This section should be entitled “Ways Other People told me about” because it pretty much comes from other people’s recent comments, with a couple of extras thrown in. But the best part? We’re up to 25 ways!
In an effort to appear organized, I created a single page with links to all of the ways to multiply. If anyone writes up any additional ways, post a link in the comments and I’ll add them.
On to the final(?) ways!
14 is 1110 in binary and 12 is 1100. Multiply the two in binary: 1100 x 1110 (think of 1110 as being on the bottom — I can’t draw it that way since the formatting won’t work). The partial products, in order, are 0000 (0), 11000 (24), 110000 (48), 1100000 (96). The nonzero products are copies of the top number, 12, shifted left — doubled — an appropriate number of times. Adding the partial products gives 10101000 (168).
(This is also mentioned by Jason Dyer in these comments.)
(20) per Repiego, described by Pat Ballew of Pat’sBlog in the comments here. This is listed by Pacioli in the Trevisio, and is essentially breaking a number into factors and then mulitplying by each of those in turn. So to multiply 14 by 12, for example, you might multiply 14 by 2 (getting 28), then multiply that product by 2 (getting 56), then multiplying THAT by 3, to get a product of 168.
(21) This method from Pappus, also described by Pat Ballew in this article. It starts off:
I will instead use 257 x 62 to shorten the process.
Write the first number on a piece of paper, and then the second number should be written backwards on a seperate piece of paper.
Align the second sheet under the first as shown below, and multiply only the numbers aligned vertically (the 7 and 2) to get 14. Record the four, and keep the one that is to carry in the mind (this is the mental part referred to in the title):
The bottom number is shifted, more pairs are multiplied and added, and you can read about the rest of the process here.
(22) This method from YouTube, which Jason Dyer of The Number Warrior pointed out in this post and in the comments here. It’s like a visual depiction of grid multiplication. It’s listed as Mayan multiplication, but I think that’s likely to be false: In Victor J. Katz’s A History of Mathematics he says that the Mayan documents that were not destroyed don’t show how the Mayans did calculation. (Hmmm. Katz also says the Babylonians used tables, and doesn’t mention those formulas I mentioned earlier.)
(23) This is described as the Prosthaphaeretic Slide Rule in this article (brought to my attention in these comments by Jason Dyer), but it really doesn’t use prosthaphaeresis at all. Rather, it’s a physical item that creates the similar triangles that the Greeks used for multiplication.
(24) Repeated Addition. Simple enough. It only works for integers, but that’s true of several of the other methods as well.
(25) On the fly shortcuts to Repeated Addition. For example, Jason Dyer’s example:
49 x 11 = (50 x 11) – (1 x 11) = 550 – 11 = 539
And we’re at 25 ways! Hooray!