I’d argue that most people use some kind of number-to-image system when doing mental math with a few more significant digits that require keeping track of. That could be the major system, number shapes, or what have you… most people doing blindfolded Rubik’s cube use a double letter system with speffz. So whilst interesting maybe, that he picked the major system, it’s really not that surprising.
Okay, so there’s 89 squares that he knows (11-99) but did you ever try to structure them in any kind of fashion to see any patterns?
Case 0 (your first 8)
- 20² = 40
- 30² = 90
- etc.
Case 1 (your next 9)
- 15² = 225
- 25² = 625
- etc.
I explained squares ending in five in my previous post. So you already know 17 out of 89 squares, which is almost 20%
Case 2 (your next 8)
- 99² = 9801
- 98² = 9604
- etc.
Here you’re really close to the base 100, so you just subtract the distance from 100 for the left hand side and then multiply the difference for the right hand side. Here with 96 x 96 as an example:
96 - 4
96 - 4
left hand side is 96 - 4 or you could do 100 - 4 - 4 to get to 92. The right hand side is 4 x 4 = 16 and put together for the answer you get 9216. Might be easier to look at 8 x 7 to make things more clear:
8 - 2
7 - 3
Using 10 as a base (instead of 100) you get either 8 - 3 across or 7 - 2 across or 10 - 2 - 3 for a left hand side of 5 and 2 x 3 = 6 for the right hand side… answer 56.
Case 3 (your next 8)
- 11² = 121
- 12² = 144
- etc.
Similar to what we just did, but now adding instead of subtracting. Here with 13 x 13 as an example:
13 + 3
13 + 3
For the left hand side it’s 13 + 3 or 10 + 3 + 3 for a total of 16 and for the right hand side it’s 3 x 3 = 9 and put together 169. You now know 33 of 89 squares or close to 40%
Case 4
- 21² = 441
- 22² = 484
- etc.
Same example that @Kinma gave above but here without recursion… let’s do 23 x 23:
23 + 3
23 + 3
The left hand side is again 23 + 3 or 20 + 3 + 3 for a subtotal of 26. “Subtotal” because our base 20 = 2 x 10, so we need to double 26 now for a total of 52. The right hand side is simple 3 x 3 = 9 (don’t double this). Put together you get 529. One more with 28 x 28:
28 - 2
28 - 2
left hand side: 28 - 2 = 26 subtotal. Using 30 instead of 10, so triple the subtotal 26 x 3 = 78 and the right and side is simply 2 x 2 = 4. Putting them together 784.
I’ll leave case 5 for you as an exercise, but you can image that instead of five times 10 for squares close to 50 you could also use half of 100 instead. Generally, you’d know say 30 from 3 x 3 and 40 from 4 x 4 as well as 35 as 3 x (3 + 1) & 5 x 5, so use base 20 for 21, 22, 23, 24 and base 30 for 29, 28, 27, 26.
You can mix and match of course and for 71 x 71 the method that @Kinma suggested will probably be faster. There’s a few more shortcuts but this will have to do for now… hope you can see that “knowing” all the squares from 1 to 100 is not as big a deal as it would appear at first. Of course if you want, you can always memorize a handful that take you too long to “calculate” this way.