Wow! Just wow! Well done. And in a short amount of time.
Excellent.
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Thanks, Kinma. 
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You really can’t do better than get detailed advice from Kinma. IMO those posts are worth money!
Methods are part of a toolkit. With experience you learn to spot which problems are easiest done this way or that. Every method is useful though you will favor some. Each one takes time to learn.
I say, learnt the squares up to 100. Turns out it’s not that much work and it’s a real boon.
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Somewhere in here, it was suggested it’s okay to bump an old thread. Man, am I putting that to the test, here.
I just discovered mental math, because of this exact thread. I was researching something else and stumbled upon it. Not only was I intrigued by the concept of “Mental Math,” especially since I was researching something similar, the idea of getting to choose a method appeals to people like me. Or people like us, right?
As an added bonus, I hadn’t participated in this forum for too long.
I learn through reading, so I went to Amazon, and discovered a book called “Mental Math: Tricks To Become A Human Calculator .” It’s not only the top rated in its category, it’s also ranked pretty high overall. Seemed a good place to start.
it uses something the author calls Ofpad, which I haven’t seen referenced here. It’s solid, but googling it seems to send me to mostly self-referential information.
And it is. I’m up to multiplication and progressing well, as long as I can see the original problem. Holding the numbers mentally is somewhat problematic. I can do it with effort, for the easier problems. I use a variation of the Dominic system for numbers, with nearly half of the two-digit combos being a number association link instead. GREAT for memorizing cards or long strings of numbers. But when I visualize Neil Patrick Harris shaving a dog, it may be easy to know that is 989165. It doesn’t help me to do calculations with the numbers.
Wondering if I should switch to a shape number system for mental math. It might help me visualize the actual numbers. I also noted in this thread people using the Major system. I’m familiar with it, but not practiced at making words from the sounds. I preferred Dominic with PAO.
Any thoughts on the best way to hold the numbers in memory? Does anybody use ofpad at all?
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It’s always okay (and encouraged) to bump old threads.
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We’ve just been saying how we want to go through the old threads and collect all the useful stuff.
I use the Major System. I never saw it as having any real advantage for computation. I can’t think of any reason why another well tested system should make a difference. If you are already part way down one track, I wouldn’t switch.
Keeping track of powers of ten is tricky for most. One approach is to break free of columns and use a | symbol so that you don’t have finalize each intermediate computation.
eg 522 = 25 | 200 | 4 = 2704
This allows overflows and even negative numbers. 9 can be written as 10-1.
There is a fair bit of discussion about this in some other thread which I can’t find right now.
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Not using memory techniques, what is the largest amount of digits you can do? Let’s say for a multiplication?
2x2?
3x3?
4x4?
5x5?
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Hope I’m understanding the question right.
I’m up to multiplication in the learning. (My progress slowed down due to work and writing.) I’ve got my tables down up to 12x12.
I’m doing 2 digits times 2 digits without aid, but I’m not always getting the right answer when not looking at the question.
The method is one of two, depending on the problem. In some cases I’m breaking up the multiplier.
53 x 24 =
(53x20) + (53x4) =
1060 + 212 = 1272
It’s when I begin adding the last step that I’m messing up unless I use a memory technique. And sometimes even with that, as I’m just beginning this step and still practicing.
The other way is rounding up.
87x99 =
(87x100) -(87x1) =
8700 - 87 =
8613.
I’m finding rounding up much easier.
I’m pretty good at single digit multiplication now, up to 5 digits times 1 digit.
Have a look here if you want to: How can I start mental calculating - #33 by bjoern.gumboldt
…or just half of 2400 and then adding 3 x 24
…using 100 as a base:
87 -13
99 -1
either across 87 - 1 = 86 or 99 - 13 = 86 or as 100 - 13 - 1 = 86 is your left and side and 1 * 13 is your right hand side.
Dear Scott,
I would advise you to play around with the criss cross method. As it is easier on the brain (less stuff to memorise).
In the case of:
53
24 X
You start with the numbers that result in the hundreds. So multiply 50 X 20 = 1,000.
Then the digits that result in the tens or 3 X 20 + 4 X 50 = 260.
Add to 1,000 gives 1,260.
Last step the 3 X 4 = 12.
Add to 1260 and get 1,272.
Always do either (or both) the 9 proof or 11 proof. I’ll start with 11:
53 mod 11 = 9 (subtract 44 from 53)
24 mod 11 = 2 (subtract 22 from 24)
9 X 2 = 18 and 18 mod 11 = 7
So the original numbers you started out with mod 11 or 9 and 2 multiplied together and. then also mod 11 ends up is 7.
Now the answer needs to also end up in 7 and it does:
1272 mod 11 = 172 mod 11 (subtract 1100)
172 mod 11 = 62 mod 11 (subtract 110)
62 mod 11 = 7 (subtract 55)
The 9 proof is easier to do, but I personally like the 11 proof because of the continued subtraction:
53 mog 9 = 8 (add the digits 5 and 3)
24 mod 9 = 6 (add the digits)
8 x 6 = 48 and 48 mod 9 = 3 (add digits: 4+8=12. Again add digits 1 and 2 = 3)
Now the answer mod 9 also needs to end up in 3 and it does.
1272 mod 9 = 3 (1+2+7+2 = 12 and 1+2 = 3)
Choose either the 9 or 11 proof or - and this is excellent training - do both.
As always a lot to type out, but with some proficiency it goes quick!
This is how you check your answer without a calculator.
Comparing the 2 methods.
Both work and both are excellent.
For you, I advice the criss-cross method. The reason is that you can do it digit by digit.
You first work out the hundreds, then the tens and last the ones.
In your method the first result is 1060, which means you need to need to keep both a thousand and a ten in your short term memory.
In the (criss) cross multiplication method you r first result is 1,000.
The second result is 26 (tens) or 260 which can be effortlessly added to 1,000.
And finally 12 can also easily be added to 1,260.
looking ahead
If you do this for a while you quickly start to look ahead.
Since your first result is 10 (hundreds) and your next step will be evaluating the tens there is zero change there will be a carry from the tens into the thousands. There will be a carry from the tens into the hundreds but not into the thousands.
Because there is no change of a carry into the thousands you know for sure that the first digit is 1.
So call out ‘one thousand and …’ and continue calculating.
Since you called out one thousand, there is no need to keep this in short term memory.
Just drop it! You will find that with practice your brain will still hold it.
If you ask yourself ‘what didd I just say?’, it will come back.
Next result is 260 and you look ahead at the 3x4. There is a carry into the tens, but not into the hundreds. The hundreds are certain. Now call out “.. two hundred and …”.
Now drop the 200 from memory and think of the 60.
Last result is 12. Add to 60 and call out ‘seventy two!’.
Now, from the result the most amount of digits you are working with is 2!
You dropped the 1,000 and the 200 and you only needed to keep 60 in short term memory in order to add to the 12.
So the most taxing thing is working out the cross or the tens (3 X 2 (tens) + 4 X 5 (tens) = 26 tens or 260). Form the 260 you only need to keep 60 in your brain while working out 3x4.
So with looking ahead you can call out the numbers that will not change in the calculation while working out the rest.
This alleviates the need to keep the whole answer in memory.
No need for any memory system until maybe you do 4 digit multiplication.
Love it!
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I don’t know if this is clear for everybody.
Check out Vedic Mathematics with a base.
Basically this is advanced cross multiplication.
in the previous example of:
53
24 X
we split 53 into 50 and 3 and 24 into 20 and 4.
Mentally try to see this as:
50 | 3
20 | 4 X
in your example of:
87
99 X
Try to see this as:
100 | -13
100 | -1 X
And start with 100 X 100 = 10,000.
Next step -13 x 100 + -1 X 100 = (-13-1) X 100 = -1,400. 10,000 - 1,400 = 8,600.
Last: -13 X -1 = 13
In the Vedic method you postpone the first multiplication with 100.
So you do: 100 - (13-1) = 86.
Then multiply with 100 or add 2 zeros: 8600.
So you look at the numbers from a “base 100”, meaning you look at 87 from 100 or -13.
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I just wanted to comment on a few unresolved or not very well summarized topics. The bottom line is that the human memory is limited to the point that it is extremely difficult to work with more than around 10 digits without some sort of mnemonic system, and as other people have commented, mnemonic systems can make it easier to remember things but they don’t store the information in a form that is convenient for calculating, so if you are going to use mnemonics, you will inevitably have to slow down a bit. The bottom line is that If you have to do a computation that requires keeping more than 10 digits in mind, it will probably be on the slower side.
As a practical consequence of this, it’s extremely hard for humans to do more than 3 digit x 3 digit multiplication entirely in their heads and get it right fast, emphasis on fast. Even calculation super geniuses find doing 4dx4d problems that are given to them verbally in their head fast to be hard. If you practice a lot and get to the point where you can do 3dx3d problems verbally in your head in less than 20 seconds, that is world class level, though still notably slower than world record level though.
If you remove that requirement that you have to do it fast, then it’s definitely possible with mnemonic techniques to compute massive products or any massive calculation slowly. Just use a memory palace for numbers to store inputs, intermediate results and answer digits in a logical fashion. And practice operating on digits stored in your memory palace and temporarily pulled into working memory.
Things become drastically easier if you have the problem written down in front of you and you can write the answer down as you go. Then suddenly speed and doing massive computations quickly is possible. The world record speeds for the various standard competition tasks correspond to averaging less than 0.1s per single digit operation required to compute the answer. Something like 8dx8d multiplication with the problem written down in front of you requires up to 64 multiplication 1d operations and 49 2d+2d or 3d+2d operations and up to 15 carries. World record speed for an average of 10 is around 14s to compute the equivalent of around 200 single digit operations.
If you want to do multiplication of big numbers, then the cross multiplication aka criss cross method is indisputably the only way to go. It can be done right to left or left to right. It’s lightly easier right to less because carries can’t affect the already computed results. 2dx2d had a special case where doing the middle part of the cross multiplication can arguably be an ok strategy, but rtl is indisputably theoretically the best.
There are special algorithms for square roots, cube roots, and division to make those easier and break them out into single digit operations that modify a running remainder.
And as already stated, there are an abundance of special cases and techniques for simplifying special cases.
But the bottom line if you want to be super fast is that you have to practice a lot and practice intelligently. Special cases only cover some small fraction of the whole space, and you just need to practice to the point that you can do everything, even brute forcing the worst possible cases fast.
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