# Calculated 9472 × 3367 in 1min 47 sec

It’s my best score. Do you believe I can lower this score with training?

I do this closing my eyes.

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Can you describe your method more specifically than “With eyes closed”? Did you take advantage of the “11” in 33" I can’t do this mentally yet. Will lose my path. Are you using pegs? I don’t claim I can do this but this is how I would approach it IF i could keep the damned subtotal straight in my head.

Sadly, I can barely get this right using paper and pen…

Did you split into pairs?
The 1rst multiplication leverages 11’
9400 x 3300 = 3102[0000]

The 2nd multiplication leverages proximity to 100.
9400 x 67 = 6298[00]

The 3rd multiplication leverages 11’
72 x 3300 = 2376[00]

The fourth is an easy base to 70.
72*67 = 4824

\begin{align} 31,892,224 &= 9472 * 3367\\ &= (94 * 33)[0000] + (94 * 67)[00]+ (72 * 33)[00] + (72 * 67) \\ &= (94 * 11 * 3)[0000] + (( 100 * 67 ) - ( 6 * 67 ))[00] + (72 * 11* 3)[00] + (72 * 67) \\ &= (1034 * 3) [0000] + (6700 - 402)[00] + (792*3)[00] + (72^2 * 67^-3) \\ &= 3102[0000] + 6298[00] + 2376[00] + (70*69 - 6 ) \\ &= 3102[0000] + 6298[00] + 2376[00] + (70*70 - 76 )\\ &= 3102[0000] + 6298[00] + 2376[00] + 49[00] - 76 \\ &= 31,892,224\\ \end{align}

No way in heck I am currently doing this in 1min 47. But If you used this method I suspect you might beat your number if you did do a single digits multiply.

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Hi Robert.

I use cross-multiplication brute-force digit by digit with no split.

As student of Geography degree I remember the track due the fact I have memorized for numbers 0 to 9 geographical associations.

Example: 1 is UNO in my language, so doing the add I remember UNiverso (universe)

3 is TRES in my language, so I remember TRipoli, a town in Lybia.

Impressive! Congratulations!

Of course with training you can lower the amount of time used.

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Малкия го изчисли за 30 секунди

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Translation; my kid did this in 30 seconds.

Dear Rumyana Gesheva,

Or, like you did earlier, add a Google Translate.

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Please enter your kid to the World Cup of Mental Calculation. It would be a pity waste such a talent!

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I think we might need to be considerate here, guys.

AFAIK, Bulgarian women are not permitted to write English without special approval. But please don’t quote me on that. However, other people are allowed to translate on their behalf.

So, maybe the next responder can continue to provide the translation until we have more info on the legal or religious stuff.

I was just about to provide the translation myself, as a courtesy to women. But @Kinma beat me to it. Thanks for that.
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If there was only one earlier post, I think @Josh supplied the google translation. But there may have been other posts since.

Thanks.

What are the rules on mental calculation like this? Do you write the answer down, or recite it? Does the timer stop when you start writing/reciting, or when you finish? Are you allowed to write/recite the digits in reverse order?

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No problem. As a moderator I am allowed to add the translation. So if I see a Bulgarian text I will add a translation. If you see one that has no translation, just pm me.

Well I recited the answer but I think in officially competition you must write down the answer.

Do you have a palace for this calculation? If so what does it look like?

Are you doing 9000 * 3000 + 9000 *300 + 9000 * 60 + 9000 * 7 (and then) 4000 * 3000, etc?

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Hi. I dont know what a palace is.

I imagine, closing my eyes, the multiplication like its written down on a sheet. Then I do cross-multiplication digit by digit.

I remember the numbers in the answer because for each number write down I associate it with a geographical place.

For example 3 is TRes in Spanish and I remember TRipoli, a town in Lybia. The association is so non-sense that you wouldnt forget!

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If you are allowed to look at the question more than once, and you are allowed to write the answer from right to left, there’s no need to remember anything - you can just write down the digits as you calculate them. Using cross-multiplication working right to left, it’s just a matter of 16 single-digit multiplications, and adding some of the 2-digit results together.

Holding the complete question and answer in your head as @benjamin1990 did seems much harder!

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Holding the complete answer in your head is damned annoying. If I am not allowed to stare at the original question and I have to keep track of the subtotal I almost always stumble once things get long and the OP is definitely calculating using the all the steps (no shortcuts).

When I have familiarity, the mental load of adding the subtotal isn’t bad but/and maybe doing it the long way relieves you of counting 0’s BUT it is very easy to stumble if you look at the number, close your eyes, and recite the answer. I think this is the way it “should” be done in competition but I have no idea how it is done.

Using 2 digit numbers and the method I quoted above I suspect that pushing this specific calculation to 20 seconds to identify the calculations, 20 seconds for the 4 base calculations and another 20 seconds to try to keep the sums right is not outrageous. If you have strong familiarity with 2x2 multiplication 60 seconds should be achievable but using 1x1 multiplication I suspect 1min45 is extremely fast and there are a lot of opportunities to incur an error where it will be tough to use a checksum as you go.

Some of you guys are probably under 3 seconds for 2x2 multiplication for the ones identified. That brings the potential to around 30 to 45 seconds with some reasonable training. With the method pre-established I wouldn’t be shocked if one of you can do it in 20 seconds elapsed time but much under that might make me a little weepy as I can’t imagine touching that in this lifetime.

@albinoblanke and any of you quiet others, can I ask you to give your various approaches a try with this 4x4 and post your times so that we can get a few data points.

We don’t often chat about 4x4 and it would be good to have some practical comparisons on the actual effectiveness of the various methods once there are more things to think about than we might like.

I’m also interested in how you visualize your subtotal. In my world, I currently just keep the big number in my head and my intermediate calculation is fast enough not to lose the subtotal. If I stumble I have to start again. I have been practicing soroban and the major system on the side but I suspect that it will be a very long time before either of these offers an efficient/effective alternative.

I’m too chatty tonight…

When we are talking about the “Long” method here I believe we are talking about the following left to right method for 4 digits ( criss cross ) without the aid of a pen or looking at the question. “Short” methods use 2,3 and 4 digit numbers, recognize the properties of numbers and algebra, or leverage memorized facts to reduce steps or mental load.

16 discrete multiplications, some mental saving on not having to manage 0’s, nice simple facts. left to right.

Another way to think about the same thing

example of 2, 3 and 4 digits

and 5x5

This is NOT what I mean when I say the long way (God forgive the education system for the sins that the students will never recover from):

In Art Benjamin’s TTC - Secrets of Mental Math - 11. Advanced Multiplication he does a nice piece on 3 and 4 digit squares using knowledge of two digit squares and bases. He also does a nice job of showing how to cheat a little by saying and dropping the initial digits. I would never do that, but if I did I can see how even the opportunity to verbalize your result is going to help you hold it faster. The stunned look in my eyes and my slow motion speaking would certainly give me away. ( Yes that’s my boy Bob, he’s a little stupid… “You mean savant”… “Nope, he’s definitely an idiot.” )

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I actually have been lurking this whole time, haha. I was interested to see other people’s times and how they would approach this problem.

My time was 40 seconds. I could have done it faster but 3367 is not a nice number. I feel numbers and when I encounter a number I don’t like, it throws me off a bit. Especially a number with repeating digits, like the “33” in 3367. On the other hand, numbers I do like, I can do much faster than usual. I have done a small 4 by 4 digit multiplication in 15 second once.

I did the calculation in my head and recited it. I also only looked once at the problem for a second, then turned around and continued in my head. No method involved, no criss-cross or memory palaces etc.

How I calculated was simple:

9000×3000+400×3000+70×3000+2×3000.
9000×300+400×300+70×300+2×300.
9000×60+400×60+70×60+2×60.
9000×7+400×7+70×7+2×7.

It sounds strange but keeping all these numbers in my head is relatively easy. 2x2 digits and 3x3 digits multiplications are very easy, they are my beginner mode. 4 x 4 is like a normal mode to me. 5 x 5 digits is advanced/intermediate and at 6 x 6 starts the expert level.

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@albinoblanke … 16 multiplications … That is how I would have approached it a couple of years ago (without your speed and success) but I read a book by Euler that he wrote in 1765 called “Elements of Algebra”. - there is an English translation. (I wish all his works were translated I don’t know German) I highly recommend it. Since then I have been unable to unsee the numbers. My first real introductions to the drug wasn’t so much undergraduate calculus, linear algebra and statistics and probabilities as a book by Carl Boyer called A HISTORY OF MATHEMATICS by Princeton University Press, 1968. It was spell binding. People who prattle on about Goedel, Escher, Bach have no idea about this masterpiece of writing. You may wish to avoid googling on any of these as they will destroy your “purity”

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I hope to study mathematics at a university one day.
It is actually one of my dreams. I know I can do it, I am more than intelligent enough and with a memory like mine I am more than capable of learning a lot of material, I think I could bring a lot to the table. Unfortunately, I have fear of failure. It prevents me from taking any action to pursuit my dream. I am worried that I will never be able to learn mathematics because of this.

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