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Published 2001 Revised 2010
Freddie Manners, of Packwood Haugh School in Shropshire, has e-mailed us about the problem "Cayley" from the February 2001 magazine. He has noticed that we had given more information than was needed to solve the problem. He is absolutely right: in fact when the problem was originally invented, it was:
C | A | U | C | H | Y |
C | A | U | C | H | Y |
- | - | - | - | - | - |
E | U | C | L | I | D |
Given that the letters in this sum represent the digits 1 ... 9, what number is represented by "CAYLEY"?
The people trying to solve it then were members of the Cambridge, Oxford and Warwick University maths societies, but they only had a limited amount of time in which to do it. We thought that maybe you would need some help, which is why we told you the numbers represented by A and D, but Freddie has proved us wrong!
Here is his reasoning:
4 | 1 | 2 | 4 | H | 3 |
4 | 1 | 2 | 4 | H | 3 |
- | - | - | - | - | - |
8 | 2 | 4 | 9 | I | 6 |
4 | 3 | 7 | 4 | H | Y |
4 | 3 | 7 | 4 | H | Y |
- | - | - | - | - | - |
8 | 7 | 4 | 9 | I | 2 |
4 | 3 | 7 | 4 | 5 | 6 |
4 | 3 | 7 | 4 | 5 | 6 |
- | - | - | - | - | - |
8 | 7 | 4 | 9 | 1 | 2 |
Well done to Freddie, both for spotting that the hints were unnecessary, and for explaining so clearly why.
A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?