BrightKidz Library
Subjects
A row of tiles with one gap in the middle of the pattern Five tiles in a row. The first shows one dot, the second two dots, the third three dots and the last five dots. The fourth tile is empty with a dashed outline, because that is the one to work out. Curved arrows hop from each tile to the next one along.

Number Puzzles: The Mystery of the Missing Number

About 5 minutes

A number sequence is a list where each number is made from the one before it by the same rule every time. Somebody removes one number and hands it to you. Your job is not really to find the missing number — it is to find the rule. Once you have the rule, the number falls out on its own.

Finding the Rule

  1. Look at the gap between the first two numbers. Did it go up or down, and by how much?
  2. Check the next pair. If the change is the same, the rule is adding or subtracting.
  3. If the numbers grow fast, try dividing one by the one before it. If you get the same answer each time, the rule is multiplying.
  4. Test your rule on a pair you have not used yet. A rule that only works on the first two numbers is a coincidence, not a rule.
  5. Apply it to the number before the gap.

Worked Example

2, 4, 6, __, 10

From 2 to 4 is up 2. From 4 to 6 is up 2 as well. So the rule looks like add 2. Test it: 6 add 2 is 8, and 8 add 2 is 10 — which matches the number after the gap. The rule holds, and the missing number is 8.

That last check matters. Working out what should come after the gap, and seeing it agree with the sequence you were given, is how you know you got the rule right rather than merely plausible.

Another Kind

15, 12, 9, __, 3

These go down, not up. From 15 to 12 is down 3, and 12 to 9 is down 3 too. The rule is subtract 3, so the gap is 6 — and 6 subtract 3 is 3, which is exactly what comes next. It checks out.

Now One That Grows

1, 2, 4, __, 16

Try adding first, the way you did before, and it falls apart straight away: 1 to 2 is up 1, but 2 to 4 is up 2. Not a constant gap, so it is not an adding rule.

Now try dividing. 2 ÷ 1 = 2, and 4 ÷ 2 = 2. Each number is double the one before it.

In 1, 2, 4, __, 16, each number is double the one before. What is missing?

Double 4 is 8, and double 8 is 16 — which is the number after the gap, so the rule holds all the way through. If you answered 6, you were adding 2; that is the adding rule you already ruled out at the start.

Make Them For Each Other

Number puzzles are more fun to write than to solve.

  1. Pick a rule and keep it secret — add 7, subtract 4, double, anything.
  2. Start at any number and use your rule five or six times to build the list.
  3. Rub one out, in the middle rather than at the end.
  4. Hand it over. If they cannot solve it, ask them what rules they tried and why they ruled each one out.

That last question is the useful one. Watching how somebody else attacks a puzzle teaches you moves you would not have thought of.