Ambiguous Children's Number Puzzle

This morning's Kidspot in the Richmond Times-Dispatch features a number puzzle entitled "Sum Fun". The puzzle is given as follows. An array of circles is given as:

	O
O	O	O
	O

The numbers 2, 3, 4, 5, and 6 are also shown. The instructions say "Here are five numbers to play with. Put one number in each circle so that the numbers down and across add up to twelve."

Call the numbers N, C, W, E, and S, for the cells in the northern, central, western, eastern and southern circles. Then we can write down two equations:

N + C + S = 12
W + C + E = 12

We have one other equation:

N + C + S + W + E = 20

as that is what the numbers sum up to. Further, all five of them have to be integers between 2 and 6. If we subtract the third equation from the first two, we get:

N + S = 8
W + E = 8

So how can one express 8 as the sum of two numbers? 1+7 is no good. 4+4 does not work because there is only one 4. There are no 1s or 7s in the puzzle. 2+6 and 3+5 are the possibilities, but because of commutativity, 6+2 and 5+3 are also possible. Therefore, one of N+S and W+E has to consist of 2 and 6 and the other one of 3 and 5.

The solution given in the puzzle is:

ANS: ACROSS: 2, 4, 6. DOWN: 5, 4, 3.

Yes, that solves the puzzle. But all the puzzle implies is that one of N+S and W+E is either {2,6} or {3,5} and the other one is the other of these. But then that means that Across: 6, 4, 2; Down 3, 4, 5 is just as good a solution. In fact there are eight solutions, because one could take either 2+6 or 6+2 and either 3+5 and 5+3 and one can put the 2 and 6 across or one could put the 3 and 5 across. The solutions are:

ANS: ACROSS: 2, 4, 6. DOWN: 5, 4, 3.
ANS: ACROSS: 2, 4, 6. DOWN: 3, 4, 5.
ANS: ACROSS: 6, 4, 2. DOWN: 5, 4, 3.
ANS: ACROSS: 6, 4, 2. DOWN: 3, 4, 5.
ANS: ACROSS: 5, 4, 3. DOWN: 2, 4, 6.
ANS: ACROSS: 5, 4, 3. DOWN: 6, 4, 2.
ANS: ACROSS: 3, 4, 5. DOWN: 2, 4, 6.
ANS: ACROSS: 3, 4, 5. DOWN: 6, 4, 2.

There is a chess-problems term for this type of puzzle that is given as having The Solution, but instead has many solutions. Such a puzzle is said to be cooked. This puzzle is cooked. The authors should have checked this before putting it in the paper. The answer they give is correct, but it is not the only one. A special danger of this type of misproblem is that it teaches children that there is only one way of doing things, that there is only One Answer. This stifles creativity in children. We have enough institutions in our society that insist that there is only One Answer, including government institutions, corporations, special interest groups, and especially religions. United Feature Syndicate should check their Kidspots before they publish them.