Ferrero Rocher Quiz Puzzle Answer

We have all seen the Ferrero Rocher advert with the pile of chocolates at the Ambassador's party, and the sort of people we are here in Quiz Master Shop Towers, we started thinking.

If we assume that the pile is a tetrahedron (that is with one chocolate at the top resting on three chocolates in the layer below, and so on) how many chocolates are in a pile ten layers tall?

And following on from this, is there a formula to calculate the number of chocolates in a pile of any height?

Each of the layers in the pile (or tetrahedron) is a triangle, and so the numbers of chocolates in each of the layers are successive triangular numbers. That is, 1, 3, 6, 10 and so on.

So taking the two parts of the puzzle in reverse order, to calculate the number of chocolates in a tetrahedron of n layers, we need to sum the first n triangular numbers. And there is a formula for that:

$$ \frac{n(n + 1)(n + 2)}{6} $$

We can see it is correct when n = 1

$$ \frac{1 x 2 x 3}{6} = 1 $$

and when n = 2

$$ \frac{2 x 3 x 4}{6} = 4 $$

So to answer the first part last, when n = 10

$$ \frac{10 x 11 x 12}{6} = 220 $$


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