Sunday Afternoon Maths XXVIII

The problem can be expressed using the following probability tree:

The probability that the card turned over from C is an Ace of Spades is:

(i) Let \(a,b\in S\). Then \(\exists \alpha,\beta\in \mathbb{N}\) such that \(a=3\alpha+1\) and \(b=3\beta+1\).

This is a multiple of three plus one, so \(a\times b\in S\).

(ii) No, as \(36\times 22=4\times 253\) and 36,22,4 and 253 are all irreducible.

Try the task again with \(S=\{a+b\sqrt{2}:a,b\in\mathbb{Z}\}\).

\(2009=7\times 7\times 41\), so there are four possible sets of dimensions of the cuboid:

In the first three cuboids, there is a face with an area of 2009 units (\(2009\times 1\), \(7\times 287\) and \(41\times 49\) respectively) and so 2009 stickers will not be enough. Therefore the cuboid has dimensions \(7\times 7\times 41\) and a surface area of 1246, leaving 763 stickers left over

For which numbers \(n\) can a cuboid be made with \(n\) unit cube such that \(n\) unit square stickers can cover the faces of the cuboid?