02 — The Square Engine
SSC CGL TIER-I & II / CALCULATION FOUNDATION
The important goal is not: "I have memorised a list of squares."
WHEN I SEE 24², 35² OR 49², THE NUMBER BECOMES FAMILIAR IMMEDIATELY.
Squares appear throughout simplification, number systems, algebra, geometry, mensuration, and trigonometry.
A square means multiplying a number by itself.
n2 = n × n
The symbol ² means "Multiply this number by itself." So 12² means 12 × 12, not 12 × 2.
BEFORE LEARNING TRICKS, LEARN THE BASIC SEQUENCE. INSTANT RECALL.
MEMORY LANDMARKS. READ THEM FORWARDS AND BACKWARDS (289 → 17).
PARTICULARLY IMPORTANT LANDMARKS
Stop thinking of squares as isolated facts.
Look at the differences between consecutive squares:
(n+1)2 - n2 = 2n + 1
The difference between consecutive squares is always the next odd number.
Every square can be built from the previous square.
e.g., 20² = 19² + 39 = 361 + 39 = 400
Multiply the number before 5 by the next number, then attach 25. This deserves special attention because it provides perfect anchor points for mental calculation.
The Method (e.g., 35²)
1. Take 3. Next number = 4.
2. 3 × 4 = 12.
3. Attach 25.
352 = 1225
45² = 2025
55² = 3025
65² = 4225
02.9 Why It Works (The Proof)
Take (10a + 5)² and expand:
= 100a² + 100a + 25
= 100a(a+1) + 25
That is exactly why we multiply a × (a+1) and append 25. You are not memorising magic. You are understanding the structure.
KNOW THESE, BUT YOU CAN ALSO RECONSTRUCT THEM.
This becomes powerful mental mathematics. Use algebraic expansion $(x \pm a)^2 = x^2 \pm 2xa + a^2$ to calculate squares near convenient bases.
02.10 Near 10
12² = (10+2)²
= 100 + 40 + 4 144
9² = (10-1)²
= 100 - 20 + 1 81
02.11 Near 100
103² = (100+3)²
= 10000 + 600 + 9 10609
97² = (100-3)²
= 10000 - 600 + 9 9409
02.12 Near 50
52² = (50+2)²
= 2500 + 200 + 4 2704
48² = (50-2)²
= 2500 - 200 + 4 2304
472
02.13 — The Three Ways to Know a Square
Route 1
Recall
You instantly know 47² = 2209.
Route 2
Previous Square
46² = 2116. Difference = 2(46)+1 = 93. Therefore 2116 + 93 = 2209.
Route 3
Near 50
(50 - 3)² = 2500 - 300 + 9 = 2209.
That is the numerical fluency we are building.
Perfect Square Elimination Rules
A perfect square in base 10 can end only in 0, 1, 4, 5, 6, or 9.
It cannot end in 2, 3, 7, or 8.
02.16 — Squares and roots are connected. Every square you learn strengthens your future square-root recognition.
Target 01: Instant
1² – 20²
Target 02: Fast
21² – 30²
Target 03: Reconstruct
31² – 50²
Ultimately, 1²–50² should live in your head. But we do not demand mechanical memorisation on the first reading. We combine: memory + pattern + derivation + repeated recall.
02.22 Your Master Rule
A square is n × n.
Consecutive squares differ by consecutive odd numbers.
Numbers ending in 5 have a special squaring method.
Numbers near 10, 50, 100 and other convenient bases can be squared through expansion.
Every square is simultaneously a square-root fact.
02.23 Mastery Standard
Do not declare "I finished Squares" just because you have read this page.
You have finished when you can recognise 1²–50², recall important squares randomly, reconstruct unfamiliar squares rapidly, and use this knowledge effortlessly inside actual SSC questions.