Sunday Afternoon Maths XXIV Answers & Extensions

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

Square Cross

Label and orient the shape as follows:

The smaller diagonal square is made up of a 1×1 square and four 1×2 right-angled triangles. Therefore its area is 5.

The line FM, and therefore the line DN, has gradient 2. The line JM, and therefore the line ON, has gradient -½. ON passes through (1,2) and DN passes through (2,0). Therefore, DN has equation $y=2x-4$ and ON has equation $y=\frac{5}{2}-\frac{1}{2}x$. These lines intersect at $(\frac{13}{5},\frac{6}{5})$, these are the co-ordinates of N. By the same method, the co-ordinates of P are $(-\frac{8}{5},-\frac{1}{5})$.

By Pythagoras' Theorem, The diagonal of the larger square is $\frac{7\sqrt{10}}{5}$ and so the area of the larger square is 9.8.

Exact Change

In the UK, eight coins are needed: 1p, 1p, 2p, 5p, 10p, 20p, 20p, 50p.

In the US, ten coins are needed: 1¢, 1¢, 1¢, 1¢, 5¢, 10¢, 10¢, 25¢, 25¢, 25¢.

Extension

Source: @AlexDBolton on Twitter

In a far away country, the unit of currency is the #, which is split into 100@ (# is like £ or $; @ is like p or ¢).

Let C be the number of coins less than #1. Let P be the number of coins needed to make any value between 1@ and 99@. Which coins should be the country mint to minimise the value of P+C?