All my life I've done mental arithmetic the 'wrong way': I've calculated from left-to-right, instead of right-to-left. So when I do something like

Turns out I'm not so weird after all (well, apart from the finding doing mental arithmetic fun bit). I've been reading Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Mental Math Tricks and the author, Michael Shermer, is just like me: he works from left-to-right.

He, like me, has found this to be a good system because it lets you discard digits early and not hold some enormous calculation in your head. For example, in the calculation

The books is filled with tricks for doing all sorts of mental calculations, including a nice section about estimation (I've always been an estimator) and working out things like tips and sales taxes. But the most fun part to me was a trick to let you do two-digit squares in your head really, really fast.

Quick, what's

Observe that

So the rule is that if you want to square number

This works because

Example:

The trick is especially impressive/easy with numbers near 100 because the multiplication becomes dead easy. For example,

`24 + 35`I'll see the`50`first and then the`9`. This even applies when there's a carry and I'll do something like`36 + 56`as`80 + 12`. I do the same thing for multiplication as well.Turns out I'm not so weird after all (well, apart from the finding doing mental arithmetic fun bit). I've been reading Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Mental Math Tricks and the author, Michael Shermer, is just like me: he works from left-to-right.

He, like me, has found this to be a good system because it lets you discard digits early and not hold some enormous calculation in your head. For example, in the calculation

`124 + 353`you can immediately say "four hundred" before doing the rest of the sum. This seems to free up headspace (at least for me) and let's me do the rest of the calculation. I'd do it like this:`124 + 353 = 400 + 24 + 53 = 400 + 70 + 4 + 3 = 477`.The books is filled with tricks for doing all sorts of mental calculations, including a nice section about estimation (I've always been an estimator) and working out things like tips and sales taxes. But the most fun part to me was a trick to let you do two-digit squares in your head really, really fast.

Quick, what's

`27`. Of course, the brute force way to do that is to calculate^{2}`27 x 27`which is a bit of a pain because it involves doing something like`27 x 20 + 27 x 7 = 540 + 189 = 729`. But there's a much faster way.Observe that

`27`. Since you probably know that^{2}= 30 x 24 + 3^{2}`3`this means you have to calculate^{2}= 9`30 x 24 + 9`which is relatively easy because the multiplication involves a multiple of ten which means it's really`3 x 24`and then add a zero.So the rule is that if you want to square number

`X`you first round it to the nearest multiple of 10, called that`X + r`, and then calculate`X - r`(i.e. round the same amount in the opposite direction). You calculate`(X + r) x (X - r)`and add back the square of the amount you rounded by,`r`, which will be 1, 4, 9, 16 or 25.^{2}This works because

`( X + r ) x ( X - r ) + r`.^{2}= X^{2}- rX + rX - r^{2}+ r^{2}= X^{2}Example:

`67`is^{2}`70 x 64 + 3`which is fairly easy to do in your head. And naturally the same trick works no matter how many digits you have, it's just that the multiplication gets harder.^{2}The trick is especially impressive/easy with numbers near 100 because the multiplication becomes dead easy. For example,

`96`is^{2}`100 x 92 + 4`which you (or at least I) can almost instantly see is^{2}`9216`.
## Comments

Have also put that book on my 'To read' list.

A two digit number can be represented as 10x + y, where x and y are the first and second digits. Squaring 10x + y gives us 100x(2) + 2*10xy + y(2). This is very simple now to compute. The result of y(2) will always be the last digit since the first two numbers have been

offsetby their multiplications by 10. The second digit will always be 2xy since it's offset by 10. The first digit will be x(2) since it's offset by 100.Example: say 13 squared. 3 is the last digit, so the last digit of the square will be 9. 2x1x3 = 6, which will be digit before 9, giving us 69 so far. 1 squared is 1, and the answer is 169.

Another example: 57 squared. 7 squared is 49, so 9 is the last digit, and we carry 4 over. 2x5x7 = 70. We add the carried over 4 to get 74. So far we have 49 then, with 7 carried over. 5 squared is 25. +7 to give 32. The answer is 3249.

100x(2) + 2*10x*5 + 5(2)

= 100x(2) + 100x + 25

= 100x(x + 1) + 25

So, the last digit is always 25 since there's an offset of 100 contributed by the first digit. The first digit is x(x+1), which is very simple to compute.

For example, 25 squared. 2x3 = 6. The answer is 625.

Another example: 85 squared. 8x9 = 72. Answer = 7225.

The answer always ends in 25, and the first part is handled by taking the first number and multiplying it by the number higher.

So, 95x95 is 9025. 9 * 10 = 90 and put 25 on the end.

My favourite one -- actually, the only one I remember from the presentation I saw 8 years ago -- is how to remember really larger numbers: phone numbers, digits of pi, etc. Assign each digit to a vowel sound (e.g. 0 = long o, 1 = short a etc.) and then make sentences of which the vowel sounds match the digits of the number you want to memorize. Never needed to use that in practise though.

When I square 2-digit numbers I always round down to the nearest 10. Then, add the double of the criss-cross product. And, add the final square.

Examples:

27*27

Round down to 20*20 = 400.

Criss-cross is 20*7 = 140.

Double it = 280. Now, we have 400+280 = 680, and we just need to add the final square = 7*7 = 49.

For me, addition is always easiest, and doubling something is also very natural.

96*96 = 8100 + Double(540) + 36 = 8100 + 1080 = 9180. Add 36 = 9216.

It probably depends on the person which method is simpler. But, this is the way I've always done it.

I noted that 27 = 3x9 so I squared 3 and 9 to get 9 and 81. Multiplying together gave 9 x 80 = 720 and 9 x 1 = 9 result 729. Perhaps 5 seconds thought.

Prime factoring usually brings the numbers back into the times tables range - barring larger primes being present.

So, say for example you want to approximate the square root of 750. From your trick above, you could find that the 750 is between 27^2 and 28^2, or 729 and 784. So you know the square root of 750 is at least 27.x.

Notice that the difference between 28^2-27^2 = (28+27)-(28-27) = 28+27 = 55. Also that the difference between are target and the lower number is 750-729 = 21.

So if we fit a line between our two square numbers and compare the slope between our two squares and our lower square to our target, we get 55/1 = 21/x. Solve for x, you get 21/55. That's slightly more than 20/55, which is 4/11, which is .36. That 1/55 slightly less than .02, so we'd approximate our x to be .38. So then our square root approximation is 27.38.

Checking myself, the square root of 750 = 27.3861, so I got close. This approximation always falls a tad short, especially the closer you get to 0. In fact, it's really a bad approximation anything less than 100, but it's close. It's because the curve of the parabola y=x^2 will actually go under your line you have created, and that difference gets worse the smaller x is.

Another example, 600 is between 24^2 and 25^2 or between 576 and 625. So square root of 600 is 24.x. That x is (600-576)/(24+25 or 625-576) = 24/49. 24.5/49 is .5, so 24/49 is .5-.5/49 = .5-1/98 which is approx. .5-.01=.49. So my guess is 24.49. Actual is 24.4948.

I love the (x-r)*(x+r)+r^2 method… this actually makes squaring in your head so much quicker, and this is one of those tricks that not many people know of. Thanks!

Dan Sherman

http://www.tenmarks.com

I've observed this pattern.

Hope it would be helpful

https://youtu.be/zFTnPvS4sio

https://youtu.be/zFTnPvS4sio