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 1 of 4: Build the semi-invariant

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In plain words

Include enough consecutive sums so that the operation merely permutes most terms; the only changed term will be controlled by its sign.

T=\sum |\text{each selected consecutive partial sum}|

Detailed analysis

Let $T$ be the sum of absolute values of the five one-term sums, the five two-term sums, the five three-term sums, and the five four-term sums, omitting one redundant representative in the standard list. This is a nonnegative integer.