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: Compute one move

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In plain words

Only the local triple changes, but cyclic expansion lets all terms cancel except a simple multiple of the total sum.

(x_3,x_4,x_5)\mapsto(x_3+x_4,-x_4,x_5+x_4)

Detailed analysis

Assume the negative middle entry is $x_4<0$ , so the move is $(x_3,x_4,x_5)\mapsto(x_3+x_4,-x_4,x_5+x_4)$ . Expanding the ten squares and cancelling gives $f_{\rm new}-f_{\rm old}=2Sx_4$ .