July 2018 little math problem

[Albert Chan]

I like to show why Complement Symmetry work, and why it sometimes does not.

I use a simpler 3-sided zigzag to illustrate.
Instead of using numbers 1 to 7, think of it shifted, from -3 to 3.
So, complement symmetry is just changing the sign of all numbers.

If a solution with a non-zero sum S, complement symmetrical solution has sum -S
Swapping values, reversing ... is not going to change the new sum back to S
Thus, the complement solution is unique.

If all cases were like that, we could return half as many solutions, and let user build the other half.

If the solution had zero sum, it's complement symmetrical solution also had zero sum.
Example: [-2,3,-1,0,1,-3,2]

Above solution, complement (sign change) + reversing digits = itself.

The 2 symmetries overlap (*) :-(

In order to "cut" primary in half, above situation cannot happen.
We required half of solutions to be able to derive from the other half.

Mini-Challenge:

(*) 2 symmetries does not overlap for sides = 4, 6, 8, 10, 12, 14, ... Why ?

In other words, for even sided zigzag, we can indeed reduce primary solutions in half.

You do not need a calculator to prove this ...