A triangle is made of wood sticks of lengths 8, 15 and 17 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. How many inches are in the length of the smallest piece that can be cut from each of the three sticks to make this happen?
Our current triangle lengths are 8, 15, and 17. Let us say that $x$ is the length of the piece that we cut from each of the three sticks. Then, our lengths will be $8 - x,$ $15 - x,$ and $17 - x.$ These lengths will no longer form a triangle when the two shorter lengths added together is shorter than or equal to the longest length. In other words, $(8 - x) + (15 - x) \leq (17 - x).$ Then, we have $23 - 2x \leq 17 - x,$ so $6 \leq x.$ Therefore, the length of the smallest piece that can be cut from each of the three sticks is $\boxed{6}$ inches.