Here is the solution I promised for “A Problem” – Grandfather’s Breakfast. I am sorry that it is so late. I have not posted lately as I have been getting ready for this weekend. O was gone Saturday through Tuesday at Summit. (Maybe a post about it later.) As I am not the best at explaining things myself, I will just give the book’s solution.
To restate the problem:
Grandfather is a very hard-boiled customer. In fact, his eggs must be boiled for exactly 15 minutes, no more, no less. One day he asks you to prepare breakfast for him, and the only timepieces in the house are two hourglasses. The larger hourglass takes 11 minutes for all the sand to descend; the smaller, 7 minutes. What do you do? (Grandfather grows impatient!)
You must begin by turning over both hourglasses. Turning over only one can only return you to the initial situation! Now there are two simple solutions:
(a). After turning over both hourglasses, put the egg into boiling water right after the smaller one (7 minutes) empties. Let the larger one run out (4 more minutes) and turn it over (11 more minutes). After it runs out again, take the egg out of the water. The egg will have been boiled for exactly 15 minutes (4 + 11), but this solution has one flaw: the whole procedure takes 22 minutes, and your grandfather is both impatient and a retired mathematician. You need an optimal solution, so you boldly
(b). put the egg in the water at the instant you turn both hourglasses over. Now, after seven minutes the small hourglass is inverted, and there are 4 minutes left in the large one. When the large one runs out 4 minutes later, there are 3 minutes left in the top of the small one, but more important, there are 4 minutes worth of sand in the bottom! With a sigh of relief, you invert the small hourglass at this instant, wait for it to run out, and present your grandfather with his breakfast. This solution takes exactly 15 minutes, and you can do no better.