South African Mathematics Olympiad

Question

Pyramids with square bases were built by stacking magnetic balls. A pyramid with a 1 1 base contains one ball, a pyramid with a 2 2 base contains five balls, a pyramid with a 3 3 base contains 14 balls, a pyramid with a 4 4 base contains 30 balls. How many balls are there in a pyramid with a 10 10 base?

Accepted Answer

The n-th pyramid is made by putting the (n - 1)-st pyramid on an n n square of balls. Thus the number of balls in the n-th pyramid is 1^2 + 2^2 + + n^2. The number of balls in the 10th pyramid is therefore 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385.