Patterns & Puzzles

Finding the rule, not just the next number

Compare & Order already asks what comes next. This board starts where that stops: what the rule is, and what the tenth number is without writing out the nine before it. Continuing a pattern and finding its rule are different skills, and the second one is the one that becomes algebra.

The repeating patterns are drawn as a row of shapes, because a repeating pattern a six-year-old has to read as a sentence is not a pattern they can see. The question is to spot the core — the shortest piece that rebuilds the whole row — and once that is found the row can be carried on for ever without drawing any of it.

Then growing and shrinking patterns, which are the same idea with the step added or taken away, and stating the rule in words: what you DO to one number to get the next. A rule that adds climbs in a straight line; a rule that multiplies climbs by more and more.

The third gear holds the off-by-one that trips up nearly everyone. The tenth number of a pattern has taken nine steps, not ten, because the first number has taken none. It is also where input-output machines live: two examples are enough to pin down a machine that multiplies and then adds, which is why a table only ever needs a couple of rows filled in.

The Open Road is the patterns worth knowing for their own sake. Square numbers really do make squares, and the gaps between them are the odd numbers in order. Triangular numbers add a longer row each time, and a run of counting numbers adds up to the count times the next one, halved. And the diagonal of the times table is the squares — the one line of it worth learning first, because knowing it halves the table.