1$ 的正方形纸。

在每一轮中，亚历克斯在纸上横切 $a - 1$ 刀，竖切 $b - 1$ 刀，将其分成 $a \times b$ 块长方形纸片。刀与刀间的距离无需相同，而且可以相互重合，此时对应的长方形长度或宽度为零。随后，从纸的左上角开始，从上至下、从左至右依次将长方形纸片编号为 $1, 2, \dots, a b$ 。

随后，比安卡选择其中一张长方形纸片并下一轮游戏，但比安卡所选纸片的编号不能与之前轮数中的选择相同。

比安卡的目标是使游戏结束时纸片的面积尽可能小，而亚历克斯的目标则是使之尽可能大。记 $S (a, b)$ 为双方都采取最优策略时最终的纸片面积。

例如， $S (2, 2) = 1 / 36$ ， $S (2, 3) = 1 / 1800 \approx 5.5555555556 e - 4$ 。

求 $S (5, 8)$ ，并将你的答案用科学计数法表示（用小写字母e分隔尾数和指数），保留十位小数。