What comes next, and what came before

The gap at the end changes the job

A row of figures changing by a fixed rule, and the question is which one comes next.

That sounds like the same question as a series with a hole in the middle, and it is not — for one practical reason. With a middle gap, the square on the far side checks your answer for free. Put your choice in and it either follows on or it doesn't.

With the gap at the end, there is nothing beyond it. Nothing checks the answer except the working that produced it.

So the checking has to be built in, and it is done backwards.

Establish the rule on three figures, not two

Almost anything fits two figures. A shape that gains a side between the first and the second might be gaining one every time, or alternating, or running a cycle of three.

Three figures pin the rule down. Four leave no doubt.

The instinct to decide from the first pair is where nearly every wrong rule comes from, and it is worth interrupting explicitly: what does the third one tell you?

Count from the last one you can see

Then the arithmetic, which is where the off-by-one lives.

The rule is applied once, from the last figure you can see — not from the first, and not twice. It sounds too obvious to state until you watch a child anchor on the beginning of the row and count the whole way along.

Anchor on the final figure. Move exactly one step.

When the gap opens the row

Occasionally the missing figure is the first one. Everything about a sequence feels left to right, so this is disproportionately hard — and the fix is only to name what is happening.

Same rule, opposite direction. Work it out from the figures you can see, then take one step backwards from the second figure.

Where do pupils lose marks?

Deciding the rule from two figures, when the third would have contradicted it.

Counting from the wrong end, and applying the rule twice or not at all.

Continuing one rule and freezing the other, when two are running.

Reading the options first, so that a plausible wrong figure arrives before the prediction does.

How can we practise this at home?

Sequences are the easiest thing to make up, so make them up. Draw four figures on paper and ask what the fifth is. Then, without redrawing anything, ask what the zeroth one would have been — the one before the row started.

That second question is the whole of the reverse skill, it takes ten seconds, and almost nobody practises it.