I found the following puzzle in a book called the Moscow Puzzles by Boris A. Kordemsky. (It's problem number 34, which you can also find at Google books.)
Write the numbers from 1 through 19 in the circles so that the numbers in every 3 circles on a straight line total 30.Take a minute to try and solve it on your own before reading on. (It's not that hard, so it really should take just a minute or so.)
The key to this puzzle is figuring out which number goes in the center. It's easy to show what numbers can't be in the center. The 1 can't be in the center because, although there are a few combinations you can come up with that add up to 30, there aren't enough with 1 in the center position. You could have the combinations
19, 1, 10
18, 1, 11
17, 1, 12
18, 1, 11
17, 1, 12
But you'll soon be left with numbers too small to form a group that sums to 30. For example, what number do you place across from the 2?
There aren't any numbers large enough to fill the third spot in this combination. You can try placing each of the numbers from 1 through 9 in the center and find the same shortcoming.
Similarly, the number 19 is too large to go in the center. There's no number you can place opposite the 18. In fact the sum of 18 and 19 is already too large, even without a number in the opposite circle.
Through trial and error, you can find that all of the numbers from 11 through 19 are too large.
That leaves only the number 10 to occupy center position. After that, all of the other numbers fall into place.
A magic square is an arrangement of n2 distinct integers (from 1 to n2) in a square such that the n numbers in each row, each column, and in the two long diagonals all have the same sum.
Using the principles from the solution to the From 1 Through 19 puzzle above, can you construct a 3 x 3 magic square?
I'll give you the additional piece of information that each row, column, and diagonal must add to the sum of 15 (to save you the trouble of looking it up on Wikipedia, which would be cheating.)
I'll have the complete solution in an upcoming post.