快猫短视频

#48 Rock, paper, scissors

Set by Christopher Dearlove

In the game of rock, paper, scissors, two players simultaneously pick and display one of these options to see who wins. Paper wraps rock, scissors cut paper and rock blunts scissors 鈥� each scoring 1 point to the winner and -1 to the loser. If both pick the same option, both score 0.

If both players pick rock, paper or scissors at random, what is the largest the total of their two scores could be after 10 games?

After paper wraps rock, we still have a rock; and although scissors cut paper, paper is cheap. But the rock blunting the scissors is more serious. So, in a variant, we decide to award 2 points in this case and -2 to the loser. Instead of choosing at random, what is the best strategy for each player in this case?

How does this change if we assign 3 and -3 points to the 鈥渞ock blunts scissors鈥� outcome? Or, in general, N points?

Solution next week

#47 Squares and circles

Solution

Rotating the inner square by 45 degrees reveals that it touches the larger square at the midpoints of each edge; it can be seen that the area is half the area of the larger square. A copy of the circle centred at the midpoint of each edge of the square will cover it totally 鈥� four circles are sufficient.

The circle inside the square takes up 蟺/4 of the area, while the square inside the circle takes up 2/蟺 of the area, which is smaller, and so the circle in a square is a better fit. (Mathematician David Singmaster studied this problem and discovered that this continues to hold for spheres in cubes and for higher dimensions up to 8, beyond which it swaps around.)