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.
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:)
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
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.
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
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.
Do you mean that you would start all over again when you lose track of the numbers? Oof, thankfully I don’t have to do that, I can just do it one go. I can look back at my steps until I reach 20-25 steps, after that I just have to go forward and don’t try to look back because that costs too much time. It does drain me a lot, I think because I see the numbers and am calculating.
I hope to know all of that one day. I really enjoy math but because of anxiety I have a hard time forcing myself to study it. Just like numbers, I can see the equations in my head and manipulate them.
Sometimes, other times I am in a better condition and don’t have to start all over again. By losing track I mean when I am unable to recall which number was in that position, I don’t have to recall the entirety while doing the calculations, but I kind of feel it when its lost and often automatically do checks after every digit I have correctly calculated.
When you start over again it becomes a lot simpler than it was the first time since some of the memory remains. I also see the numbers when I am calculating but I usually have the answer digits darken out as I calculate the other answer digits. I don’t really find it functioning with sound, but at the time I did use sound to remind myself of the numbers as I went along. It’s simply faster to see the numbers to calculate the numbers, I find, rather than saying the calculation.
I’m sure you will, you can take things at the pace you find best , perhaps have a look at a few books until one really clicks with you. I also see the equations in my head when I manipulate them. I think the only time I don’t see things visually in my head with mathematics is when I am using proofs or definitions or part of my own verbal reasoning.
Not entirely right.
Difficult to explain though. On 25 X 36 the result forces itself. So I see 900 quicker than trying to actually do the calculation. Also the memorisation occurs when doing the calculation first. But it is not the answer that is memorised. It is the path to the answer.
What I suspect happens is, that my brain the path to the last time I did this calculation finds and then just redoes the same calculation.
What I did was 25 x 36 = 100 x 9.
Here is how I would do 253 X 368:
In this case , since 25x36 is easy to do I split the numbers in 2:
25 | 3
36 | 8 X
The “|” is just there to show where I split the number.
Then I do the criss cross using 250, 3, 360 and 8 as the numbers.
So to drive the point home, I treat it as a 2x2, when it is actually a 3x3.
Also I do this from left to right.
So first step is 25 X 36 = 900. Actually, in my mind I call out 250 X 360 = 90,000.
Doing it this way also gives me the ballpark number.
Then comes the cross.
8 X 250 + 3 X 360
8 X 250 = 2,000 and
3 X 36 = 1,080
Add this to 90,000 gives 93,080.
3 X 8 = 24
Add to 93,080 gives 92,104.
Then I check it with 9 or 11 proof.
On doing the above calculation, I realise I brute force the 11 proof.
1: I think “93,104 - 88,000” and immediately 5,104 comes up. No calculation. Maybe my mind does a quick 93-88, but the process is so quick that I don’t know for sure.
2: 5,104 - 4,400 = 704. Again; no calculation.
3: 704 - 660 = 44
4: 44 - 44= 0
This happens when I visualise the numbers. Only when I see both numbers, the result comes up.
Same with the numbers in the multiplication:
253 - 220 = 33
33 - 33 = 0
(actually I would stop here, since I now know that we will be multiplying with zero, so the end result needs to be zero. But for now let’s finish the 11 proof:)
368 - 330 = 36
36 - 33 = 3
3 x 0 = 0
Later today I realised this process does not always happen.
On a simple 45 - 18, nothing came up immediately.
When I raised the 18 to 20, then immediately 25 came up.
And then cascading, 27 came up as the answer.
For people who wonder why I am writing this; I am trying to find out when my brain immediately gives an answer and when not. See this thread and others.
Your brain immediately calculated the subtraction, this was not memorization. This is calculation because you probably haven’t memorized that 93,104 - 88,000 is 5104, that would be ridiculous and random. Brute forcing would be to try to do this every time with any calculation by concentrating and trying to mimic the instantaneous calculations, like the one mention above, that your brain does sometimes.