Skip to content
PiRush

LearnPatterns

Two patterns in one

Patterns that take turns, signs that flip from plus to minus, and how to spot the one number that does not fit.

Some patterns look messy because they go up, then down, then up again. Often there is no mess at all. There are just two simple patterns mixed together. Think of a street where the houses on the left are numbered 1, 3, 5, 7 and the houses on the right are numbered 2, 4, 6, 8. If you walk along and read the numbers left, right, left, right, you get 1, 2, 3, 4, 5, ... Two patterns, taking turns. This page shows four ways that patterns mix, and how to pull them apart.

Jumps that take turns

In the first kind, the jumps take turns: one jump up, one jump down, again and again. Write the jumps under the numbers.

Pattern5869710?
Jumpstart+3−2+3−2+3−2

The jumps are +3, −2, +3, −2, +3. The last jump was +3, so the next one is −2, and the next number is 10 − 2 = 8.

Example 1

What number comes next? 20, 16, 19, 15, 18, 14, ?

  1. The jumps are −4, +3, −4, +3, −4.
  2. The last jump was −4, so the next jump is +3.
  3. 14 + 3 = 17.

Answer: 17

Two patterns that take turns

In the second kind, the numbers in the odd places (1st, 3rd, 5th, ...) make one pattern, and the numbers in the even places (2nd, 4th, 6th, ...) make another. Split them into two rows.

Place1st2nd3rd4th5th6th7th
Number110220330?
Odd places1·2·3·4
Even places·10·20·30·

The odd places count 1, 2, 3. The even places count 10, 20, 30. The missing number is in the 7th place, which is odd, so it belongs to the first pattern: 4.

Example 2

What number comes next? 1, 20, 3, 18, 5, 16, 7, ?

  1. Odd places: 1, 3, 5, 7. They go up by 2.
  2. Even places: 20, 18, 16. They go down by 2.
  3. The missing number is in the 8th place. 8 is even, so it belongs to the second pattern.
  4. 16 − 2 = 14.

Answer: 14

Signs that flip

In the third kind, the numbers switch between positive (above zero) and negative (below zero). Look at the size of each number and at its sign separately.

The numbers 1, −2, 3, −4, 5 jump from one side of zero to the other, and each one is a step further from zero than the one before.

Example 3

What number comes next? 1, −2, 3, −4, 5, ?

  1. Without the signs, the numbers are 1, 2, 3, 4, 5. Next is 6.
  2. The signs go plus, minus, plus, minus, plus. Next is minus.
  3. So the next number is −6.

Answer: −6

A sign that flips every time can also come from multiplying by a negative number. Remember the rules: a positive times a negative gives a negative, and a negative times a negative gives a positive.

Example 4

What number comes next? 3, −6, 12, −24, 48, ?

  1. Without the signs, the numbers are 3, 6, 12, 24, 48. They double.
  2. The signs flip every time. So each number is the one before times −2.
  3. Check: 3 × (−2) = −6, and −6 × (−2) = 12.
  4. 48 × (−2) = −96.

Answer: −96

The one that does not fit

In the fourth kind of question, you are told that one number is wrong, and you must find it. Do not look for the one strange gap. Find the rule from the numbers that agree with each other, and then test every number against it.

  1. Look at most of the numbers and guess the rule: adding, doubling, squares, cubes, or "add the two before".
  2. Write out the pattern as it should be, using that rule.
  3. Compare it with the given numbers. The one that is different is the odd one out.

Example 5

One number does not fit. Which one? 2, 5, 8, 12, 14, 17

  1. The gaps are 3, 3, 4, 2, 3. Most of them are 3.
  2. With the rule +3, the pattern should be 2, 5, 8, 11, 14, 17.
  3. Only one number is different: 12 should be 11.

Answer: 12

Example 6

One number does not fit. Which one? 1, 8, 27, 64, 120, 216

  1. 1, 8, 27, 64 are 1³, 2³, 3³, 4³, the cube numbers. And 216 is 6³.
  2. The 5th number should be 5³ = 125.
  3. It is 120, so 120 is the odd one out.

Answer: 120

The same method works for any kind of pattern. In 3, 6, 12, 25, 48 the numbers double, so it should be 3, 6, 12, 24, 48, and 25 is wrong. In 1, 1, 2, 3, 5, 9, 13, 21 each number should be the sum of the two before: 3 + 5 = 8, so 9 is wrong.

Try it yourself

  1. 1.What number comes next? 2, 7, 4, 9, 6, 11, ?

    Show the answer

    8

    The jumps take turns: +5, then −3. The last jump was +5, so now take away 3: 11 − 3 = 8.

  2. 2.What number comes next? 3, 100, 6, 90, 9, 80, ?

    Show the answer

    12

    Two patterns take turns: 3, 6, 9 and 100, 90, 80. The 7th place belongs to the first one, so next is 12.

  3. 3.What number comes next? 2, −4, 6, −8, 10, ?

    Show the answer

    −12

    Without the signs the numbers go up by 2, so the size is 12. The signs take turns, and after a plus comes a minus.

  4. 4.What number comes next? 1, −3, 9, −27, ?

    Show the answer

    81

    Each number is the one before times −3. −27 × (−3) = 81, because a negative times a negative is positive.

  5. 5.One number does not fit. Which one? 5, 10, 15, 21, 25, 30

    Show the answer

    21

    The rule is +5: 5, 10, 15, 20, 25, 30. So 21 should be 20.

  6. 6.One number does not fit. Which one? 2, 4, 8, 16, 30, 64

    Show the answer

    30

    The numbers double: 2, 4, 8, 16, 32, 64. So 30 should be 32.

Ready to test yourself?

Play Patterns: 20 seconds per question, and it gets harder as you go.