How to Square Anything

Image by Radiolab

First, a disclaimer. This method is meant to be a trick to show off to people and make you seem really smart. This will not save you time on a test over multiplying the number by itself, especially with numbers that are three digits or more. You also need to be fairly good at multiplication. (A number and then ^2 is that number squared; 8^2 is 8 squared.)

To use this method, you need to know the squares of the numbers 1 through 9, and the square of higher numbers may be helpful as well. (The square of a number is the number multiplied by itself)

We’ll start with two-digit numbers. I’ll split all two-digit numbers into several categories.

  1. Numbers that end in 0. Square the first digit and stick 00 on the end. Example: 80^2. 8^2=64. 80^2= 6,400.
  2. Numbers in between 11 and 29 (inclusive). Knock the units digit off of the number and add a 0 on the end. Then add the digit to the original number. Multiply the two numbers together and add the square of the digit you knocked off. Example: 28^2. (28) with the 8 knocked off is 20. 28+8=36. 20*36=720. 720+8^2=720+64=784. To multiply a number by 10, put a 0 on the end. To multiply a number by 20, multiply the number by 2 and then by 10.
  3. Numbers in between 31 to 39 (inclusive) or 41 to 49 (inclusive). Subtract the number from 50. Then subtract the result from the original number. Multiply the result by 50 and add the square of the first result. Example: 43^2. 50-43=7. 43-7=36. 36*50= 1,800. 1,800+7^2=1,800+49=1,849. To multiply something by 50, stick two zeroes on the end and divide by 2.
  4. Numbers in between 51 to 59 (inclusive) or 61 to 69 (inclusive). Subtract 50 from the number. Add the result to the original number. Multiply the result by 50 and add the square of the first result. Example: 62^2. 62-50=12. 62+12=74. 74*50=3,700. 3,700+12^2= 3,700+144=3,844.
  5. Numbers from 71 to 79 (inclusive), 81 to 89 (inclusive), or 91 to 99 (inclusive). Subtract the number from 100. Subtract the result from the original number. Put two zeroes on the end and add the square of the first result. Example: 94^2. 100-94=6. 94-6=88. 88(00). 8800+6^2=8800+36=8836. 

So, now it’s time to expand this to higher digits! (The hardest numbers to square with this method are 31 to 39, 61 to 69, and 71 to 79.)

THREE DIGITS

Knock off the last two digits and add 00 on the end. This is Result 1. Add the two digits you knocked off to the original number. This is Result 2. Multiply Result 1 and Result 2 together. Then square the digits you knocked off using the Two-Digit method.

Example: 184^2. 184 with 84 knocked off is 100. 184+84=268. 268*100=26,800, 26,800+84^2

84^2. 100-84=16. 84-16=68. 68(00). 6,800+16^2=6,800+256=7,056.

26,800+7,056= 33,856

FOUR OR MORE DIGITS???

This is possible, but it’s very hard and requires either impressive memory or paper and pencil to write all of the subtotals down.

Replace all of the digits in the number (except the first one) with zeroes. Add the replaced digits to the original number. Multiply the two results together and add the square of the replaced digits (using either this or the Three-Digit Method)

Example: 1234^2. 1234 with all of the digits except the first replaced with 0s is 1000. 1234+234=1468. 1468*1000=1,468,000. 1,468,000+2342.

234^2. 234 with 34 knocked off is 200. 234+34=268. 200*268=53,600. 53,600+342

342. 50-34=16. 34-16=18. 18*50=900. 900+162=900+256=1,156. 2342=53,600+1,156=54,756.

GRAND TOTAL: 1234^2=1,468,000+54,756=1,522,756

So, this is a really cool trick to show off to your friends. If you keep practicing, you’ll be able to do it faster and faster! This can be faster than multiplying the numbers out (but only with certain 2-digit numbers), but you won’t be beating calculators or anything with it. So, go and practice if you’d like! Impress your friends with the trick!

All of the calculations in this article are 100% accurate, with no calculators used.

About the Author

Seth Casel
Seth is the author of Rage Inducing Puzzles, Hexudoku, and How To Square Anything. He currently takes math at the Upper School and, as his profile picture suggests, is very good at taking screenshots of virtual creatures. He enjoys reading, math, playing video games, playing chess, and coming up with more rage inducing puzzles!
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