Squaring 2 digits number from the top of your head

Note: Again, this is boring nerdy trick similar to methods to check divisibility of numbers that I posted before.

Say that you want to square the number 73 and you don't have a calculator near you ? So what can you do ? Well, that might not always be the case, but you can use this as a party trick, or to check the results of your calculations, etc.

To be able to square 2 digits number from the top of your head, you need to be able to do these two things:

1. Multiply single digit number with 2 digits number in your head.

Well, this is not really a joke. Normally when we do multiplications using pen and paper, say multiplying 7 with 76, what we would do is to multiply the number from right to left side, and writing down the carry overs.

But in order for this trick to work, you need to reverse the order of your usual multiplication methods. Do it from left to right instead.

E.g. when you are doing 76 x 7, first remember that 70 x 7 = 49, then 7 x 6 = 42, then add 42 to 490, to get 532.

2. Remember the squares of all single digit numbers

e.g. 7 x 7 = 49, 4 x 4 = 16, etc

Now we can proceed to square 2 digits number by following these steps:

1. Find the nearest multiple of ten from the 2 digit number.

Say that you're squaring 73, the nearest multiple of ten is 70, which is 3 less than 73. Or say that you're squaring 49, the nearest multiple of ten is 50, 1 more than 49.

2. Add and subtract the difference between the original number and the multiple of ten to come up with 2 different numbers

In the case of 73, nearest multiple of ten is 70, which means that you should end up with 70 (i.e. 73 - 3) and 76 (i.e. 73 + 3). Or in the case of 49, then it should be 50 and 48.

3. Now multiply that 2 numbers together.

Since one of the numbers are multiple of ten, you will end up with multiplications of 2 digit with single digit numbers. E.g. 70 x 76 is essentially 7 x 76, while appending extra 0 at the end, which ends up with 5320, or 48 x 50 = 2400, etc.

4. Square the differences from step 2, and add it to final result

So while squaring 73, the difference is 3, and the result of 76 x 70 = 5320. What you should do is to square the difference (i.e. 3 x 3 = 9), and add to final result, thus yielding 5329. Which is the square of 73.

Same applies to 49 x 49, the previous result is 2400, and the difference is 1, thus, final result is 2401.

Sample exercises:

1. 64 x 64.

first step: nearest multiple of ten = 60

second step: 60 and 68 (difference = 4)

third step: 4080

final step: 4080 + 16 = 4096

2. 45 x 45.

first step: nearest multiple of ten = 40 (and 50)

second step: 40 and 50, (diff = 5)

third step: 2000

final step: 2000 + 25 = 2025

3. 81 x 81.

first step: nearest multiple of ten = 80

second step: 80 and 82, (diff = 1)

third step: 6560

final step: 6560 + 1 = 6561

With enough practice (just do it in your mind while washing dishes or ironing your clothes), you can perform this trick fast enough to impress your peers.

Note: Again, this is boring nerdy trick similar to methods to check divisibility of numbers that I posted before.

Say that you want to square the number 73 and you don't have a calculator near you ? So what can you do ? Well, that might not always be the case, but you can use this as a party trick, or to check the results of your calculations, etc.

To be able to square 2 digits number from the top of your head, you need to be able to do these two things:

1. Multiply single digit number with 2 digits number in your head.

Well, this is not really a joke. Normally when we do multiplications using pen and paper, say multiplying 7 with 76, what we would do is to multiply the number from right to left side, and writing down the carry overs.

But in order for this trick to work, you need to reverse the order of your usual multiplication methods. Do it from left to right instead.

E.g. when you are doing 76 x 7, first remember that 70 x 7 = 49, then 7 x 6 = 42, then add 42 to 490, to get 532.

2. Remember the squares of all single digit numbers

e.g. 7 x 7 = 49, 4 x 4 = 16, etc

Now we can proceed to square 2 digits number by following these steps:

1. Find the nearest multiple of ten from the 2 digit number.

Say that you're squaring 73, the nearest multiple of ten is 70, which is 3 less than 73. Or say that you're squaring 49, the nearest multiple of ten is 50, 1 more than 49.

2. Add and subtract the difference between the original number and the multiple of ten to come up with 2 different numbers

In the case of 73, nearest multiple of ten is 70, which means that you should end up with 70 (i.e. 73 - 3) and 76 (i.e. 73 + 3). Or in the case of 49, then it should be 50 and 48.

3. Now multiply that 2 numbers together.

Since one of the numbers are multiple of ten, you will end up with multiplications of 2 digit with single digit numbers. E.g. 70 x 76 is essentially 7 x 76, while appending extra 0 at the end, which ends up with 5320, or 48 x 50 = 2400, etc.

4. Square the differences from step 2, and add it to final result

So while squaring 73, the difference is 3, and the result of 76 x 70 = 5320. What you should do is to square the difference (i.e. 3 x 3 = 9), and add to final result, thus yielding 5329. Which is the square of 73.

Same applies to 49 x 49, the previous result is 2400, and the difference is 1, thus, final result is 2401.

Sample exercises:

1. 64 x 64.

first step: nearest multiple of ten = 60

second step: 60 and 68 (difference = 4)

third step: 4080

final step: 4080 + 16 = 4096

2. 45 x 45.

first step: nearest multiple of ten = 40 (and 50)

second step: 40 and 50, (diff = 5)

third step: 2000

final step: 2000 + 25 = 2025

3. 81 x 81.

first step: nearest multiple of ten = 80

second step: 80 and 82, (diff = 1)

third step: 6560

final step: 6560 + 1 = 6561

With enough practice (just do it in your mind while washing dishes or ironing your clothes), you can perform this trick fast enough to impress your peers.

I can do this.

9119246 - 1111111 = hurr durr.

Wouldn't it be easier to do 64x64 by doing 60*64 + 4*60, rather than 60*68 + 4^2? Just seems easier to remember.

@Chavtheworld: It is, just that a generalized form that works on all 2 digits number is easier to remember. If I am handling numbers which are exponent of 2 (64 = 2^6) I would rather do calculation in powers of 2 (i.e. 64 x 64 = 2^6 x 2^6 = 2^12 = 4096)

@konig_kei: Ja, Mathe ist immer langweilig.

I'm not sure this trick seems like it would save a lot of time in this strange case I need to square numbers when I have neither a computer nor my phone. Maybe Jigsaw will be taking an educational turn in his next movie.

3 x 1 = 3

3 x 2 = 6

3 x 12 = 36 (3 and 6 is 36!)

6 x 1 = 6

6 x 2 = 12

6 x 12 = 72 (Add 6 to 1 and keep the 2!!)

9 x 1 = 9

9 x 2 = 18

9 x 12 = 108 (Add 9 to 1 and keep the 8!!!)

12 x 1 = 12

12 x 2 = 24

12 x 12 = 144 (Add 12 to 2 and keep the 4!!!!!!)

I invented this myself when I was 12

MATHEMAGIC!!

1+1=11, right?