You can use rods of integer sizes to build "trains" that all share a common length. A "train of length 5" is a row of rods whose combined length is 5. Here are some examples:

Notice that the 1-2-2 train and the 2-1-2 train contain the same rods but are listed separately. If you use identical rods in a different order, this is a separate train.

• How many trains of length 5 are there?
• Come up with a formula for the number of trains of length n. (Assume you have rods of every possible integer length available.) Prove that your formula is correct.
• Come up with an algorithm that will generate all the trains of length n.

You can use Trains as a warm up for The Simplex Lock.