The first of two clock puzzles comes from a favorite source, The Moscow Puzzles.  The object is to take a clock face and see if you can draw five straight lines on the face such that you leave two numbers in each section of the clock.  What sum could remain constant in each section of the clock?

As an example, a line might go diagonally from underneath the 4 above the 5 & 6 and finish between the 6 & 7.  Therefore, the two numbers inside that section would be 5 & 6. The sum for that section would then be 11.  Can the constant for 11 hold for the entire clock?

Is there any other number that can hold for the entire clock? The solution is provided here, but many clues have been given such that you should be able to figure it out without seeing the solution.

A second puzzle involving the face of a clock comes from one of my other favorite sources by Royal Vale Heath: Mathemagic: Magic, Puzzles and Games with Numbers. For this, you will need to take out the numbers on the face of the clock, and on the inside, replace each number with increasing multiples of 3… 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, & 36.  Now on the outside, surround the clock with increasing multiples of 4… 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, & 48.  Make sure that each set of multiples are aligned properly.  Now consider what happens if you take and subtract the smaller multiple from the larger multiple in each case… for example, in the first case, we would have 3 (the inside multiple) subtracted from 4 (the outside multiple).  The difference is 1, which is where 1 should be on a normal clock.  However, there are other curiosities you might find… for example, subtract 18 from 24, then add the result to 24 (the larger multiple). What is noticeable? Does the same hold true in each case?  Also try squaring some of the numbers and working from there.

References:

Heath, R. (1953). Mathemagic: Magic, puzzles and games with numbers. New York: Dover Publications, Inc.

Kordemsky, B. (1972). The moscow puzzles: 359 mathematical recreations. New York: Dover Publications, Inc.