Recreational Mathematics

A few especially cute problems which can be solved in a few seconds if you think about them the right way:

  1. Give three integers in arithmetic progression, whose product is prime.
  2. I give you a pile of n objects. You return it as 1 or more piles whose sizes add up to n. Your 'score' is the product of the sizes of the piles. How do you maximise your score?
  3. I have laid out a circular track, with circumference 10 miles, and put petrol cans around it at intervals of 1 mile. The cans contains various non-negative amounts of petrol, totalling 10 gallons. I give you a car which does 1 mile per gallon, and show you a map saying how much petrol is in each can. Can you guarantee to be able to choose a starting point and starting direction which allow you to go right round the track? If so, how? If not, how must I distribute the petrol amongst the various cans.
Some "I wonder what happens if" problems:

  1. If you stop people at random and ask them their birthdays, how many people, on average, will you need to stop before you've collected all 365 days in the year? (Assume leap years don't exist.)
  2. Same as above, but allow for the existence of leap years.
  3. Suppose that you have two armies of size a and b, where a is greater. Suppose that each "soldier" inflicts casualties at a constant rate. How many soldiers are left in the larger army when the smaller army has been wiped out?
See The Archimedeans for more details. I picked up most of the above problems from other people attending the Puzzles and Games Ring.