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#218 Playground parity

Set by Katie Steckles

I was eating a sandwich on a park bench when a horde of children from the local school descended on the playground in front of me and divided into two groups.

I couldn’t help overhearing the kids by the swings shout across to the kids near the climbing frame: “If two of you run over here, our group will be double yours!”

The kids by the climbing frame replied: “But if two of you come over here, the groups will be the same size!”

The mathematician in me couldn’t resist figuring out how many children there must be in each group for this to be true. But I also discovered something else. If a number of children going one way makes one group double the other, and that same number going the other way makes the two groups the same, the total number of children must always be a multiple of… what?

Solution next week

#217 Vicious circle

Solution

Neville can potentially escape. If he starts from the centre of the circular island and tries to run directly away from the monster to the other side of the moat, it will be able to catch him because it can swim half the circumference (πr) faster than he can run the radius (r). But if he is careful, Neville can keep the centre directly between himself and the monster as long as he is within a quarter of the circle’s radius. This is because he can run around this smaller circle (½πr) in the same amount of time that the monster can swim the outer circle (2πr).

If Neville does this right, he will still be on the opposite side to the monster after traversing a quarter of the circle’s radius. Once he has run from there to the moat (¾r), the monster will still have about 0.14r left to swim, leaving a sliver of time to jump across and escape.