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#73 Square dance

Set by Sam Hartburn

If you arrange five square tiles, each with side length 1, in a bigger square as shown, what proportion of the area of the square do the tiles use (to three decimal places)?

 

One of the tiles is rotated by 45 degrees and all five are arranged inside a smaller square as shown. What is this square’s area? What proportion of that area do the tiles use?

 

Ten tiles in a 4-by-4 square use up 10/16 = 0.625 of the area. But with four of them rotated by 45 degrees, they fit in a smaller square. What is this square’s area? What proportion of that area do the tiles use?

 

Solution next week

#72 Match up

Solution

After two rounds, player A has a (2/3) × (2/3) = 4/9 probability of winning the game.

Player B has a (1/3) × (1/3) = 1/9 probability of winning after two rounds. In other words, player A wins 4 times as often as player B, or 4/5 of the time. This applies regardless of the number of rounds, as if neither wins after two rounds, they return to level.

If player B is one point up, the next point will either be won by player A (2/3 of the time) – taking them back to level, where player B wins 1/5 of the time – or it will be won by player B (1/3 of the time), who then wins the game. So the overall chance of player B going on to win is (2/3) × (1/5) + (1/3) = 7/15.