Viral Maths — Ch06: Squares

📌 What This Chapter Covers

Finding squares fast is a foundational skill — it feeds directly into Multiplication (Padosan, Trishul approaches) and Square Roots (next chapter).
There are five named techniques, each optimized for a specific range or pattern: near 50, near 100, ending in 5, near 150/200/300+, and ending in 25.
The book advises memorizing the squares from 1 to 100 first, as given in Chapter 1, and then using these advanced techniques for three-digit numbers.

⚡ QUICK RECALL

Selection guide: number in the 40s–50s → Near Base 50; number in the 90s–120s → Near Base 100; ends in 5 → dedicated 5 Trick; number 150+ near a multiple of 50/100 → Near Base 150/200/300; ends in 25 → dedicated 25 Trick.

🗂️ Chapter Index

Tab	Technique	Best For
2	Near Base 50	Numbers roughly 35–65
3	Near Base 100	Numbers roughly 85–125
4	Ends in 5	Any number ending in digit 5
5	Near Base 150/200/250/300...	3-digit numbers near a multiple of 50
6	Ends in 25	Numbers like 125, 225, 325, 725, 825...
7	Master Table	All 5 techniques summarized

Square of Number Near Base 50

Example: (41)² — number LESS than 50

Step 1: 41 is 9 less than 50 Step 2: Subtract 9 from 25: 25−9 = 16 Step 3: Square of 9 = 81 Step 4: Combine: 1681

Example: (53)² — number MORE than 50

Step 1: 53 is 3 more than 50 Step 2: Add 3 to 25: 25+3 = 28 Step 3: Square of 3 = 09 Step 4: Combine: 2809

⚠ EXAM TRAP

The square of the distance must always occupy a two-digit slot. If it has one digit, as in 3²=9, write 09, not merely 9; otherwise the answer will be incorrect.

⚡ QUICK RECALL

If the number is LESS than 50, SUBTRACT the distance from 25. If the number is GREATER than 50, ADD the distance to 25. Then attach the square of the distance on the right.

Square of Number Near Base 100

Example: (94)² — number LESS than 100

Step 1: 94 is 6 less than 100 Step 2: Subtract 6 from the ORIGINAL number: 94−6 = 88 Step 3: Square of 6 = 36 Step 4: Combine: 8836

Example: (112)² — number MORE than 100 (with carry)

Step 1: 112 is 12 more than 100 Step 2: Add 12 to the ORIGINAL number: 112+12 = 124 Step 3: Square of 12 = 144 Step 4: 144 has 3 digits — carry the extra 1 to 124: 124+1 = 125 Step 5: Combine: 12544

⚠ EXAM TRAP

In the Base 50 approach, add to or subtract from 25; in the Base 100 approach, add to or subtract from the ORIGINAL NUMBER. The formulas are different, so do not mix them.
If the square of the distance has more than two digits, as in 12²=144, carry the extra digit to the left.

Square of Number Ending with 5

Example: (65)²

Step 1: Square of last digit 5: 25 (always fixed ending) Step 2: Multiply the leading digit (6) by its successor (7): 6×7 = 42 Step 3: Combine: 4225

⚡ QUICK RECALL

Formula: n5² = [n×(n+1)] | 25. Works for ANY number ending in 5 — 15², 25², 35²... even 105², 205².

Square of Number Close to Base 150/200/250/300...

Example: (207)² — number MORE than base 200

Step 1: 207 is 7 more than 200 (extra = 7) Step 2: Add extra to original: 207+7 = 214 Step 3: Since base is 200, multiply by (200/100=2): 214×2 = 428 Step 4: Square the extra: 7² = 49 Step 5: Combine: 42849

Example: (348)² — number LESS than base 350

Step 1: 348 is 2 less than 350 (negative extra = 2) Step 2: Subtract from original: 348−2 = 346 Step 3: Since base is 350, multiply by (350/100=3.5): 346×3.5 = 1211 Step 4: Square the extra: 2² = 04 Step 5: Combine: 121104

⚡ QUICK RECALL — Multiplier Rule

Multiplier = Base ÷ 100. Base 150→×1.5, Base 200→×2, Base 250→×2.5, Base 300→×3, Base 350→×3.5, Base 400→×4.

⚠ EXAM TRAP

Always place the square of the extra number in a two-digit slot. If it has one digit, pad it with 0, as in 2²=04; otherwise the digits will be misaligned in the final combination.

Square of Number Ending With 25

Example: (725)²

Step 1: Square the leading digit(s): 7² = 49 Step 2: Add half of the leading digit(s) to it: 49 + 3.5 = 52.5 Step 3: Multiply by 10: 52.5 × 10 = 525 Step 4: Square of 25 (fixed): 625 Step 5: Combine: 525625

Example: (825)²

Step 1: 8² = 64 Step 2: 64 + half of 8 (4) = 68 Step 3: 68 × 10 = 680 Step 4: Square of 25: 625 Step 5: Combine: 680625

⚡ QUICK RECALL

Formula: for n25², compute [n² + n/2] × 10, then attach 625 (the fixed square of 25).

📋 Master Table — All Square Techniques

Technique	Range	Core Formula
Near Base 50	~35–65	(25±distance) \| distance² (2-digit slot)
Near Base 100	~85–125	(original±distance) \| distance² (carry if ≥100)
Ends in 5	any n5	[n×(n+1)] \| 25
Near Base 150/200/300...	3-digit near multiples of 50	(original±extra)×(base/100) \| extra² (2-digit slot)
Ends in 25	any n25	[n²+n/2]×10 \| 625

🔑 Approach Selection Flowchart

Number ends in 25 → Ends-in-25 technique (most specific, check first)
Number ends in 5 (but not 25) → Ends-in-5 technique
Number is roughly 35–65 → Near Base 50
Number is roughly 85–125 → Near Base 100
3-digit number near a multiple of 50 (150, 200, 250, 300...) → Near Base 150/200/300 technique

⚡ QUICK RECALL

First check whether the number ends in 25 or 5. If it does, use the corresponding dedicated trick because it is the fastest. Otherwise, use proximity to the base to choose Near 50, Near 100 or Near 150+.