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 4 of 4: Force termination

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In plain words

Every legal operation strictly lowers a nonnegative integer, so only finitely many operations can occur.

2Sx_4<0

Detailed analysis

Because $S>0$ and a legal move has $x_4<0$ , we have $f_{\rm new}-f_{\rm old}=2Sx_4<0$ . An infinite run would create an infinite strictly decreasing sequence of nonnegative integers, impossible. Therefore the procedure always stops.