IBM Ponder This #309 · January 2024

This riddle was inspired by true events - thanks, true events!

A third-grade elementary school student received a riddle for the holidays:Fill in the 16 blank spaces on the board with the numbers $1,2,\dots ,16$without repetitions (every number appears exactly once) such that all the equations are satisfied.

Here is a textual representation of the board, with the empty number cells denoted by '??':

??+??-??-??=5
 +  +  -  +
??+??+??-??=10
 +  -  -  +
??-??+??+??=9
 -  -  +  +
??-??+??-??=0
 =  =  =  =
17  8 11 48

A solution can be represented as a list with 16 elements, such that the first 4 elements are the values of the variables in the first row, the next 4 belong to the second row, etc.

Your goal: Find all solutions to the above problem. In your submission, please supply only one such solution (as a list, as described above) along with the total number of solutions. If you find an interesting way to solve the problem in a way that a third-grade student might come up with, we'd like to hear how you did it.

A bonus "*" will be given for finding two solutions of the above equations such that

1. The solutions differ in all places (i.e., there is no cell with the same number in both solutions)
2. It is possible to change at least half of the plus/minus signs in the board such that boths solutions still give the same results for every row and column.

Please note: When you change at least half the plus/minus signs, you can also change the resulting numbers. The results need not be the same as the original board, they only need to be the same for the two solutions. Give your submission by writing out the two requested solutions, followed by a list of the plus/minus signs given as a row-by-row list. The original board is represented by the following list where the first row is listed first, the second row second, etc.

[+,-,-,+,+,-,+,+,+,-,+,-,-,+,-,+,+,-,-,+,+,-,+,-]

Solution

Best opened after a real attempt

To be added.