Magic T

Remember "Magic C"? The one where you had to put the digits 1 - 5 to create sums? Well, today I have Magic T. The question is nearly identical, but with a little twist (besides the other letter).

The digits 1, 2, 3, 4, 5, 6, and 7 are placed in squares that form a T shape. The top row has 3 squares and the middle column has 5 squares. The intersection square, the one that appears on both the row and the column, is shaded. The digits are placed in such way that the sum of the digits in the row and the sum of the digits of the column are equal.

Show that the shaded square must have an even number to create equal sums, and find an example for each possible digit on the shaded square.

One way to do that is find every possible combination for these squares and see that only even numbers can be in the shaded square. However, there is a mental condition that describes doing such thing: insanity. There are many possible combinations (5040 to be exact), and finding the right ones will take forever. It seems like we need a better method: creative thinking.

First, try making a correct combination. It shouldn't take you too long. A good method you can use is number replacement. Check the difference between the two sums and switch digits accordingly. Notice what happens when you change digits: the difference changes by an even number. Always. That happens because one sum goes up by a number, and the other down goes down by the same number, so combined the difference was changed by twice that number, which is always even.

Now we know that the key here is the difference between the sums, but how does the shaded square relate to it? Since it appears on both sums, it dictates how the differences will relate to each other.

If you place an odd number in the shaded square, you will have three odd numbers left. If you place two of them or none of them at the top row, you will have one or three of them left for the middle column, respectively. This means that the top row will have an odd sum and the column will have an even sum. Not good.

If you place only one of the remaining odd numbers on the row, you will have two left for the column. Once again, this situation is not good. It creates an even row and an odd column, which do not have an even difference.

But if you place an even number in the shaded square, it's perfect: you will either have both sums even or both sums odd, so you can play with the differences and create a zero difference Magic T.

Here are three examples, each with a "shaded square" of 2, 4, and 6 (they are lying on the side so they could be written here):

7
2 - 1 - 3 - 4 - 5
6

7
4 - 1 - 2 - 3 - 6
5

7
6 - 1 - 2 - 3 - 5
4

Hope you liked it.
Nadav

nadavs