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Brain Teasers

### Clock Hands II

A
pocket watch is placed next to a digital clock. Several times a day, the
product of the hours and minutes on 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 not 270°. If fractional minutes aren’t allowed, find the
times at which the product of the hours and minutes is equal to the number of
degrees between the hands.

*This brainteaser was
written by Derrick Niederman.*

**Solution:**
**3:36 and 11:20.**

The
angle between the hour and minute hands equals the product of the hours and
minutes at 3:36 and 11.20. In the latter case, the product equals 220,
which is the number of degrees between the hands—if you go the long way around!

Let
*h* = hours and *m* = minutes. The hour hand covers 30° every hour, so at the top of
any hour it has traveled 30*h* 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 6*m* – (30*h* + *m*/2) or (30*h* + *m*/2) – 6*m*. Consequently,
the relevant equations are:

Because *m* is a positive integer, the
first equation implies that 11 – 2*h*
must divide evenly into 60*h*, and that
happens only when *h* = 3.
Plugging in *h* = 3 yields *m* = 36, so (3, 36), or 3:36,
must have the desired property. Sure enough, there are precisely 3 × 36 = 108°
between the hour and minute hands at 3:36.

For
the second equation, the right-hand side must be a positive integer, and that
happens only when *h* = 11. In that
instance *m* = 20, producing (11, 20),
or 11:20, as our second time. The fact that you go the “long way around”
in this instance arises because the second equation essentially reverses the
order of the hour and minute hands.

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