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: Isolate the changed sum

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In plain words

If the operated triple is $x,y,z$ , the old and new lists differ only by sums containing the middle entry in one specific way.

u=x+z+w,\quad S=u+y>0,\quad y<0

Detailed analysis

Write $u$ for the sum of the other four entries, so the invariant total is $S=u+y>0$ . Since $y<0$ , we have $u>-y>0$ . The unmatched term changes from $|u|$ to $|u+2y|$ .