### Month Mania

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

### Doodles

A 'doodle' is a closed intersecting curve drawn without taking pencil from paper. Only two lines cross at each intersection or vertex (never 3), that is the vertex points must be 'double points' not 'triple points'. Number the vertex points in any order. Starting at any point on the doodle, trace it until you get back to where you started. Write down the numbers of the vertices as you pass through them. So you have a [not necessarily unique] list of numbers for each doodle. Prove that 1)each vertex number in a list occurs twice. [easy!] 2)between each pair of vertex numbers in a list there are an even number of other numbers [hard!]

### Euler's Officers

How many different solutions can you find to this problem? Arrange 25 officers, each having one of five different ranks a, b, c, d and e, and belonging to one of five different regiments p, q, r, s and t, in a square formation 5 by 5, so that each row and each file contains just one officer of each rank and just one from each regiment. There is an interactive version of this problem.

# Diagonal Sums Sudoku

### Rules of Diagonal Sums Sudoku

Like the standard Sudoku, this Sudoku has two basic rules:
1. Each column, each row and each box (3$\times$3 subgrid) must have the numbers 1 to 9.
2. No column, row or box can have two squares with the same number.
The puzzle can be solved with the help of clue-numbers which are written after slash marks on the intersections of border lines. Each clue-number is the sum of two digits in the two squares that are diagonally adjacent to each other. The position of each pair of diagonally adjacent squares is indicated by either two forward slash marks // or two backward slash marks \\.

For example, the //12 on the border of the top right hand box means that possible pairs of numbers in the cells above-right and below left are:
3 and 9, 9 and 3; 4 and 8, 8 and 4; 5 and 7, or 7 and 5 respectively.

Similarly, the \\6 in the bottom left box means that possible pairs of numbers in the cells above-left and below-right are:
1 and 5, 5 and 1; 2 and 4, or 4 and 2 respectively.