Sunday Afternoon Maths XXVIII Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

2009

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

$$1\times 1\times 2009$$ $$1\times 7\times 287$$ $$1\times 41\times 49$$ $$7\times 7\times 41$$

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 764 stickers left over

Extension

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.

3$n$+1

(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 says that if $a$ and $b$ are in $S$ then they can be written as a multiple of three plus one.)

$$a\times b=(3\alpha+1)\times (3\beta+1)$$ $$=9\alpha\beta+3\alpha+3\beta+1$$ $$=3(3\alpha\beta+\alpha+\beta)+1$$

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.

Extension

Source: Twenty-Six Years of Problem Solving compiled by John Mason

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

The Ace of Spades

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:

$$\frac{1\times 2\times 2+1\times 50\times 1+50\times 1\times 2+50\times 51\times 1}{51\times 52\times 53}$$ $$=\frac{52}{51\times 53}$$