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# some clever title...not "hip to be square"...

This'll be a post probably of interest only to and , because it's about math, and , cuz she likes it when i get all didactic.

I was watching the movie Parenthood recently with some friends and in it a precocious toddler is asked to find the square root of 8649. I blurted out "93" a second or two later, to the bemusement of the others present. Later, Hutch (the Rice chemistry professor that's actually using my monkey discourses to teach catalysis in his class) asked me how i did that so fast. Thus, i wrote up this explanation of how to calculate square numbers quickly in one's head.

The first step is that you need to have the squares from 1-25 memorized. Then, using that knowledge, calculate subsequent squares by "FOILing" out the sum/difference between that number and the nearest number divisible by 50 and things will work out nicely. This is how it works out for a number between 25 and 50:

x2 =
(50 - (50 - x))2 =
502 - 2 * 50 * (50 - x) + (50 - x)2 =
2500 - (50 - x) * 100 + (50 - x)2 =
(x - 25) * 100 + (50 - x)2

That may look somewhat complicated to do in one's head, but it's not too bad when dealing with an actual number:

372 =
(50 - 13)2 =
2500 - 13 * 100 + 132 =
1200 + 169 =
1369

So basically, it's (37 - 25) hundred + 132. Similarly, for a value between 50 and 75, it would be (25 + (x - 50)) hundred + (x - 50)2:

642 = (25 + 14) * 100 + 142 = 3900 + 196 = 4096

Finally, for to do the square from the movie:

932 =
(100 - 7)2 =
1002 - 2 * 100 * 7 + 72 =
(100 - 2*7) * 100 + 49 =
8649

Square roots are even easier, since with the ending of 49 i knew the answer has to be ± 7 from an even 50 value. As 8649 is clearly between 902 and 1002, i was able to impress the ignorant masses.

One final trick that i found when i discovered the above (in high school during (English?) class) is that it's even easier to do squares of numbers that end in 5. All you need to do is lop off the 5, take the rest and multiply it by one plus that number, that's your hundreds value, and append a "25" to the end and you have your answer.

Again, it's easier to see by example:

152 = (1 * 2) * 100 + 25 = 225
252 = (2 * 3) * 100 + 25 = 625
352 = (3 * 4) * 100 + 25 = 1225

Enjoy.

----

If needed, i'd use the above method for even larger numbers (rewriting squares as ((500x ± y)2 or (5000x ± y)2). For example, when watching a video about a savant named Daniel Tammet, they showed him calculating 276 in his head. It took us both around 90 seconds. No idea how he did the calculations, but i did it by breaking it down as follows: 276 = 318 = (3 * 812)2 = (3 * 6561)2 = 19,6832 = (20,000 - 317)2 = (40,000 * 9683) + (300 + 17)2 = 387,420,489.

Again, by breaking it down that way (starting with the 3172 and working up), i didn't have to remember too many digits at a time, except at the end. Moreover, any addition i did had no more than a couple digit overlap between non-zero digits (e.g. the final addition was 387,320,000 + 100,489). I confess, that this was close to my limit on mental calculation. The fact that 39 happened to be so close to 20,000 helped a lot.
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