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: Conclude termination

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In plain words

A strictly descending sequence of nonnegative integers cannot be infinite.

T_0>T_1>T_2>\cdots\ge0

Detailed analysis

Every legal move strictly lowers the nonnegative integer $T$ . Therefore an infinite run would produce $T_0>T_1>T_2>\cdots\ge0$ , impossible. The procedure always ends.