Sometimes I start newcomers to MathsJam on timing puzzles. Here are five classics in increasing order of difficulty and implausibility, each with a distinct flavour. The first is distinctly steaky…
1. Some friends are coming over for a steak dinner. You want to cook three steaks as quickly as possible, but your grill pan only holds two at a time. Each steak must be cooked for five minutes on each side. What is the fastest you can have all three ready? Show that you can’t do any better.
Alas, when this precise situation once arose with my family at home (who says puzzles are never useful?), I was unable to convince my father not to cook the steaks in the obvious order. But at least that meant I got my steak sooner.
2. I give you two sand timers, one that measures four minutes, the other seven. Time nine minutes when I say “now”… Now!
Next, your mission, should you choose to accept it…
3. I give you two fuses that each burn take one hour to burn through from one end to the other, though not at a uniform rate. Given some matches, can you time fifteen minutes, starting whenever you want?
This problem tortured our tutor throughout a university maths summer garden party:
4. Four explorers come to a narrow rope bridge which they judge can only hold two of them at once. It’s night and they only have one torch between them, which must always be used when crossing to avoid certain death. The most foolhardy explorer can cross in one minute, the next in two minutes, the cautious one in five, and finally the limping explorer who was injured by a diabolical trap takes eight minutes. When any pair traverse together the bridge together, they must move at the slower’s pace. What is the soonest they can all safely get across to the other side?
5. A boy, a girl and a dog are trying to get back to their home ten miles away before their dinner gets cold. The boy and girl walk at 2 miles per hour, while the dog trots along at 4mph. But they do have a single skateboard that at most one of them can use at a time: the boy and girl each riding along at 12mph, but the skilful dog can propel itself on the skateboard at an impressive 16mph. With careful planning, what is the earliest they can arrive back home?
Solutions and references some other time. If you’ve done all the puzzles (possibly before), try to generalise some!