Patterns with growing gaps
When the gaps between numbers are not the same, look at the gaps themselves, or try "double it and add one".
In many patterns the gaps between the numbers are not all the same. That does not mean there is no rule. It means the rule is hiding in the gaps. The trick is simple: write the gaps under the numbers and look at them as a new, smaller pattern. In the 1820s the English inventor Charles Babbage designed a great calculating machine, the Difference Engine, that worked in exactly this way. It was meant to make long tables of numbers using nothing but gaps and adding. It was not finished in his lifetime, but a working one was built from his later plans in London in 1991.
Gaps that grow by the same amount
Take the pattern 1, 2, 4, 7, 11. Write the gaps under it. Then write the gaps between the gaps on a third line.
| Pattern | 1 | 2 | 4 | 7 | 11 | ? |
|---|---|---|---|---|---|---|
| Gaps | start | +1 | +2 | +3 | +4 | +5 |
| Gaps of the gaps | — | — | +1 | +1 | +1 | +1 |
The gaps are 1, 2, 3, 4. They grow by 1 each time, so the next gap is 5, and the next number is 11 + 5 = 16. When the gaps of the gaps are all the same, you have found the rule.
Example 1
What number comes next? 10, 12, 15, 19, 24, ?
- The gaps are +2, +3, +4, +5.
- Each gap is 1 bigger than the last, so the next gap is +6.
- 24 + 6 = 30.
Answer: 30
Gaps that grow by 2, 3 or more
The gaps can grow by any amount. Find it the same way: subtract each gap from the next one.
Example 2
What number comes next? 4, 6, 10, 16, 24, ?
- The gaps are +2, +4, +6, +8.
- Each gap is 2 bigger than the last, so the next gap is +10.
- 24 + 10 = 34.
Answer: 34
Example 3
What number comes next? 2, 3, 7, 14, 24, ?
- The gaps are +1, +4, +7, +10.
- Each gap is 3 bigger than the last, so the next gap is +13.
- 24 + 13 = 37.
Answer: 37
The pattern can also go down with growing gaps. In 100, 99, 97, 94, 90 the gaps are −1, −2, −3, −4. The next gap is −5, so the next number is 90 − 5 = 85. Here a common wrong answer is 86, which takes away 4 again.
Gaps that double
Sometimes the gaps do not grow by the same amount. Instead each gap is double the one before. Then the gaps of the gaps are not all the same, but the gaps line shows the rule clearly.
| Pattern | 1 | 3 | 7 | 15 | 31 | ? |
|---|---|---|---|---|---|---|
| Gaps | start | +2 | +4 | +8 | +16 | +32 |
Example 4
What number comes next? 1, 3, 7, 15, 31, ?
- The gaps are +2, +4, +8, +16. Each gap is double the one before.
- The next gap is 16 × 2 = 32.
- 31 + 32 = 63.
Answer: 63
The same idea works with any start. In 3, 4, 6, 10, 18 the gaps are 1, 2, 4, 8, so the next gap is 16, and the next number is 18 + 16 = 34.
Multiply, then add
Some patterns grow because each number is made from the one before in two moves: first multiply, then add or take away a little. You can spot these because each number is a bit more (or a bit less) than double or triple the one before.
Example 5
What number comes next? 2, 5, 11, 23, ?
- Each number is about double the one before. Test it: 2 × 2 = 4, and 5 is 1 more.
- 5 × 2 = 10, and 11 is 1 more. 11 × 2 = 22, and 23 is 1 more.
- The rule is "multiply by 2, then add 1".
- 23 × 2 + 1 = 46 + 1 = 47.
Answer: 47
Notice that the gaps here are 3, 6, 12: they double. So "gaps that double" and "multiply by 2, then add" are often two ways to see the same pattern. Either way gives 47.
Example 6
What number comes next? 1, 4, 13, 40, ?
- Each number is about 3 times the one before.
- 1 × 3 + 1 = 4, 4 × 3 + 1 = 13, 13 × 3 + 1 = 40.
- The rule is "multiply by 3, then add 1".
- 40 × 3 + 1 = 120 + 1 = 121.
Answer: 121
The small move can also be a take away. In 4, 7, 13, 25, each number is double the one before, minus 1: 25 × 2 − 1 = 49. In 1, 4, 10, 22, each number is double plus 2: 22 × 2 + 2 = 46.
Two steps that take turns
In the last kind of pattern, two different steps take turns. First one step, then the other, then the first again. Write the step under each number and the turns become clear.
| Pattern | 1 | 2 | 3 | 6 | 7 | 14 | 15 | ? |
|---|---|---|---|---|---|---|---|---|
| Step | start | ×2 | +1 | ×2 | +1 | ×2 | +1 | ×2 |
Example 7
What number comes next? 5, 10, 8, 16, 14, 28, ?
- 5 × 2 = 10, then 10 − 2 = 8.
- 8 × 2 = 16, then 16 − 2 = 14.
- 14 × 2 = 28. The steps take turns: × 2, then − 2.
- The last step was × 2, so now it is − 2: 28 − 2 = 26.
Answer: 26
A plan for any pattern
- Write the gaps under the numbers.
- If the gaps are all the same, add one more gap.
- If the gaps grow by the same amount, find the next gap from the gaps of the gaps.
- If the gaps double, double the last gap.
- If each number is about double or triple the one before, try "multiply, then add or take away a little".
- If the steps seem to go up and down, check whether two steps take turns.
- Test your rule on every number before you trust it.
Try it yourself
1.What number comes next? 3, 4, 6, 9, 13, ?
Show the answer
18
The gaps are 1, 2, 3, 4, so the next gap is 5. 13 + 5 = 18.
2.What number comes next? 50, 48, 44, 38, 30, ?
Show the answer
20
The gaps are −2, −4, −6, −8, so the next gap is −10. 30 − 10 = 20.
3.What number comes next? 1, 3, 8, 16, 27, ?
Show the answer
41
The gaps are 2, 5, 8, 11, which grow by 3. The next gap is 14, and 27 + 14 = 41.
4.What number comes next? 5, 6, 8, 12, 20, ?
Show the answer
36
The gaps are 1, 2, 4, 8, and each one doubles. The next gap is 16, and 20 + 16 = 36.
5.What number comes next? 2, 7, 22, 67, ?
Show the answer
202
Each number is the one before times 3, plus 1. 67 × 3 + 1 = 201 + 1 = 202.
6.What number comes next? 3, 6, 4, 8, 6, 12, ?
Show the answer
10
The steps take turns: × 2, then − 2. The last step was × 2, so now take away 2: 12 − 2 = 10.
Ready to test yourself?
Play Patterns: 20 seconds per question, and it gets harder as you go.