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 3 of 4: Show the absolute value decreases

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In plain words

The positive total places $u+2y$ strictly between $-u$ and $u$ .

-u<u+2y<u

Detailed analysis

The upper inequality $u+2y<u$ follows from $y<0$ . The lower inequality $u+2y>-u$ is equivalent to $u+y>0$ , namely $S>0$ . Hence $|u+2y|<u=|u|$ , so $T$ strictly decreases.