Reasoning topic

Missing number reasoning: find the rule, then test it

These questions give a figure or a grid with one number missing. Every part can follow the same rule, or the numbers can run round the figure as a series. Find the rule in one part, test it on the others, then use it on the blank.

What the question looks like

A typical question gives a grid with one number missing and asks for that number. Other questions use circles, boxes or a wheel, with the same rule in every part.

Find the missing number in the figure. Its three rows read 25, 2, 50; then 5, ?, 500; then 50, 50, 2500.

  1. 110
  2. 50
  3. 70
  4. 100
Answer

100

In each row the third number is the first times the second: 25 × 2 = 50, 50 × 50 = 2500, so 5 × 100 = 500.

How it works

Find the rule in one part of the figure: a row, a circle or a box. Try the plain operations first: +, −, × and ÷.

Test it on every other part before you use it. A rule that fits one part only is not yet the rule.

If plain operations do not fit, look for landmarks such as squares, cubes and primes, or for digits that are reversed or rearranged. Numbers that run round a figure can be a series: write the gaps. When a rule has several steps, work it out by BODMAS.

A wheel's sectors, read anticlockwise starting from the sector with 1, hold these numbers, and the last sector is empty. What number goes in it? 1, 3, 6, 11, 18, 29, 42, ?

  1. Numbers that run round a figure can be a series. Write the gaps: 2, 3, 5, 7, 11, 13.
  2. The gaps are not steady, and their own gaps, 1, 2, 2, 4, 2, settle nothing. But 2, 3, 5, 7, 11 and 13 are the primes, in order.
  3. The next prime is 17, so the empty sector holds 42 + 17 = 59.

Answer: 59

Try three

1. The figures below follow the same logic. Find the missing number. In each, two numbers stand above a third: 8 and 4 above 21, 9 and 6 above 51, 7 and 7 above 41, and 9 and 8 above the missing number.

  1. 17
  2. 89
  3. 27
  4. 71
Hint

Add the two top numbers. How does the total compare with the number below?

Answer

71

8 + 4 = 12, and 21 is 12 written backwards. 9 + 6 = 15, and 51 is 15 backwards. 7 + 7 = 14, and 41 is 14 backwards.

So 9 + 8 = 17, written backwards: 71.

2. The figures below follow the same logic. Find the missing number. In each circle, two numbers stand above a third: 16 and 9 above 7, 49 and 25 above 12, 81 and 4 above 11, and 36 and 64 above the missing number.

  1. 14
  2. 10
  3. 100
  4. 28
Hint

What kind of number is every one of the top numbers?

Answer

14

Every top number is a square, and the bottom number is the sum of their square roots: 4 + 3 = 7, 7 + 5 = 12, 9 + 2 = 11.

So the last circle gives 6 + 8 = 14.

3. Find the missing number in the figure. Each row of the table pairs one number with another: 348 with 495, 941 with 792, 526 with 396, and 517 with the missing number.

  1. 594
  2. 908
  3. 396
  4. 234
Hint

Look at the digits of 348 and 495. Rearrange them.

Answer

594

348: the largest order of its digits is 843 and the smallest is 348, and 843 − 348 = 495.

941: 941 − 149 = 792. 526: 652 − 256 = 396. So 517 gives 751 − 157 = 594.

Missing Values in the app

Practise Missing Values in the app. The questions follow your level, and what you practise comes back just before you'd forget it.

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The concepts behind it

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