 Double digit squares... useful applications?

Can anybody think of some useful applications of knowing the squares for 11² to 99² beyond the obvious two:

• difference of squares to multiply 2-digit numbers
• skipping a couple of digits when doing square roots

…can’t help but think there must be other useful things you can do with that.

By the way, this post is not about learning how to calculate or memorize the squares...

…this has already been answered elsewhere on the forum multiple times; here one of my contributions in case that interests you more than the question above:

How can I start mental calculating

For continued impractical applications, 2 digit squares allow you to leverage.

ab^2 = a^2 + 2ab + b^2
which gets you to 4 digit squares in a very tidy way.

\begin{align} 3456^2 &= 34^2 + 2*34*56 + 56^2\\ &= 1156|0000| + 3808 |00| + 3136\\ &= 11,943,936 \end{align}

Similarly as you said difference of squares give you a fast path to multiplication using your well known facts

\begin{align} 55 * 59 &= 57^2 - 4 \\ &=3249-4\\ &=3245\\ \end{align}

Between the two you have a strong set of additional tools for multiplication of larger amounts pairing up digits and reducing the number of actual calculations to half or better in some cases.

From a practical sense, I suppose it allows you to know how many square feet of paint you need quickly I suppose.

From a third spot, I find that thinking about numbers in different ways develops “practiced” numeracy. It doesn’t seem to stick around if you don’t practice but while you are practicing you can get a bit of a taste for how smart people play with abstractions.

There is a similar kind of feeling to the Soroban.

If you could, but I can’t, visualize your calculations effectively while having an intuitive grasp of algebra it might be very entertaining. I wouldn’t compare it to playing the piano but it would be nice to be adept with numbers and digging into learning squares, exponents, primes, logarithms, integration, algebra as mental exercise is a pleasant hobby. Squares are kind of a minor gateway drug.

Oh nice, didn’t think of that… that’s pretty convenient and looks faster than doing duplex… given you know your squares well.

Too practical for my taste… not that much into DIY Preaching to the converted, but not the best selling point if you are explaining the benefits to other people.

By the way, I think something went wrong copy/paste-ing or someplace… 57² is not 6249… it should read:

\begin{align} 55 * 59 &= 57^2 - 2^2 \\ &=3249-4 &=3245 \end{align}

…didn’t know we could use LaTeX here. That’s cool. I always thought people uploaded pictures in their posts.

was typing on the ten key and hit the 6 instead of the 3 and didn’t review my post. fixed. MathJax is very convenient