Puzzles

The smallest number you can make with the digits in the red boxes is 123.

Because 13 and 11 are prime and larger than 7, no-one can have taken them, as their is no way the other person's product could then be the same. Once 13 and 11 are discarded, the prime factorisation of the product of all the other cards is \(2^{11}\times3^5\times5^2\times7^2\). In order to be shared to give the same product, the powers must be even, and so the other card not taken must be 6 (as 2 and 3 are the numbers with odd powers).

Therefore the product of the three cards is 858.

The diagonal of the blue square is \(\sqrt{28}\). The radius \(R\) of the large circle satisfies \(R^2+R^2=(R+\sqrt{28})^2\). Solving this, we find that \(R=\frac{\sqrt{28}}{\sqrt2-1}=\sqrt{28}(\sqrt2+1)\).

The radius \(r\) of the small circles satisfies \(r+r\sqrt2=R\), and so \(r=\frac{\sqrt{28}(\sqrt2+1)}{\sqrt2+1}=\sqrt{28}\).

The area of the square is \(4r^2=4\left(\sqrt{28}\right)^2=4\times28\). This is 112.

To win this game, you need to always end your turn on a multiple of 901, as whatever you opponent does, you can always get back to a multiple of 901.

Therefore, you should take £791 on your first turn to get to a multiple of 901.

For the first and last digits of the result to be different, there must be a 1 carried from the middle column. Therefore the middle digit of Eve's number is 9. Eve's number could be 292 or 193, but it cannot be the first as reversing this is not larger than her original number, so here number is 193.

30030 is 2×3×5×7×11×13, so the answer is the number of ways of sharing six elements into unlabelled boxes (A000110).

The answer is 203.

There are 1 time starting 00; 2 times starting 01; 3 times starting 02; 4 times starting 03; 5 times starting 04; 6 times starting 05; 6 times starting 06; 6 times starting 07; 6 times starting 08; 6 times starting 09; 2 times starting 10; 3 times starting 11; 4 times starting 12; 5 times starting 13; 6 times starting 14; 6 times starting 15; 6 times starting 16; 6 times starting 17; 6 times starting 18; 5 times starting 19; 3 times starting 20; 4 times starting 21; 5 times starting 22; and 6 times starting 23.

Therefore there are 112 ways in total.