5X5 mulitiplications in 85 sec

Video demonstrations:
I may take a long break from calculatoin from now on, it’s really stressful keep thinking about breaking the record all the day.
There are still six months left in the year, wish after my mindset fully reset I could get under 60 sec by the end of the year.

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That’s a great score! You’re working at about 3 operations per second, which means you could theoretically do 8×8 multiplications in about 20 seconds each, and practically probably do 10 such questions in about 4–5 minutes, with some practice specifically on that size of question.

If you get to that stage, you would be a strong contender for a medal at Memoriad or the Mental Calculation World Cup.

Well done!

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Thank you Daniel

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I’m not sure what is criss cross. I use the vertical method I learned from kindergarten.

Since Nodas points out that there are only criss cross and its variations, maybe mine is a variation?

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That is a bold statement though. The criss cross method is the multiplication and addition of single digits.

345x673 starts with 3x5=15 or 3x6=18.

I start with 345x600: 345x100= 34,500 → 34,500x6= 207,000.

And you sort of follow the same method but without the zero’s, so 345×6= 2070. Please correct me if I am wrong.

To me, the way you and I calculate is very different from the criss cross method.

There is a reason why the criss cross method is the prefered method for most mental mathletes and not the way you and I calculate and that reason is because they are not the same or even a variation.

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I guess it depends on what you still consider a variation… you are going “criss” (x6) when you multiply with the first one and “cross” when you add for the subtotal; “criss” (x7) again when you multiply with the second one and “cross” again with the next subtotal.

If you went unit (3), tenth (7), hundredth (6) instead when doing x673, you’d even be able to write down the result (right-to-left that is) as you calculate; albeit, carrying still quite a bit after each digit you write down. The commonly used criss cross let’s you write down a digit and then carry less ‘subtotals’ forward… maybe not an issue with a 3x3 digit multiplication, but definitely not so much fun when it’s 8x8 digits.

…but maybe ask @Nodas what he means by that statement and if he still considers the above a variation.

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You pointed out a key difference between the methods; the amount of subtotals one has to hold in his mind.

I take into account the difficulty. To me, the difference seems too great to call it a variation.

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The 3 most recent record-holders for multiplication use variations of cross-multiplication:

  • Freddis Reyes (right-to-left, digits)
  • Jeonghee Lee (left-to-right, soroban)
  • Marc Jornet Sanz (right-to-left, digits)

So this corroborates Nodas’ claim that it’s the only way to go. There are some alternatives possible, such as described by albinoblanke, but in my experience cross-multiplication is faster.

I expect Nodas was referring to claims by people like Daniel Tammet that they can use other senses to avoid doing the logic required in a calculation. These claims are indeed false—everyone who has success in (standard categories of) mental calculation competitions is using a specific method. For multiplication, this is usually cross-multiplication (but doesn’t have to be!) but there aren’t people getting serious scores who can e.g. smell the answer.

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I agree. I have synesthesia but it plays no role when I am calculating, only when I am memorizing digits or just looking at digits.

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Absolutely, I was just trying to point out that it is not day and night but rather more day and later that day. Here’s another way this could be done… definitely after dark:

If you look at the problem for a second (time you don’t have in competitions), you’ll see that 500 is almost halfway between the two numbers. In fact, this would be a really easy problem using difference of squares if the problem were 345*\color{red}655 instead… just do 500^2-155^2 which is easier than it appears.

First for the obvious one: 500^2=250,000 and since 15|5 ends in 5 you can just multiply the left-hand-side (15) by “one more than itself” (see Vedic Maths), so basically 15^2=225 and then add another 15 on top for 15*16=240 and append the 5^2 to get 155^2=24,0|25. Just take the difference and you get:

345*655=250,000-24,025=225,975

Leaves you with the missing 345*\color{red}18 or more easily 345*(20-2) because then you just have to double 345 to get to 690, add two zeroes and then subtract the same number shifted by one decimal point.

345*(20-2)=6,900-6,90=6,210

In the very last step you just add your two results together:

345*673=225,975+6,210=\color{green}232,185

Admittedly, the problem was playing along nicely, especially the doubling minus 10% part because of the 18 in the second step. Also, the second square ending in a 5 in the first step made this a plausible way for people who haven’t memorized the “bigger” squares; however, ultimately you’re no better off using this method than doing what you were talking about because of the subtotals. Anyway, it’d call this way more different by comparison.

In competitions you’re not getting points for finding “pretty” solutions, so a simple criss cross here will be more efficient than the above. You’re probably halfway done by the time you see the pattern in the problem and by the time you get your first subtotal, you’d be done using a simple criss cross approach.

Anyways, figured this solution would be a nice one to read for people that can appreciate a bit of imagination. :wink:

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For 345 * 673 there is another “pretty” solution.
We have 345 * 673 = 345 * 672 + 345
Now we can factor 345 = 15 * 23
Therefore 672 * 345 = (672 * 15) * 23 = 10080 * 23 = 231840
The last multiplication is very easy because of the zeros in the middle and zero at the end
Finally 231840 + 345 = 232185

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@Kamil

Another easy way to do this -

673 × 345

There is a simple technique , we use when both numbers end with 5.

First factor of 345 = 115 × 3 (23 × 15 is also good but this factor came in my mind first and I think it’s easy)

We assume 673 to 675 (and in the last step we can easily subtract the remaining 2 multiplication)

Now here is our question -
675 × 345 (115 × 3)

Method :
Except last digit we have to multiply the remaining no and have to add the average of it.
Like this

Multiply - 67 × 11 = 737
Average - 67 + 11 / 2 =39
Add (776)

And because there is a 5 in the last we have to put 25 in the end (note if the average is odd we have to put 75 in the end)

So our answer is 77625 (and because we multiplied by 675 rather than 673 so we have that extra 2 , just subtract it)

77625 - 230 (115 × 2)
77395

Summary : multiply the number and add average of it (and forget 5 when calculating, if the condition - both no has 5 in the end.
And put 25 or 75 in the end)

And if you remember we factor like this
345 = 115 × 3

So last step is to multiply the answer by 3 .

77395 × 3 = 232185

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Compare my 360 seconds with the 85.48 of Flou.

Besides, he have 10 correct answers, when I have 5.

He’s amazing!

Bjoern and Albino. Nice discussion guys.

By 'criss - cross ’ and variations, I meant all the methods that every competitor uses in calculation competitions. I never met anyone who does NOT use criss-cross in competitions, and I have been in 8 competitions already (5 World Cups, 2 International Memoriads and 1 MSO )

Albino’s method is more intuitive and it reminds me of the method of Jacques Inaudi in the 19th century. That can be done for 3x3 or 4x4 sometimes

But when we are talking about 5x5 or 8x8 it is extremely hard to invent a method that is not criss cross.

But of course, multiplication is not only criss-cross.

Multiplication itself as an operation, is just an applied addition.

Example : Addition: 3+4 = III + IIII = 7 sticks
but Multiplication: 3 x 4 = IIII + IIII + IIII = 12 sticks

So in theory, you don’t even need any mathematics or any algorithms to find multiplications such as 5x5 digits.

You could just as well arrange sticks in rows and columns and count them all.

That 5x5 digit result is 9 or 10 digits long, and the average human lifespan in seconds is also 10 digits ( around 3 x 10^9 seconds = 3 billion seconds). So, it would take a human lifetime to count those sticks, when counting 1 stick per second.

But in theory, it’s possible. You can just write a program who will count these sticks in 10 nanoseconds.

But in mental multiplication the goal is to find the best personal algorithm so that you can make that operation faster, without having to count rows and columns of sticks. And of course, without using any computing power.

΄Mental΄ comes from Mente , which is the Latin word for Mind/Brain

Nodas

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Hot dayum comrad- how are you on division?

Hi @MagicallyDelicious,
I record it on another post

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Woo!