快猫短视频

#49 Squares and cubes

Set by Howard Williams

Take two positive whole numbers that are both less than 20. The difference in the squares of these numbers is a perfect cube and the difference in their cubes is a perfect square. (Perfect means that the root of the square or cube is a whole number.)

What are the two numbers?

There is a number between 50 and 100 that you can multiply both these numbers by and get another pair with the same property. What is it?

What property does this number have that makes this work? How can you use this idea to find more of these pairs of numbers?

Solution next week

#48 Rock, paper, scissors

Solution

Since each play results in one player鈥檚 score increasing and the other decreasing (or staying the same), the scores will always be N and -N, for some value of N, so the total after any number of plays will be 0. For the modified game, a player should pick paper twice as often as scissors (since it is twice as good against rock) and scissors equally as often as rock. All probabilities must add to one, so each time we should choose rock with probability 1/4, paper with probability 1/2 and scissors with probability 1/4. If we award 3 and -3 points when rock blunts scissors, the probabilities for paper, rock and scissors become 3/5, 1/5, 1/5. If we award N and -N points, the ratio must be N : 1 : 1.