SSC CGL Reasoning · Analogy

Square and Cube Rules

When a number pair or a set jumps too fast for multiplying or adding, the rule is usually a square, a cube or a root, often with a small number added or taken away.

All Analogy notes

Why this matters

Ninety questions in the bank, forty-one EASY, forty-five MODERATE and four HARD, eleven of them from the 2025 shifts. Every one is quick once the squares to 30 and the cubes to 15 are known by heart, and slow without them; the near-square habit (50 is 49 plus 1) does most of the work.

Concept 1 of 3: Squares, square roots and a number times its neighbour

If the second number is roughly the first number multiplied by itself, look for a square. Write the nearest square next to the number and read off what was added or taken away.

Definition

Know the squares up to $30^2 = 900$ . Write any number near a square as square plus or minus a little: $50 = 7^2 + 1$ .
Common pair rules: $n^2$ , $n^2 \pm k$ , $k \times n^2$ , $(n + 1)^2$ , $\sqrt{n} \pm k$ , and the next square ( $n^2 + 1$ to $(n + 1)^2 + 1$ ).
A number times its neighbour: $n(n + 1) = n^2 + n$ , so 132 is $11 \times 12$ and 42 is $6 \times 7$ .
Fast route: a perfect square ends only in 0, 1, 4, 5, 6 or 9. A number ending in 2, 3, 7 or 8 is never a square, so an option ending that way fails a square rule at once.
Without the printed note (several 2025 questions give one pair and no note), a rule may use the digits, such as squaring the reversed number. With the note, it may not.

Neighbour product and next square

n(n + 1) = n^2 + n \qquad (n + 1)^2 = n^2 + 2n + 1

Worked example

9 : 84 :: 12 : 147 :: 15 : ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 3 · Q8Easy

Example 1 · Analogy · Square and Cube Rules

Find the missing number: 6 : 42 :: 9 : ?

Reading a neighbour product as a square

11 \times 12 = 132

, not

11^2 = 121

. When a number sits just above a square by the square root itself, it is

n(n + 1)

, and the next term in the pattern is

(n + 1)(n + 2)

Squaring the next number instead of moving to the next square

'24 goes to 35' is

5^2 - 1

going to

6^2 - 1

. Working out

24^2

25^2

wastes the time; first ask whether each number is a square plus or minus a small amount.

Drill 41 more on squares, square roots and a number times its neighbour

Concept 2 of 3: Cubes and cube roots

A pair like 7 and 343 is a cube; 9 and 1458 is twice a cube. Know the cubes well enough that a three- or four-digit number makes you think of one, then look for the small multiplier or constant around it.

Definition

Learn the cubes: $6^3 = 216$ , $7^3 = 343$ , $8^3 = 512$ , $9^3 = 729$ , $11^3 = 1331$ , $12^3 = 1728$ , $13^3 = 2197$ , $14^3 = 2744$ , $15^3 = 3375$ .
Pair rules: $n^3 \pm k$ , $k \times n^3$ , $n^3 \div k$ , $n^2(n + 1) = n^3 + n^2$ , $n^3 - n$ , the cube root, and $n^2$ going to $n^3$ .
Fast route 1: when a big number is a multiple of a small one, divide first and look for a cube: $686 \div 2 = 343 = 7^3$ .
Fast route 2: the last digit of a perfect cube fixes the last digit of its root. Cubes of 1, 4, 5, 6, 9 and 0 end in the same digit; 2 and 8 swap, and so do 3 and 7. This works only for perfect cubes.

Two forms SSC uses

n^2(n + 1) = n^3 + n^2 \qquad n^3 - n = (n - 1)\,n\,(n + 1)

Worked example

4 : 66 :: 6 : 218 :: 7 : ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 1 · Q7Easy

Example 2 · Analogy · Square and Cube Rules

Find the missing number: 17 : 4914 :: 19 : ?

Using the last-digit rule on a number that is not a cube

The last digit tells you the root only if the number is a perfect cube. 3,400 ends in 0 but is not

10^3

20^3

; check the size first (between

15^3

and

16^3

Taking k times a cube for a plain cube

1,458 is not a cube, but it is

2 \times 9^3

. If a big number is not on your list of cubes, divide it by 2, 3, 5 or 6 before you give up on a cube rule.

Drill 24 more on cubes and cube roots

Concept 3 of 3: Squares inside a number set

In a set of three, squares appear in two ways: the result is built from the squares of the other two, or every number in the set is itself a square plus or minus a little.

Definition

Result from squares: $c = a^2 + b^2$ , $c = (a + b)^2$ , $c = a^2 - b^2$ , middle $= a^2 + c$ , and the same with a constant added.
Whole set of squares: consecutive squares plus or minus k, or the set $(n, n^2, n^3)$ .
Fast route: $a^2 - b^2 = (a - b)(a + b)$ , so $15^2 - 13^2$ is $2 \times 28$ with no squaring at all.
Two options can fit a square rule; check every option before you mark one.

Square identities

a^2 - b^2 = (a - b)(a + b) \qquad (a + b)^2 = a^2 + 2ab + b^2

Worked example

(3, 7, 58) and (6, 2, 40). Which set is like them? (a) (5, 4, 41) (b) (8, 1, 63) (c) (2, 9, 87)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 9 Sep 2024, Shift 3 · Q23Moderate

Example 3 · Analogy · Square and Cube Rules

Select the set in which the numbers are related in the same way as are the numbers of the following sets. (34, 47, 62) (119, 142, 167) (NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding /deleting /multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed)

Treating the square of a sum as a sum of squares

(3 + 4)^2 = 49

, but

3^2 + 4^2 = 25

. The two rules give different results for every set, so test the given sets with the exact one.

Stopping at the first option that fits

Square rules are loose enough that two options sometimes fit. Check all four; if two still fit, look again for a stricter rule that the given sets also obey.

Drill 22 more on squares inside a number set

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Squares, square roots and a number times its neighbour
Neighbour product and next square
$n(n + 1) = n^2 + n \qquad (n + 1)^2 = n^2 + 2n + 1$
Cubes and cube roots
Two forms SSC uses
$n^2(n + 1) = n^3 + n^2 \qquad n^3 - n = (n - 1)\,n\,(n + 1)$
Squares inside a number set
Square identities
$a^2 - b^2 = (a - b)(a + b) \qquad (a + b)^2 = a^2 + 2ab + b^2$

Watch out for (6)

Reading a neighbour product as a square
→ Squares, square roots and a number times its neighbour
Squaring the next number instead of moving to the next square
→ Squares, square roots and a number times its neighbour
Using the last-digit rule on a number that is not a cube
→ Cubes and cube roots
Taking k times a cube for a plain cube
→ Cubes and cube roots
Treating the square of a sum as a sum of squares
→ Squares inside a number set
Stopping at the first option that fits
→ Squares inside a number set

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