Problem 3

To each vertex of a regular pentagon an integer is assigned, with positive total sum. If three consecutive vertices carry

x,y,z

and

y<0

, replace them by

x+y,-y,z+y

. This operation is repeated whenever some number is negative. Must the procedure always end after finitely many steps?

Step 2 of 4: Define a nonnegative integer potential

In plain words

The potential measures squared differences between vertices two positions apart. It is always a nonnegative integer, so it cannot decrease forever.

f(x_1,\ldots,x_5)=\sum_{i=1}^5(x_i-x_{i+2})^2,\quad x_6=x_1,\ x_7=x_2

Detailed analysis

Define $f(x_1,\ldots,x_5)=\sum_{i=1}^5(x_i-x_{i+2})^2$ with cyclic indices. Clearly $f\ge0$ and $f$ is integer-valued.