 # Puzzler's Prime Number Connection

I’ve been listening to back-dated podcasts, and imo Car Talk didn’t take full advantage this puzzler, posted earlier in the year. Prime numbers play an important role in the solution, and this would have been a good opportunity to show how understanding prime numbers can work to the puzzler solver’s advantage.

Puzzler: Draw 2 lines across a standard clock face, numbered around the circle from 1 to 12. Depending on whether or not the lines cross, the lines divide the clock face into either 3 or 4 segments. Are you able to draw the two lines so the sum of the numbers in each segment is the same?

However you do it, the sum-total must add up to 1 + 2 + 3 … +12. This is easier to calculate as (1+12) + (2+11) … + (6+7) = 6 X 13. Or in prime number form the total must be 2 X 3 X 13.
The prime number breakdown for the sum eliminates the possibility that the lines cross and there are 4 segments. Why? B/c 4 = 2 X 2, so there is no integer k that satisfies 2 X 2 X k = 2 X3 X 13.
That means there can only be 3 segments, the lines definitely don’t cross, and each segment must add up to 2X13 = 26. Where to draw the lines at that point is pretty easy.