Monday puzzle: two clock puzzles

Puzzle 1: clock hands
The trick is the hour hand is exactly on one of the minutes. Since the hour hand moves from one number to the next (5 minute markings) in a span of 60 minutes, that means the hour hand is only on minute markings for every 12 minutes: the times 00, 12, 24, 36, and 48.

From here it is just an exercise in trial an error to figure out the right times. If the minute hand is at 00, the hour has to be near 11, 12, or 1 to solve the puzzle. But in these times the two hands are separated by either 5 or 0 markings.

For the minute hand at 12, the candidate time would have the hour nearby at the number 2. But at 2:12, the hour hand has moved one marking, and the minute hand is two markings past the number 2. The two hands are separated by just one minte marking.

We can proceed to figure out the minute hand at 24 will work. If the minute hand is at 24, the candidate hour hand would be nearby at 4. We can check 4:24 exactly works: the hour hand is 2 markings past the clock “4″, and the minute hand is 4 markings past the clock “4″. So does the next candidate of 7:36.

Finally, you can check that the minute hand at 48 does not work.

So the two times are 4:24 and 7:36. Notice the symmetry of the answers.

If this puzzle solving technique was not rigorous enough for you, then you can read the answer here.

Puzzle 2: sums on a clock

The way I solved this was to use a bit of number theory.

I added up the digits on the clock and the total sum is 78.

The prime factorization if 78 is 78 = 2 x 3 x 13. If we want the clock divided into 3 pieces, then you need three segments where the numbers add up to 2 x 13 = 26.

It takes a small bit of creativity to realize this is possible. Clearly the numbers 11 + 12 + 1 + 2 = 26. And also the numbers 5 + 6 + 7 + 8 = 26.

The remaining numbers are 3 + 4 + 9 + 10, which also add up to 26. So we can draw a clock as follows:

The middle section is the tricky part of the puzzle, but that’s what makes it fun.