BrightKidz Library
Subjects
One list of numbers, two different answers for what comes next Five solid discs climb steadily from the lower left towards the upper right, joined by a quiet line, standing for the numbers already known. From the last disc two bold lines set off in slightly different directions and end at two different marks: a disc higher up and a square lower down. The list so far allows either of them.

A Rule That Fits Is Not Always the Rule

About 12 minutes

There is one step in the method you already have that is doing more work than the others: test your rule on a pair you have not used yet. It is the step that turns a hunch into something you are willing to write down, and it is the right thing to do.

It is also not proof, and it is worth knowing exactly what that means — because there is a sequence that passes that test five times in a row and still has a different rule underneath it.

Dots on a circle

Here is where the numbers come from, and you can do all of this with a pencil and a round lid.

Draw a circle and put some dots on the edge. Join every dot to every other dot with a straight line. Those lines cut the inside of the circle into pieces. Count the pieces.

Dots round a circle, every pair joined, and the pieces counted Six stages of the same circle. At each stage one more dot is added round the edge and every dot is joined to every other dot by a straight line, cutting the inside of the circle into pieces. The pieces are counted underneath: one, then two, then four, then eight, then sixteen. The sixth stage looks as though it should give thirty two and gives thirty one. 1 dot, 1 piece 2 dots, 2 pieces 3 dots, 4 pieces 4 dots, 8 pieces 5 dots, 16 pieces 6 dots, 31 pieces

Count the pieces yourself at each stage before you move on. The lines are all drawn for you; the counting is the part that matters.

  1. One dot. No lines to draw, so the circle is still in one piece. That is 1.
  2. Two dots, one line across, and the circle falls into 2.
  3. Three dots make a triangle, and now there are 4 pieces — the triangle plus the three curved bits around it.
  4. Four dots. Six lines, and they cross in the middle. Count carefully and you will find 8.
  5. Five dots, ten lines, and a pentagon shape appears in the centre. There are 16.
  6. Six dots. Everyone says 32. Count them properly and there are 31.

Look at what the first five stages did: 1, 2, 4, 8, 16.

That is doubling, and it is not a near miss or a rough pattern. It is exact, four times over. Every check the method asks for passes: the gaps are not constant, so it is not adding; dividing gives 2 every single time; and testing the rule on a pair you had not used works at every stage you could reach.

And then the sixth dot gives 31.

Why it breaks, which is the useful part

A surprise on its own is only a trick. This one has a reason, and once you see it the doubling stops looking mysterious and starts looking like a coincidence — which is exactly what it was.

Adding a new dot does not double anything. It adds some new lines, and here is what one new line is worth:

A new line adds one new piece, plus one more for every line it crosses on the way.

Draw a line straight across an empty circle and it cuts it into two: one new piece. If that line crosses two lines already there, it gets chopped into three sections, and each section splits a piece it passes through: three new pieces.

So the total grows by however many lines there are, plus however many crossings they make. For the first few dots those two amounts happen to add up to exactly the number of pieces you already had. From the sixth dot on, they stop keeping up. Nothing changed about the rule; the numbers just walked away from doubling.

So what is a sequence puzzle really asking

Not which number is forced, because usually none is. It is asking which rule is the best explanation of the numbers you can see — and that is a judgement, not a calculation.

The best explanation is normally the simplest rule that fits everything. For 1, 2, 4, 8, 16, doubling is much simpler than counting pieces in a circle, so doubling is the right answer to give. It is the right answer even here, where it happens to be the wrong rule.

You find a rule that fits every number you were given. What have you shown?

Two things that make you surer

More numbers. Each extra one you are given is another chance for a wrong rule to fall over. Five numbers left room for two answers here. Six did not.

Knowing where the numbers came from. This is the big one, and it is the thing a list of numbers can never tell you. 1, 2, 4, 8, 16 means doubling if somebody built it by doubling, and means something quite different if somebody built it by drawing circles. The numbers are identical. Only their history separates them.

What to do with this

Keep doing exactly what you were doing. Find the gap, test the rule, check it against a pair you have not used. That method finds the setter's rule nearly every time, because a setter picks a rule first and builds the list from it — so the simplest fit really is theirs.

The only thing to change is two words. Instead of the rule is doubling, say I think the rule is doubling — and then say why you think so. That is not hedging, and it is not being unsure of yourself. It is the difference between an answer and a reason, and the reason is the part somebody can argue with, which is the part worth having.