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#211 Cross purposes

Set by Peter Rowlett

Debbie and Hoi are playing a game where they take turns to cross out numbers written on a piece of paper.

Each player must cross out a divisor or a multiple of the number most recently crossed out. The first player who is unable to cross out a number loses.

Hoi goes first and crosses out 11. Debbie smiles, knowing she can now win in three moves. What number does she cross out?

2 3 4 5

7 11 15 19

33 35 38 51

55 76 77 95

Solution next week

#210 Action station

Solution

I will catch the train… just!

If I am walking or running at a constant speed, I lose or gain time at a constant rate per unit distance. If I were to walk the whole way, I would get to the end point 10 minutes later than I wanted to, and I would get to the midway point 5 minutes behind. Similarly, if I ran the whole way, I would reach the end 15 minutes ahead and the midway point 7.5 minutes ahead. That means I am a net 2.5 minutes ahead by running half the distance and walking the rest of the way. I lose 2 minutes when hunting for my running shoes, which means I arrive half a minute before the train leaves.