Here is a pattern that keeps going: 4, 7, 10, 13, and on and on. Roughly how big do you think the 100th number would be?

A rule is what a machine does to a number: usually a multiply step, then an add or take-away step. Each machine turns an IN number into a different OUT number, but the rule inside is hidden. Study the pairs and see if you can work out what each machine is doing.

One pupil works at the board while the rest of us say the step aloud together. The rule inside this machine is hidden. We feed a number in, read what comes out, and use the pairs to work out the secret rule. Find the step between the outputs first; then work back to what the machine does to each number. Once we think we have the rule, we predict the next output before we check.

Today we work through five hidden-rule machines together, each a step trickier than the last: first 6, 11, 16, 21; then 4, 8, 12, 16; then 5, 8, 11, 14; then 7, 17, 27, 37; and finally a stretch with 7, 13, 19, 25. Every one has a constant step, so a multiply-and-add (or take-away) rule will crack it. Feed numbers in, read the outputs, find the step first, then crack the rule before we check.

Today we found the constant step in a pattern, cracked hidden rules from pairs, and used a rule to jump straight to any term. We will now go back to our very first question and answer it in one move — watch how the rule reaches the 100th term without listing all the ones before it.