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Monday, 15 September 2014

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

2 comments:

1. Surely $2009 \neq 7^4$...

1. Yes, I mixed up 49 and 41. I've corrected it