My digital alarm clock has many responsibilities. First, it has to wake me up at 6:30 A.M. to get my kid to school. Later, I reset it for the various 10-minute naps I take during the day.

Resetting the alarm clock for a new time entails changing the hour one hour at a time on a 24-hour clock and then changing the minute value one minute at a time. In the worst case, the minute value must be changed by clicking 59 times. (As you can see, it's not an iPhone, Android or other device that allows me to key in the right time.)

So, I thought that it would be nice to have an extra button that would advance the minutes by more than 1 with each click. The question then is: What number of minutes, as an interval, would be most efficient?

Warm-up:
Suppose the multi-minute button (MM for short) always advanced the time by 5 minutes. What, then, is the greatest possible number of clicks necessary to reset to the correct minute value?
Warm-up solution

Problems:

1. Still, 5 minutes may not be the best interval to choose for MM. What might be better for minimizing the worst-case number of clicks? What is that worst case?

2. Suppose that the first click on MM advanced a certain number of minutes, the second click on MM a possibly different number of minutes, and so on. What should those numbers be to minimize the worst-case number of clicks?

3. Suppose you were given two multi-minute buttons (MM1 and MM2), and each advanced the time by a fixed number of minutes. What should those two fixed numbers be for the two buttons to minimize the worst-case number of clicks? How many is that worst number?

Hint: Suppose one button advanced the minutes by, say 32, then two clicks of that button advanced the minute hand by 4 without changing the hour value.

We now have three buttons that advance the time, one by 1 minute and the others by other amounts. Many variants are still possible. For example, we could take away the constraint that all buttons advance the time and the constraint that one of the buttons must advance by 1 minute. If you have a cool variant that you can solve, then please post it, with the solution, as a comment on this story.

ABOUT THE AUTHOR(S)

Dennis Shasha is at the Courant Institute of Mathematical Sciences, New York University. His most recent puzzle book, Puzzles for Programmers and Pros, was published in May 2007 by John Wiley and Sons/Wrox.