Ones and Twos - Futility Closet

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From the Kanja Otogi Zoshi (“Collection of Interesting Results”) of Nakane Genjun, 1743:

Your friend has 30 go stones. He lines them up out of your sight, placing down either one or two stones with each deposit and calling “here” so you’ll know this has been done. When all 30 stones have been placed, you have heard him say “here” 18 times. How many deposits contained one stone, and how many two?

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If the number of deposits of one stone is x, then the number of deposits of two stones is 18 – x. The total number of stones placed is x × 1 + (18 – x) × 2 = 30. Solving this gives x = 6, so in this case 6 deposits contained one stone and 12 contained two. More generally, if a is the total number of stones and n is the number of cries of “here,” then x × 1 + (n – x) × 2 = a x + 2n – 2x = a x = 2n – a Hence the number of times that two stones are placed is n – x = n – (2n – a) = a – n The number of deposits of one stone: 2n – a The number of deposits of two stones: a – n (From Sakurai Susumu, Wasan: The Fascination of Traditional Japanese Mathematics, 2009.)

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