# A very small thought

Sad but true… I’m out of practice so I may have realized many times before but be too senile to remember (but maybe not, I could actually be dumb enough that this idea is new to me.).

I love my squares. 16*18 = 288 because 17^2= 289 - 1, done.

Yet when I can’t remember 23^2 I keep saying 20^2 + 2 * 20 * 3 + 3^2, or criss-cross. or skip a step 460+60+9… When I could just be saying 20 * 26 + 9 = 529. Funny how obvious that is. The other ways are fun but when resolving a square you are always within 4 of the 0, 5 doesn’t really count although with bigger numbers may you could lump it in 1295^2 , 1300*1200 + 25 seems pretty tidy (if a bit contrived).

I don’t think you would go wrong very often and find an easier method ( 1 digit x 2 digits + 1,4,9,16 ) for the first 99 squares. Ending in 0 and 5 are the only possible easier solutions assuming you have memorized a * (a+1) the n-1 number of digits that you are squaring kind of thing.

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If the units digits sum = 10 life is easy

67*53 = (60+7)(60-7) (6,+7)(7,-7)

depending on whether the tens digit difference is odd or even you get the sq of a multiple of 10 or 5

77*53=(65+12)(65-12) (6,7)(6,-7)

It is always better to “know” than to calculate. The more fixed facts you have, the fewer mistakes you make and the cognitive load decreases allowing you to more accurate where you do have to think. Thinking is a risky business, IMO. Could be wrong

PS

The Squares method is a specialization of the Anchor Method (sometimes misspelled, I think, as ‘Anker’) or the Base Method in Vedic. This allows you to fold the multiplication around an arbitrary number, even one that does not lie between the two multiplicands.

This is probably my typo. I am Dutch and anchor in Dutch is Anker.

Maths is my out of mind, I can’t understand

The solution is easy. For a while just calculate using the ‘Base’ method.
Either that or always make 2 or 3 calculations when practicing:
1: Criss cross
2: Anchor
3: Base

It makes great practice anyway.

I made a small sheet for training purposes:

The idea is to open this on a phone. The columns are so wide that you cannot see the answer.
Scroll right to see the answer.

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Good to know. If it’s a Dutch spelling, IMO that’s perfectly valid. If the term was butchered by illiterate English speakers, I am not so tolerant I have a son who speaks Dutch. I should have checked with him LOL

I recommend memorizing the squares up to 100 it’s less work than you might think

You probably know up to 20.

In the remaining list you can knock 16 numbers which end in 0 & 5

Numbers in the range 40-50 are so quickly calculated that I don’t memorize them
That knocks out another 16

90-110 are similarly trivial - that’s another 8 (not counting sqs >100)

That leaves about 40 and you probably know some of those - 322 & 64^2 and others.

Not that daunting.

I have a spread sheet of Major System mnemonics for this, if you want raw material to start from. Some numbers are hard to encode.

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I have done my first 110 squares by rote a few times now so no need for the major system. will be solid in a couple of weeks. Complements takes me a couple of months of daily practice to trust implicitly. Then I will work criss cross with 2 digits till it’s reliable then 3 and 4 digit numbers. Then roots, division, and logs and I will basically be back up to speed. Then linear algebra and calculus and more of the same till I quit again or get my math skills back up to an undergraduate level

How do you train this?

I am currently examining how I calculate and when I go for complements and when not.
Will write about it when my thoughts are more clear.

Brute force. I spend an hour a night explicitly practicing using the method till my brain accepts it as true.

I might be a bit slow today, but I still don’t get it!
Can you give an example of how you do complements training?

Some numbers just scream ‘complement’.
When I see ‘98’ for example I immediately also see ‘2’.
There is no calculation, the ‘2’ just presents itself.
Mentally, it is like a bifurcation. When calculating with 98, I see both 98 and 2.

On 56, nothing like this happens. If I ask my brain for its complement, only then 44 just presents itself. Again, no calculation.

In the other thread we are doing 5600 + 3700.
Nothing immediately presents itself. So I go to work:
80, 13 => 93; 9300

If you give me 17 and 18, then 35 comes up immediately. No calculation.
If you give me 56 and 37, then nothing comes up immediately. So I go to work.

If I force 56’s complement: 44 and force my brain to work with that, I go 44 minus 37 = 7. 7’s complement is 93.
This works, but my brain actively resists the double complement. Maybe that is why no complement comes up on 56 + 37.

I am trying to find out when my brain does what in which circumstance. Most of times it goes so quick I don’t even know. But I want to know.

I’ll write more when I know more.
Let me know if this is interesting.

Very interesting but probably not relevant :). I suspect once I have the skill nailed the rationale will not much matter. It is a gap in my technique. I could never do linear algebra by hand and actually get the right answer. Knew all the steps but -1 + 1 -1 -1 + 1 etc always killed me. I had a similar issue with computer programming tests on paper. I like to blame laziness but as always it is mostly a lack of horsepower. Just smart enough to make fire. Not smart enough not to get burned.

If you treat every multiplication problem as a Base 10^x problem then all of them are complements problems. I just work em till they sit right.

Do you mean symbolic LA or computing actual coefficients? Inverting or reducing a large matrix by hand takes formidable skill. In the engineering dept there were legends about Phd students in the days before computers, who were admitted into the program for no other talent than their ability to solve large systems by hand. They were never allowed to graduate :).

The thing is; I have no idea how I can teach somebody when to use a complements and when not.
One can reach an answer either way, so “do” or “don’t” is a matter of taste and speed.

On 98+15 my brain forces me to use them and I can almost not prevent the use of complements.
On 56+37 no complement comes up and I wonder what algorithm my brain uses. Is this just experience or can I find out what exactly is happening?

For training I can advise you to - for a short time - always use both ways and let your brain come up with what the best moment to use either one is.

5,6 + 3,7 = 5,6 + 4,-3 = 93

I try complements when a digit is >5

Exactly. Yours is a mathematically sound algorithm. Even then; my brain decides differently.

When observing my thought process, I see: 80, 13, 93. Apparently no complements used.

However; when I do 88 + 15, I immediately see 103.
I assume my brain does a quick complement and subtraction. However; since the result comes up immediately. I don’t know for sure.

If I even look at 88, a faint 12 comes up. It feels like a bifurcation.

Keep in mind; I never practiced this.

I always liked calculating and when I was young I did a lot mentally. Now, at 52, somewhat less.

Finding out exactly how I get to a result is very interesting for me.

I do believe that it is inherently easier for the brain to deal with digits <=5 but that advantage probably disappears after years of training. Like with typing on an English QWERTY kbd - an inefficient layout. There is DVORAK which is supposed to be much smoother but that this point, I don’t even ‘see’ the keyboard. It’s like driving, I don’t realize I’m doing it unless I think about it. I don’t know where the letters are on the keyboard, I have to ask my fingers. People who do switch don’t report significant speed gains.

… completely off topic… The constraint to speed become mental rather than finger movement efficiency quite quickly… DVORAK, WORKMAN, and several others all make claims of “efficiency” but in practice there is no significant reduction in injury or increase in speed. I have an infinity steno machine here collecting dust. Steno writers exceed 200 wpm regularly while capturing court transcripts verbatim. Another skill I haven’t mastered.