Mental math trick for squared 20s and squared teen numbers?

Question

I'm working on a pullup progression workout. I'd like to mental math how many pull-ups done in one session. I do decrements from $20s$; e.g., $20,19,18,17,16,15,14,13,12,11,10$. This is like the summation version of a factorial, the nth triangular number:

$\frac{(n^2+n)}{2}$

n = 20

(400+20)/2 = 210

n = 9

(81+9)/2 = 45

$\sum_{n=10}^{20} n = 210-45 = 165$

So $165$ pull-ups in one workout.

But each week I usually increase the amount, and end earlier; e.g., $21,20,19,18,17,16,15,14,13,12$.

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I'm not sure if this is appropriate for this SE, but I'd like to figure out the best way using mental math. The goal is to do 50 continuous pullups, then maybe 80...

Anyway, I want to learn a method to the mathness for 9 or 10 sets. (Sometimes I don't feel like doing the 10th set.) I think once I'm able to do 29 decrement for 9 sets, I'll be able to do 50 continuous pullups. Here's a list of n²:

21² = 441

22² = 484

23² = 529

24² = 576

25² = 625

26² = 676

27² = 729

28² = 784

29² = 841

Mental math for squared 20s?

11² = 121

12² = 144

13² = 169

14² = 196

15² = 225

16² = 256

17² = 289

18² = 324

19² = 361

Mental math for squared teens?

Note how these 18 numbers could be memorized. Although, the 18 results (total pull-ups) would be better memorize in the first place:

I also just want to do the mental math rather than memorize. So if you know a better way than the two-step triangular number mental math (which is more like three-steps if a subtraction method is worked in), that'd be better to know.

Answer 1

Teen numbers

11*11 = 121 (memorize)

12*12 = 144 (memorize)

13*13 = (1*1).(3+3).(3*3) = 1.6.9 = 169

14*14 = (1*1).(4+4).(4*4) = 1.8.(1)6 = 196 (carry one from 16 to 8)

15*15 = (1*1).(5+5).(5*5) = 1.(1)0.(2)5 = 225 (carry 1 higher, & carry 2 higher)

16*16 = (1*1).(6+6).(6*6) = 1.(1)2.(3)6 = 256 (carry 1 higher, & carry 3 higher: 3+2)

etc.

20s method

XY (two-digit number), want (XY)²

X² = J

Y² = K

Concatenate J and K for JK.

(X*Y*2, and put zero on end) = L

(XY)² = JK+L

Example

26²

→ 4=J

→ 36=K

→ JK = 436

→ 2*6*2 = 24 → 240

436+240

= 676 = 26²

Method 2

https://medium.com/i-math/find-perfect-squares-mentally-with-this-trick-b8fa2c116d73

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Supplementary

1. Faster mental calculations
2. https://www.soa.org/news-and-publications/newsletters/compact/2013/april/com-2013-iss47/mental-mathbe-there-or-b2-squaring-two-digit-numbers/
3. https://en.wikibooks.org/wiki/Mental_Math

Answer 2

You can very easily calculate the squares up to $75^2$ if you've memorised the squares up to $25^2=625$. I'll give an example.

To square $37$, first square the difference from $37$ to $50$, that is $13^2=169$. Then add as many hundreds as $37$ differs from $25$, that is, add $1200$. Add both to obtain $37^2=169+1200=1369$.

Another one: For $56^2$ first square $50-56$ to obtain $36$ and add $56-25=31$ hundreds to obtain $56^2=16+3100=3136$.

Proof: Expand $$a^2=(50-a)^2+100(a-25).$$

For numbers ending in $5$ it's even easier. Example: to square $75$, calculate $7\cdot8$ and concatenate $25$ to obtain $75^2=5625$. For $125^2$ do $12\cdot13=156$ and concatenate $25$ to get $125^2=15625$.

Proof: Left to the reader.