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鸡兔同笼

September 12, 2010

鸡兔同笼,这个问题,是我国古代著名趣题之一。大约在1500年前,《孙子算经》中就记载了这个有趣的问题。书中是这样叙述的:“今有雉兔同笼,上有三十五头,下有九十四足,问雉兔各几何?”

《孙子算经》的解法没有用到二元一次方程组。解法如下:如砍去每只鸡、每只兔一半的脚,则每只鸡就变成了“独角鸡”,每只兔就变成了“双脚兔”。这样,(1)鸡和兔的脚的总数就由94只变成了47只;(2)如果笼子里有一只兔子,则脚的总数就比头的总数多1。因此,脚的总只数47与总头数35的差,就是兔子的只数,即47-35=12(只)。显然,鸡的只数就是35-12=23(只)了。

In the ancient Chinese book Chiu-chang Suan-shu (Nine Chapters on Arithmetic, 九章算术), estimated to have been written some time around 200 B.C., there appears a problem:

A bunch of roosters and rabbits are prisoned in the same cage. One counts from the top, and sees thirty-five heads. One counts from the bottom, and sees ninety-four feet. How many roosters and rabbits are there?

A solution by Sun Zi is as follows:

Suppose one chops off half of the feet for each rooster and rabbits, and is left with forty-seven feet, which accounts for one foot each rooster, and two feet each rabbit. If all the thirty-five heads belong to roosters, then there are supposed to be thirty-five feet. However, for each rabbit, the count for feet increase by one. Since there are twelve more feet in the actual count than thirty-five, the number of rabbits is twelve. There are twenty-three roosters.

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