Squaring numbers and a forgotten book

I happened on a demonstration of a mental math trick on Reddit for squaring numbers in your head and was immediately reminded of a technique I learned in 1972 from a great book on speed arithmetic that I have unfortunately forgotten the name of.

The video’s formulation uses the identity n^2 = (n^2 - a^2) + a^2 to make the multiplication simpler, but the book had an extremely elegant way to notate a different identity that works nicely for doing the squares of two-digit numbers in one’s head, and rapidly doing multi-digit squares on paper.

The Reddit example squared 32 by changing it to 32 * 32 = 30 * 34 + 4 = 1024, which is clever, but check this out!

Start with the identity (a + b)^2= a^2 + 2ab + b^2 and treat 32 * 32 as (30 + 2)^2.

Visualize this in your head:

0904 
 12

That’s the a^2 + b^2 on the first line, and 2ab on the second. Now just add it up normally, with blank spaces equal to zeroes, and you get 0, 10, then 102, then 1024.

The left-to-right add means you never have to remember the carry value, just the changed result. Let’s try 47.

1649
 56

1, 21, 220, 2209. Simple.

The Wikipedia page on mental arithmetic is a great resource that has this technique, but lacks the notation visualization shown here which honestly is what makes it easy. The same technique works for larger numbers too. There’s more to remember, which may make it too hard to do in your head, but it makes squaring large numbers on paper trivial.

Let’s say we want to square 123:

010409 
 0412 
  06

1, 14, 151, 1512, 15129. (a^2 + b^2 + c^2 + 2ab + 2bc + 2ac). Squares on the top row, 2ab on the left in the middle, 2bc on the right in the middle, 2ac on the bottom.

I will admit that I didn’t properly get how to do the multi-digit notation right 45 years ago, but I hadn’t really understood the mapping of the identity to the positions on the page and was doing it by rote. The notation is the slickest part of this, as it automatically handles the proper number of multiplications by 10 for you.

The left-to-right addition and a trick of doing mental addition by repeating the current total to oneself when adding the next number to keep from losing one’s place (ex. 45 + 37 + 62 – 45, 75, 75…82, 142, 144 and cast out 9s — 0, 1, 9, and 1+4+4 =9) were all also in that same book. I really wish I could remember what it was!