How to Square Numbers in Your Head (2 and 3 Digits)

7 min read · IQ Tester

To square numbers in your head, pick the trick that fits the number. If it ends in 5, multiply the tens digit by the next number up and put 25 on the end (35² = 3 × 4 = 12, so 1,225). If it's near 50 or 100, use its distance from that anchor. For anything else, round up and down by the same amount, multiply those, and add the distance squared (43² = 40 × 46 + 9 = 1,849).

If you keep losing the pieces halfway through a calculation like this, IQ Tester checks whether mental arithmetic or working memory is the weak spot and trains it daily: Find my weak spots.

Squares ending in 5

This is the classic squaring trick, and it's the best place to start because it's almost instant.

The rule: take the digits before the 5, multiply by one more than itself, then write 25 after it.

• 25²: 2 × 3 = 6 → 625
• 35²: 3 × 4 = 12 → 1,225
• 65²: 6 × 7 = 42 → 4,225
• 85²: 8 × 9 = 72 → 7,225
• 115²: 11 × 12 = 132 → 13,225

It works because every number ending in 5 squares to something ending in 25, and the front part always comes out to n × (n + 1). That makes it a great first step in learning how to square large numbers in your head, too: 205² is 20 × 21 = 420, then 25, so 42,025.

Numbers near 50 and 100

Squaring numbers near 50

For numbers from about 40 to 60:

1. Find the distance from 50 (positive if above, negative if below).
2. Add that distance to 25. That gives the hundreds.
3. Square the distance. That gives the last two digits.

• 53²: 25 + 3 = 28, and 3² = 09 → 2,809
• 47²: 25 − 3 = 22, and 3² = 09 → 2,209
• 56²: 25 + 6 = 31, and 6² = 36 → 3,136
• 44²: 25 − 6 = 19, and 6² = 36 → 1,936

If the distance squared is more than two digits (distance 10 or more), carry the extra into the hundreds. But by then, the next method is usually easier.

Squaring numbers near 100

1. Find the distance from 100.
2. Add it to the number (or subtract, if below). That gives the front part.
3. Square the distance for the last two digits.

• 96²: 96 − 4 = 92, and 4² = 16 → 9,216
• 104²: 104 + 4 = 108, and 4² = 16 → 10,816
• 93²: 93 − 7 = 86, and 7² = 49 → 8,649
• 98²: 98 − 2 = 96, and 2² = 04 → 9,604

Round up, round down

This method works for any two-digit number, which makes it the real answer to how to square two digit numbers in your head.

The rule: move the number up to the nearest ten, move it down by the same amount, multiply those two, and add the distance squared.

Example: 43²

• Distance to 40 is 3. Go down 3 to 40, and up 3 to 46.
• 40 × 46 = 1,840
• Add 3² = 9
• 1,849

Example: 67²

• Go up 3 to 70, and down 3 to 64.
• 70 × 64 = 4,480
• Add 9
• 4,489

Example: 28²

• Up 2 to 30, down 2 to 26.
• 30 × 26 = 780
• Add 4
• 784

Why it works: (a + d)(a − d) = a² − d². So a² = (a + d)(a − d) + d². You're trading one hard multiplication for an easy one with a round number. If you've used the difference of squares trick in how to multiply in your head, this is the same idea backwards.

The method is easy to follow on a page. In your head, you have to hold 1,840 while you work out 3², then add them, and that's where many people drop a digit. If that's you, it may be working memory rather than arithmetic, and it's hard to know which by yourself. IQ Tester's 16 questions tell them apart and give you one short daily drill, like "Chunk it," aimed at whichever one is slowing you. Find my weak spots

Three-digit squares

Round-up-round-down scales up nicely to square 3 digit numbers in your head. Pick the nearest round hundred or fifty.

Example: 212²

• Down 12 to 200, up 12 to 224.
• 200 × 224 = 44,800
• 12² = 144
• 44,800 + 144 = 44,944

Example: 997²

• Up 3 to 1,000, down 3 to 994.
• 1,000 × 994 = 994,000
• 3² = 9
• 994,009

Example: 148²

• Up 2 to 150, down 2 to 146.
• 150 × 146 = 21,900 (100 × 146 = 14,600 plus 50 × 146 = 7,300)
• Add 4
• 21,904

Three-digit squares take real practice, mostly because of the numbers you need to hold. Say each part out loud as you go.

Squares 1 to 30 worth memorizing

Knowing some squares by heart makes every method faster, because you'll need the "distance squared" part constantly.

Checking your answer

Two fast checks catch most slips.

1. The last digit. A square's last digit depends only on the original's last digit. A number ending in 3 or 7 squares to something ending in 9. Ending in 4 or 6 gives 6. Ending in 2 or 8 gives 4. So if 67² comes out ending in 8, it's wrong.

2. The digit-sum check. Add up the digits of the original until you get one digit. Square that, and reduce again. It should match the digit sum of your answer.

• 43 → 4 + 3 = 7 → 7² = 49 → 4 + 9 = 13 → 1 + 3 = 4
• 1,849 → 1 + 8 + 4 + 9 = 22 → 2 + 2 = 4 ✓

How to square root numbers in your head (estimating)

Squares help with square roots, too. To estimate √700, find the squares either side: 26² = 676 and 27² = 729. So √700 is between 26 and 27, a bit closer to 26. That's usually enough for a test estimate.

Practice set

1. 45²
2. 52²
3. 94²
4. 38²
5. 71²
6. 103²
7. 305²
8. 189²

Answers: 1) 2,025, 2) 2,704, 3) 8,836, 4) 1,444, 5) 5,041, 6) 10,609, 7) 93,025, 8) 35,721

Hint for 8: 189² = 200 × 178 + 11² = 35,600 + 121.

Build squaring into a daily habit

That's how to square numbers in your head: 5s, anchors near 50 and 100, and round up, round down for everything else. IQ Tester's 16 questions take about two minutes and give you a reasoning tier plus your weak spots, including mental arithmetic and working memory. Then you get one tip and one drill of about 10 minutes a day, and the drill library grows as you keep going.

Find my weak spots

FAQ

What is the fastest way to square a number in your head?

It depends on the number. Numbers ending in 5 and numbers near 50 or 100 have near-instant tricks. For anything else, round up and down to a friendly pair, multiply them, and add the distance squared.

How do you square two-digit numbers quickly?

Use round up, round down. For 67², go to 70 and 64, multiply to get 4,480, then add 3² = 9 for 4,489. With practice this takes a few seconds for any two-digit number.

How do you square 3-digit numbers in your head?

Use the same round up, round down method with a nearby hundred or fifty. For 212², multiply 200 by 224 to get 44,800, then add 12² = 144 for 44,944. Holding the parts is the hard bit, so say each one out loud.

How can I check if my square is right?

Check the last digit first, since it depends only on the original's last digit. Then use the digit-sum check: reduce the original to one digit, square it, reduce again, and compare with your answer's digit sum.

Is the ending-in-5 trick really that simple?

Yes. Take the part before the 5, multiply it by the next number up, and write 25 after it. 75² is 7 × 8 = 56, then 25, giving 5,625.