Think! There is always an easier way

After all the other theoretical threads, let’s do something practical.

Sometimes the best way of multiplication might elude one for a minute.

Today, I had to calculate 52 times 45.
At first I thought, well, the numbers are close together, so the ‘base’ system might yield results fast.

If we would take 50 as the base, here is how this would play out:

(50 + 2) * (50 - 5)
= 50 * 50 + 50(2 - 5) -2 * 5
= 2500 - 150 - 10
= 2340

But I did not do that.
All of a sudden I realised I could rewrite 52X45 as 26X90 and continue from there:

26* 90
= 260* 9
=2600-260
=2340

Moral of the story; don’t always use the same algorithm.
Think first whether there is an even easier way to get to the result!

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Good example. I haven’t done much systematic work on factoring techniques. There’s a whole big field by itself.

I comment that when you write 260*9 = 2600-260,
You are implicitly using the complement form of 9, (10-1)

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This is true. Also a very interesting field.

Also true. However; I don’t realise this while calculating.
When I think of 260 * 9, I go: 2600 minus 10% = 2600 - 260 = 2600 - 300 + 40
So implicitly I do a complement of 260 from 300.

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I do the same thing w/ 22 and 33 as well as redistributing by doubling and halving with 5 when the muliplicand is even. Recognizing when things reduce to a trivial solution is nicer than grinding a method when you can but it can also slow you down if you are actively seeking alternatives. I think the payoff comes with longer calculations. We often use 2x2 as a trivial example of bigger problems but the difference between searching for an efficient algorithm and just working the problem is very small in 2x2 multiplication. Larger numbers where memory become an element of the calculations would pay dividends from investing in selecting better algorithms. I’d like to have most 2x2 around 1 second. Searching for something efficient that I don’t immediately see takes a couple of seconds and then calculating takes a couple of seconds and it doesn’t take long to be well over 5 seconds if I search and calculate (unless I get lucky)

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IMO there are too many tricks to use. I have too many tools in my woodshop. There’s a core set that I use all the time and by the time I get the plow plane out of the drawer and set it up I could have cut the groove with a chisel. Had I whole bunch of pieces to do, I would have broken out the plow.

It’s good to examine these methods and understand how they work but I don’t think you need all of them. Better to invest in learning how to adapt problems to the techniques one does know, such as subtracting one so that the difference is even and the squares / anker method can be used.

I haven’t tried my hand at larger numbers. Those are bigger problems, more steps and might require a bigger toolkit. My experience as an engineer was that 2 digits was plenty for thinking things through. It was good enough to know that the idea probably worked and was at least worth checking more carefully with the computer. In most everyday situations 2 digits is plenty too. The whole Industrial Revolution was done with slide rules which are at most 3 digits.

That was very valuable for me. To have that kind of fluency, where I can just think through a problem where numbers appeared without without distraction or extra cognitive load. But hanging around here and looking through some books, I’ve gotten curious.

My calculation books refer to this method by the quaint name Aliquot Parts

36x112 = 40x112 - 10% = 4032

They have recipes for other fractions too. But I only skimmed the chapter. Now I have more to read :slight_smile:

Indeed you do!

Today I had to calculate 359 X 247.
This screams criss cross with negative numbers:

360 -1
250 -3 X


and then criss cross multiplication:

360 X 250 = 90 X 1,000 = 90,000(making use of the aliquot parts).
90,000 -3 X 360 - 250 = 90,000 - 1,080 - 250 = 90,000 - 1,330 = 88,670
(in my mind I do 90,000 - 2,000 + 670, using the complement of 1,330 from 2,000)

88,670 + (-1X -3) = 88,670 + 3 = 88,673

Checking the result. The 11-proof forces its way up:
88,673 - 88,000 - 660 = 13
13 - 11 = 2
So ‘2’ is the result of the answer.

359 - 330 = 29
29 -22 = 7

247 - 220 = 27
27 - 22 = 5

5 X 7 = 35
35 -33 = 2
So ‘2’ is the result of the multiplication of the number we start with.

Both end in 2, so the result is probably ok.

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I wonder about something. Can you guys brute force your calculations?

I ask this because if you can then you should definitely try to practice that. When I do small calculations like 253×368, I can, if I want to, brute force the calculation and perhaps have the answer in 5 seconds. Maybe you guys can brute force the fastest method to calculate something. It comes with a cost though, which is accuracy.

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What do you mean by brute forcing my calculations? I usually give my way of calculation when I talk about these things.
How is brute forcing different?

Brute forcing is not a technique, it is more like a state of mind. Normally you are “with the numbers” when you are calculating; you know where you are in the calculation, where you were before and where you are next. Brute forcing is like letting your calculating mind run ahead of you and you just have to quickly catch the numbers that it throws at you. With brute forcing you just have to trust your gut, or in this case, your brain that it spews out the right numbers.

All of us look sometimes at a calculation twice or check the steps again, even though your first calculation is probably right. Next time, try to calculate something and as soon as you have an in-between answer, force yourself to the next step in the calculation, no looking back, trust your brain.

When I do calculations, it is not a matter of “if” but often a matter of “when” I can get the answer. So, I tried to shorten the “when” part by removing the thing that slows me down the most, myself, and it works. I used to call it “turbo-mode” when I was younger but brute forcing sounds better, haha:)

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I sometimes do. Don’t know if we are talking about the same thing.
In your example of 253 X 368, I see 250 X 360 = 90,000 first.
I don’t calculate 25X36, I just know the result.

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I guess you would continue to solve the problem after calculating 250×360. The way you should then try to continue is by trying to speed up the process, as much as you can. You probably have to add or subtract in your process (I don’t know how you guys operate) and that is something you could try to brute force, speed up.

Brute forcing to me is trying to calculate the problem faster, that’s it.

About the closest thing I might describe as brute force is criss-cross but I generally see something immediately that I can use as a fact to make my life easier with multiplication.

It is quite possible that simply training to criss-cross to a very high level will give good results without the need to develop numeracy. Different horses for different courses. I find raw calculation comes with a higher mental load than understanding the properties of numbers in all cases but that’s me.

I don’t know how criss-cross works exactly but I do know that it involves some calculation, brute forcing could be possible with criss-cross.

To me, doing the raw calculation is much easier. When I see what others are doing when they try to solve a problem, I get all confused. They have to memorize and find patterns and that looks exhausting, haha. I remember when I saw Arthur Benjamin trying to square a 5 digit number for the first time and he kept saying different words and repeating those words and weird numbers came out and he was really trying, I think this might have been the point at which I realized that my brain works in a different way than other people. I still find it hard to believe that the only way most people can solve a simple 2 digit multiplication is through memorization and training and that a 3 digit multiplication is often considered hard.

Art Benjamin is just using a simple peg system and exaggerating so people understand the process.
Memory techniques are not necessary for simple calculation but they can be handy. I don’t tend to use them but I see a day coming. You are going to be extremely hard pressed to beat someone in speed when they can calculate as fast as you but see the shortcuts when they are there (if being fast is what you are shooting for). We all play with numbers for different reasons. Some like to entertain others. Some like to entertain themselves. Feel free to play under any rules or constraints that you like. I suspect that is what makes it fun.

In order for people to be just as fast as me or faster, they would need to use techniques and/or shortcuts already. Apparently it is not normal to be able to calculate 5 digit multiplications in your head without techniques or shortcuts or training. Only some savants can do this (which I am trying to find out if I am because there are more things that suggest this).

I wish I could share my experience with numbers. It isn’t just some party trick, it actually benefits me. It makes me feel like a computer sometimes.

Brute forcing as you called it is something I often trend towards but then attempt to ‘get out of the habit’. Simply because it leaves the comfort and rhythm out. If someone were to ask me if I am sure on my answer I would give it a good 30% even though it is mostly accurate. Brute forcing is good if you have a solid ability to calculate I would say, because you won’t make errors and it speeds up. It’s definitely something I will do if I raise my ability to calculate again. It’s also logical because you don’t really question yourself when someone asks you to calculate something like 8 x 8, regardless of how fast you are doing the process even if you don’t just employ answer memory.

This is definitely true but in my younger years I have done this out of sheer determination reminding myself of the numbers and just about having it click in my head, not in reasonable time though. The kind of restarting when you lose the number and taking excessive time, I have even calculated 7 digit multiplication in my head doing this but I had always employed the method that I don’t quite know the name of to multiply digits:
81x 81 ->
1x1 = 1
8x1+8x1 = 16
8x8 +1(from 16) = 65

=6561
essentially a method where you start from the right column then keep going diagonal across the numbers and eventually get to the front column. I actually did this right after learning my multiplication tables, never knew there was another way to do it until a bit later. This still counts as using methods though I suppose.

I just generally had a habit and still do to prefer mental calculations, particularly going to higher levels of mathematics, I used to always do 3x3 inverses for matrices in my head or 6 step differentiation questions. It would start much more difficult than I would end up doing it later on but I suppose this is a training effect, it also then tends to naturally employ the brute forcing you were describing, almost as if I am doing the steps in parallel.

Fascinating.

Can you tell us exactly what you are doing with the above numbers, and any other steps.

For example, is 250 X 360 = 90,000 one of the steps?

Is 25 x 36 another step?

Thanks.

This looks a bit like the criss-cross method but still impressive because you actually calculate, brute forcing should be possible. Kinma memorized his calculations, I don’t know if he can actually brute force them. Solving from memory is the fastest way to solve a problem but the thing is… you have to memorize it first and you can’t increase the speed of recall. I don’t have to memorize anything and you to some extend don’t either.

How long did it take you to calculate it? And did you have the exact answer or did you have some mistakes? I’ve only done a 7 by 7 digit multiplication twice. One was right and the other was off by 1digit. Instead of the answer 41,255,919,647,328, I would have 41,255,919,643,328. Calculations like these are too large for me to trace back my mistakes, 7 by 7 is a 49 step process. I do like the challenge of these calculations, it feels like a 49 round boxing match with numbers :slight_smile:

I find I can increase my speed of recall though I haven’t met many other people who do this. I find for numbers its good to memorize a base so you can generate/calculate the answers using this base.

I didn’t track my time exactly but I would say less than 30 minutes for sure.

I definitely had more mistakes than exact answers but I have been right on the odd times. My mistakes tended to be the odd digit somewhere.

I often remember just starting all over again once I wasn’t able to backtrack something.

Indeed, the challenge tends to make the calculations even more interesting. I actually found that mathematics in general is a little like this. When you move from basic arithmetic to calculus, linear algebra or even analysis depending on how you look at it, it’s basically doing multiplication alike all over again if you do it in your head that is. The process is different but the essence is one and the same.