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A while back I came across an interesting method to do multiplication. I don't know what it's called and am interested in when (and who) developed this method.

I don't know if it's a mathematical curiosity or if it was developed in a time and place before hindu / place numbers became commonplace.

Say one wants to multiply 21 x 31

Draw 2 lines, leave some space and then draw 1 line. You've just written "21" (See: Figure 1)

For "31" draw 3 lines perpendicular to the original set of lines; then leave a space and draw one line.

enter image description here

To multiply one counts the intersection points at the top left to ascertain how many 100s.(In this case it's 6)

Then, to determine the tens, one counts the intersection points at the top right and bottom left (In this case it's 2+3).

To determine the ones place count the intersections at the bottom right. (In this case 1)

enter image description here

I've played with numerous variations. It gets a little tedious drawing all the lines for 52x78 but it works. (There is a carrying action when numbers pass 10).

TL/DR: What is this method called? When was it created?

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    $\begingroup$ According to the article "Chinese Stick Multiplication" this method dates from 11BCE China. Unfortunately, none of these claims include references. $\endgroup$ – Nick R Aug 7 at 23:27
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    $\begingroup$ @NickR Hi, Nick. The 11th century BC date is very doubtful, the actual source of the poster is most likely Talwalkar's May 2014 video, see below. $\endgroup$ – Conifold Aug 8 at 5:51
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    $\begingroup$ Given that this is clearly a base-10 methodology, it's extremely unlikely to be ancient. Consider the plethora of different "size" abacuses in use in various countries, for example. $\endgroup$ – Carl Witthoft Aug 8 at 12:15
  • $\begingroup$ @Conifold I believe the claim is for the year 11 BCE rather than the 11th Century BCE - which certainly would be beyond belief. Regardless, the precise date of 11BCE looks itself to either refer to a known written source (unreferenced) or is also rather suspicious in itself. Whatever the case, the source I have linked lacks credibility. $\endgroup$ – Nick R Aug 8 at 16:14
  • $\begingroup$ @CarlWitthoft I believe that this method would work in any base, would it not? It does however imply the use of a placeholder number system. $\endgroup$ – Nick R Aug 8 at 17:21
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It is a fun method but it appears to be very recent. It is characterized as Chinese, Japanese, Korean, Indian, or even Mayan method in various internet posts, all of them recent, and without attribution, naturally. The "ancient origin" story is most likely made up, it is reminiscent of the Lucas's towers of Hanoi hoax, still credited on the internet to the "ancient Indian priests from the Temple of Brahma", see How certain is it that Lucas invented the Towers of Hanoi puzzle? All current mentions seem to be traceable to a 2006 MetaCafe video by Akahad.

Of course, as some posters pointed out, the mathematical idea behind counting intersections is a truly old idea of adding partial products, see e.g. detailed explanation in Japanese Multiplication: The Real Reason Why It Works. Napier's bones (rabdology) is another implementation of the same idea. It was derived from the earlier gelosia method, described by Luca Paccioli in Summa de Arithmetica, Geometrica, Proportioni et Proportionalita (Everything about Arithmetic, Geometry, and Proportion, 1494), and transmitted to Italy from Indians or the Chinese through Arabs. The ancient Chinese did use sticks to implement partial products, but it was not the same, more table-like and suitable for abacus, see Numbers Through the Ages by Flegg, pp.190-2:

"Since the era before the birth of Christ the Chinese used little bamboo or wooden sticks as calculating pieces (chou) on their counting board... There is a lot of evidence that from the Warring States period (fourth and third centuries BC) calculations were made with stick numerals on a counting board, using a place-value system. Stick numerals are depicted on the coins of that period; they are also described in literary works. Perhaps the most famous instance of the latter is in the Tao Te Ching, where Lao Tzu says 'Good mathematicians do not use counting-rods'".

The question about the origin of the method of the Akahad's video was asked repeatedly on Math SE, see Where does the “Visual Multiplication” technique originate from? and links in the comment thread. @WillOrrick found a comment by Bill Hart to a video by Vi Hart that sheds some light on it. Apparently, Akahad learned the method from his Chinese girlfriend, who, in turn, learned it from her Chinese schoolteacher. She believed him to be the inventor. Here is the comment in full:

"Vi, you might be interested to know this method seems to have originated with a school teacher in China. It was first taught to a school girl in China. She taught it to her boyfriend, Akahad, who made a video on MetaCafe on Nov 16th 2006. Akahad was criticised for the fact that it is inefficient for numbers with large digits. However he claimed it was not intended to be an actually efficient method, but only "meant to be a little trick to show to friends and kids who hate maths". The video was so popular it made $2000 in 4 days. The school teacher who introduced it apparently did so to get kids interested in maths and the criss-cross pattern was used because it reminded the school children of the stools they sat on. It is commonly referred to as the Vedic or Mayan or Japanese method. But perhaps we should be calling it a Chinese method (though there are other Chinese methods perhaps more worthy of the appellation)!?"

In 2014 Presh Talwalkar made another (YouTube) video about the stick multiplication, which received over a million views in a month. He then wrote a book about it, Multiply Numbers By Drawing Lines, and did some research on its origins. Bill Hart's comment above is taken from his post. Here is more:

  1. From my extensive research, line multiplication dates to Nov 16th 2006 from a Chinese teacher (see links below). If you know of an earlier source, please let us all know with proper proof.

  2. To people who say this method is well-known, so why isn't line multiplication mentioned on Wikipedia? It's really, really important that research meet certain standards to be part of academic literature. I would love to see the history of the method, and its uses, as part of the multiplication algorithm – like lattice multiplication is listed as a method. I think line multiplication merits an entry and mention, but the active community of Wikipedia can let be the experts. These are the closest examples of pages where I think the method could be mentioned (or have its own page too):

https://en.wikipedia.org/wiki/Multiplication_algorithm

https://en.wikipedia.org/wiki/Lattice_multiplication

  1. To my knowledge, I am the only person's who has published this method in a book Multiply Numbers By Drawing Lines

https://www.amazon.com/gp/product/B00N85DCTU

Of course I'm interested in other sources. I've been searching for 5 years and no one has yet sent me anything.

Sources for 1

What is the origin of line multiplication? Math StackExchange post authored by me July 2014

https://math.stackexchange.com/questions/877120/what-is-the-origin-of-how-the-japanese-multiply-line-multiplication

Alas, the Chinese schoolteacher's name is not mentioned by anyone. Perhaps, he was inspired by the ancient stick numerals, but we may never know.

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    $\begingroup$ This looks like the perfect answer. Still, I'm going to wait a while before accepting it as that seems to be the preferred stackexchange way of doing things. $\endgroup$ – Mayo Aug 8 at 12:56

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