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#29 Four by four

Set by Howard Williams

By only cutting along the lines of a 4-centimetre-by-4-centimetre square grid, it can be divided into two halves of equal area and of identical shape. The simplest method for achieving this is a 4-centimetre cut, straight across the middle.

Ignoring rotational and reflectional symmetries, there are two ways (shown above) of using a 6-centimetre cut to divide the grid into equal areas of identical shape.

By making an 8-centimetre cut, can you divide the grid into two identical shapes?

With a 10-centimetre cut, it’s possible to do this in two ways. Can you find both?

Solution next week

#28 Dice differences

Solution

The initial game isn’t fair. There are 36 ways to roll two dice. Of these, 24 result in a difference of two or less; 12 have a larger difference. Under the adjusted rules, returning to the start requires at least three rolls – two moving one place left, and one two places right. As the counter is half as likely to move right but goes twice as far in that direction, movement is balanced between the players. But the player moving left still has an advantage, as they always get to use their full move. The player moving right may only need to go one space, but to do so they would have to move two spaces. In a three-space grid, the player going left will win 2/3 of the time; in a five-space grid they will still win 12/23 of the time, or around 52 per cent.