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Arithmagons
By Rachelle Najman (T4009) on Monday, March 26, 2001 - 06:58 am:

A secret number is assigned to each vertex of a triangle. On each side of the triangle is written the sum of the secret numbers at its ends. Find a simple rule for revealing the secret numbers.

For example, secret numbers 1, 10, and 17 produce the numbers 27, 18, and 11 along the opposite sides.

Also generalize to other polygons...does it only work for odd ones or even ones as well, explain?!?!
By Kerwin Hui (Kwkh2) on Monday, March 26, 2001 - 10:16 pm:
For the first question, subtract the opposite side from the sum of the other two sides, then divide by 2. I'll leave you to check the algebra.

There are only solutions to polygons with odd number of sides. For even ones, it is clear that we can assign +1 and -1 alternately to achieve 0 on every sides. For odd ones, this does not happen.

More rigorously, consider each vertex being assigned a number ai. Thus, we have n simultaneous consistent equations of n unknowns:

a1+a2=A1
a2+a3=A2
...
an+a1=An

In the case where n is even, any equation can be deduced from the others. In the case where n is odd, these equations are independent of each other.

Kerwin