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This brainteaser was written by Derrick Niederman. A pocket watch is placed next to a digital clock. Several times a day, the number of minutes shown by the digital clock is equal to the number of degrees between the hands of the watch. (The watch does not have a second hand.) As you can see, 10:27 is not one of those times — the angle between the hands is much greater than 27°. If fractional minutes aren’t allowed, at what times does this happen? Resources for Teaching Math © 2009 National Council of Teachers of Mathematics, Inc. http://illuminations.nctm.org Solution: 12:00, 3:20, and 6:40 It’s easy to see that the number of minutes on the digital clock is equal to the number of degrees between the hands of the watch on the hour at noon and midnight. At that time, the hands of the watch overlap, so the angle between them is 0°. At the same time, the digital clock shows 00 as the number of minutes. To find other solutions, let h = hours and m = minutes. The hour hand covers 30° every hour, so at the top of any hour it has traveled 30h degrees, and with every passing minute it travels another m/2 degrees. Given that the minute hand moves 6° every minute, the number of degrees between the hands is given by either 6m – (30h + m/2) or (30h + m/2) – 6m. Consequently, the relevant equations are: m = 6m − ( 30h + m2 ) → 9m = 60h m = ( 30h + m2 ) − 6m → 13m = 60h For both equations, the possible values for h are 1, 2, 3, …, 12. Therefore, the solutions to the first equation are (3, 20), (6, 40), (9, 60), and (12, 80). The first two solutions correspond to the times 3:20 and 6:40. The last two of these are solutions to the equation but not solutions to the problem—the digital clock will never display 60 or 80 as the number of minutes. There are no solutions to the second equation for h = 1, 2, 3, …, 12, because the equation implies that h must contain a factor of 13, and none of the numbers 1–12 is a multiple of 13. Finally, If we think of the “on the hour” value for the minute hand as m = 0 and then solve either equation above for h, we find that both equations yield h = 0. This gives us an additional solution of (0,0), which corresponds visually to 12:00 on the watch dial. Resources for Teaching Math © 2009 National Council of Teachers of Mathematics, Inc. http://illuminations.nctm.org