Another "15" Puzzle

A 6th-grade approach

This is an old riddle. You have an empty tic-tac-toe board, and the object is to fill every sqaure with a different number (from 1 to 9) so that all rows, columns and diagonals will always add up to 15. The purpose of this page is not to show you the answer, but to give you some insight on how to solve the puzzle by yourself.

What number belongs in the middle?

We know that any 3 squares (in-a-row, by-column or diagonal) must add up to 15. Label the first row of squares as a, b & c; then middle row as d,e,& f; and finally the bottom row as g, h & i. Now, let's concentrate on the four ways that cross through the CENTER SQUARE:


a+b+c+d+e+e+e+e+f+g+h+i = 15 + 15 + 15 + 15

which is the same as:

(a+b+c+d+e+f+g+h+i) + (e + e + e) = 60

... or ...

(a+b+c+d+e+f+g+h+i) + 3e = 60

Geez, there seems to be an excess of of the letter "e", but it IS on the middle square. Anyway, we don't know exactly WHAT "a" or "b" or "c" are, and we don't know what the other letters are either; but we DO know that one of those letters HAS to be 1, and that another HAS to be 2, and that another HAS to be 3, and so on and so on. Which means that:

(a+b+c+d+e+f+g+h+i) = 45

So, no matter what the "secret code" is, those 9 letters HAVE to add up to 45, simply because:

1+2+3+4+5+6+7+8+9 = 45

Earlier, we said that:

(a+b+c+d+e+f+g+h+i) + 3e = 60

...and because (a+b+c+ ... +i) = 45, then:

45 + 3e = 60

3e = 60 - 45

3e = 15

e = 15/3 = 5

(remember, "e" was the middle square)


The middle square is 5


A lot of people guess that the middle square is indeed 5, just by good-ol'-fashion inituive logic; if the number "5" is in the middle of the sequence (1,2,3,4,5,6,7,8,9), then maybe... JUST MAYBE, it also happens to be in the middle square of this puzzle. Intuition proves to be correct some of the time; and in this case, IT DOES.

So now we know what the middle square is. A lot of simple but tedious math was involved in solving that square, so you're probably wondering if the rest will be that much harder? The answer is NO... maybe. You DO have eight more squares to solve. The next step is a little tricky, but not eight times harder!

Let's look at the rules of adding ODD and EVEN numbers together:

Fifteen is an ODD number, so all 8 directions (rows, columns or diagonals) must be either:




If an ODD number was at a corner, then the opposite corner would also have to be ODD, since the middle sqaure is already ODD ("tic-tac-toe, three-in-a-row") ... or... If one of corners is EVEN, then the opposite corner would also have to be EVEN for the same reason.

And get this; the same rule applies for the edges. If one edge is ODD, then the opposite edge must also be ODD; or if one edge is EVEN, then the opposite edge must also be EVEN.

At first it sounds like a no-brainer, but then you have to consider that all the other rows and columns must also add up to 15, which is an ODD number. You have 5 ODD numbers and 4 EVEN numbers to work with, and one of the ODD numbers is already used up.

Let's make one of the corners ODD. That means that the opposite corner must also be ODD.

So now the rest of the top row must be filled with either ODD numbers or EVEN numbers. If the top row is all ODD, then the bottom row is forced to become all ODD. As a matter of fact, all nine squares are forced to be ODD!

Already, you have used up all the ODD numbers, and then some (way too many odd numbers)! But, if the rest of the top row was EVEN instead, then the rest of the bottom row must also be EVEN; and the two blank squares in the middle row are also forced be EVEN (which is way past the even-numbered limit).

As it turns out, all four corners are EVEN, and the four edges are ODD. There are many solutions, and it is up to YOU to figure out just ONE of them. You don't even have to use the screwy ODD-EVEN logic, if you just rely on your gut instinct.


A lot of people usually guess that ALL the corners must be either EVEN or ODD (but not both), and that all the edges are visa-versa. A quick trial-and-error session will prove that. Once again (in this case), intuition has proven to be correct without using a single ounce of logic. Other times, it never works; but don't let that discourage you.

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