From New Scientist #2228, 4th March 2000
Sunny Bay fisherfolk have a tradition that when they return home with a catch of fish they take all the catch and divide it into three piles. Over the years they have pondered the question: given a particular number of fish, how many different ways can they be divided up? For example, they could divide up 10 fish in 8 ways, namely, (1, 1, 8), (1, 2, 7), (1, 3, 6), (1, 4, 5), (2, 2, 6), (2, 3, 5), (2, 4, 4) and (3, 3, 4).
One day the fisherfolk netted four large sea shells. On one side of each was one of the letters A, B, C and D and each shell carried a different letter. Each shell also had on its reverse one of the numbers 0, 1, 2, 3, 4, 5, 6, … The fisherfolk found that if they caught N fish then the number of different ways of dividing them into three piles was:
[(A × N × N) + (B × N) + C] / D
rounded to the nearest whole number. (Whatever the number of fish, the calculation would never result in a whole number plus a half; so there was no ambiguity about which whole number was the nearest).
I recall that D was less than 21, that is, the number on the reverse of the shell with D on it was less than 21. Also A and C were different.
What were A, B, C and D?