By: Tao Steven Zheng (鄭濤)

【Problem】

In the "Mengxi Bitan", the Chinese scientist and statesman Shen Kuo (1031–1095 AD) calculated the number of different configurations on a weiqi board[1]. The weiqi board is a 19×19 grid in which black and white stones are placed on the intersecting points.

If there are three possible configurations per point (black, white, empty), how many different configurations are possible? Express the answer in scientific notation.

(Note: This problem does not take into account the game rules.)

[1] According to the Mengxi Bitan, this problem was first studied by the Tang dynasty mathematician, astronomer, and Buddhist monk Yi Xing (683 – 727 AD). Yi Xing’s original name was Zhang Sui. Weiqi (more commonly called Go in the West) is a two-player board game in which the aim is to capture more territory than the opponent. The game was invented in China more than 2500 years ago and is believed to be the oldest board game continuously played to the present day.

【Solution】

There are 361 points on the weiqi board.

According to Shen Kuo’s calculations, the number of different configurations on a 361-point weiqi board is approximately the number wan (10000) written consecutively forty-three times

This answer is in the correct order of magnitude, but it is imprecise.

Every intersection point can be either black, white, or empty, so the number of possible positions is exactly . The value expressed as a power of ten is easy to determine using logarithms[2].

[2] The theory of logarithms was invented by the Scottish mathematician and theologian John Napier (1550 – 1617) in 1614.

Thus,