Missing numbers for SSC GD

A missing-number question shows numbers arranged in a table, a circle, a triangle or a square, with one number replaced by a question mark. All the numbers follow one hidden rule. You find the rule from the complete parts and use it on the incomplete one.

It is a close cousin of the number series: the same adding, multiplying and squaring, only laid out in rows and shapes instead of a line. This type appears regularly in the SSC GD reasoning section, along with its partner, symbol substitution, where the signs +, −, × and ÷ are swapped around.

The checking order

Try these in order. Stop as soon as one rule fits every complete row or figure.

1. Row rule. Is the third number the sum, difference or product of the first two?
2. Column rule. Does each column add up to the same total? Is the bottom number made from the two above?
3. Equal totals. Do all rows (or all figures) add up to the same number?
4. Squares and cubes. Is the result a sum of squares, or a square plus or minus something?
5. Product plus or minus. Is it first × second + 1, or first × second − first?

Common rules at a glance

Look at rows 4 and 5. The same pair, 3 and 4 or 2 and 3, can lead to 13 by two different rules. That is why you must test the rule on a second row before trusting it.

Solved examples

Example 1 (row sum).

1. Row 1: 2 + 3 = 5. Row 2: 4 + 6 = 10. The rule is third = first + second.
2. Row 3: 7 + 8 = 15.

Example 2 (row product).

1. Row 1: 3 × 4 = 12. Row 2: 5 × 2 = 10.
2. Row 3: 6 × 3 = 18.

Example 3 (equal column totals).

1. Rows give no simple rule: 5 + 8 is not 7, and 5 × 8 is not 3. So check columns.
2. Column 1: 5 + 9 + 6 = 20. Column 2: 8 + 4 + 8 = 20. Column 3: 7 + 10 + 3 = 20.
3. Column 4: 3 + 11 + ? = 20, so ? = 6.

Example 4 (sum of squares in circles). Three circles each have two outer numbers and a number in the centre. The outer numbers are (2, 3), (4, 1) and (3, 3). The centres are 13, 17 and ?.

1. Try a sum: 2 + 3 = 5, not 13. Try a product: 6, not 13.
2. Try squares: 2² + 3² = 4 + 9 = 13. Check the second: 4² + 1² = 16 + 1 = 17. It works.
3. Third: 3² + 3² = 9 + 9 = 18.

Example 5 (corners and centre). Three squares have four corner numbers and a centre number. Square 1: corners 3, 8, 2, 5, centre 18. Square 2: corners 6, 1, 7, 9, centre 23. Square 3: corners 5, 4, 8, 7, centre ?.

1. Square 1: 3 + 8 + 2 + 5 = 18. Square 2: 6 + 1 + 7 + 9 = 23. The centre is the sum of the corners.
2. Square 3: 5 + 4 + 8 + 7 = 24.

Example 6 (symbol substitution). If ÷ means +, − means ×, + means − and × means ÷, find the value of 20 × 4 ÷ 6 − 2 + 5.

1. Replace every sign first: 20 ÷ 4 + 6 × 2 − 5.
2. Now apply BODMAS. Division and multiplication first: 20 ÷ 4 = 5 and 6 × 2 = 12.
3. Then left to right: 5 + 12 − 5 = 12.

Common mistakes

Practice set

1. Rows: (1, 2, 3), (4, 5, 9), (6, 7, ?)
2. Rows: (2, 5, 10), (3, 4, 12), (5, 6, ?)
3. Rows: (9, 3, 6), (14, 4, 10), (15, 7, ?)
4. Rows: (3, 4, 13), (5, 2, 11), (6, 3, ?)
5. Columns: (4, 7, 9), (6, 5, 9), (8, ?, 9)
6. Each outer number leads to the centre number: 4 → 16, 6 → 36, 9 → ?
7. If − means + and + means −, find 10 − 5 + 3.
8. If × means −, − means +, + means ÷ and ÷ means ×, find 18 + 3 ÷ 4 × 5 − 2.

Answers:

1. 13. Third = first + second: 6 + 7.
2. 30. Third = first × second: 5 × 6.
3. 8. Third = first − second: 15 − 7.
4. 19. Third = first × second + 1: 3 × 4 + 1 = 13, 5 × 2 + 1 = 11, so 6 × 3 + 1 = 19.
5. 3. The first two columns each add to 20, so 8 + ? + 9 = 20.
6. 81. The centre is the square of the outer number: 9² = 81.
7. 12. Replace: 10 + 5 − 3 = 12.
8. 21. Replace: 18 ÷ 3 × 4 − 5 + 2. Then 18 ÷ 3 = 6, 6 × 4 = 24, and 24 − 5 + 2 = 21.

What to do next

• Learn squares to 30 and cubes to 15; see squares and square roots.
• Solve 15 missing-number questions a day for a week, testing every rule on two rows.
• Do five symbol-substitution questions a day, always rewriting the expression first.
• Revise series, which uses the same number patterns in a line.