Schemes like this multiplication technique are fascinating, the patterns one can see are beautiful. The fact they are not taught in schools is scandalous. Their use is actively discouraged.

My Dad bought me a book about the Trachtenberg System of Basic Mathematics[1] when I was a kid, it had all sorts of funky new ways to solve maths problems just like this. My primary school hated it, they marked me down for not showing my working and doing it the wrong way. I was brilliant at maths mainly because of this book, but didn't get the extra marks for doing it the "right" way.

http://en.wikipedia.org/wiki/Trachtenberg_system

Thank you. I too read that book as a child but I couldn't remember what it was called.

The only beef I had with the idea in it was that it was difficult to understand why they worked, compared to standard multiplication.

I used this a lot while writing a 16bit math system for the NES. A similar trick to this is Egyptian Division (http://everything2.com/title/Egyptian+division)

This can also be used for powers. Square instead of doubling, multiply instead of adding, and it works for matrices as well as numbers.

Of course this also works with bases other than two for your underlying number system. Or with mixed approaches.

In general, there's no known algorithm for finding x^n with the fewest numbers of multiplications in time polynomial in O(log n), i.e. the size of the representation of n.

(I can't remember whether the problem was (co-)NP complete, or even how to construct the certificate to proof that the problem is in NP or perhaps co-NP. Though I'd bet on it being in NP.)

And then you can also think about balancing the number of multiplications and the number of intermediate results you have to store.

Is this what you meant: Addition Chain Exponentiation; in general, performs better than binary exponentiation, but it could be very hard to find an addition chain for a given exponent: :-)

http://en.wikipedia.org/wiki/Addition-chain_exponentiation

Yes, thanks!

Take a look at Vedic Mathematics http://www.vedicmaths.org/introduction/tutorial/tutorial.asp. There are sixteen Sutras or principles that provide interesting hacks for quick "in your head" Arithmetic.

As an interesting aside, the algorithm for multiplication is algebraically identical in both methods.

Somewhat related: An elementary method of fast multiplication of large integers:

http://en.wikipedia.org/wiki/Karatsuba_multiplication

Polish Hand Magic is also an interesting trick to do multiplications: http://www.scientificamerican.com/article.cfm?id=parents-cor... A comic illustrating the trick: http://www.smbc-comics.com/index.php?db=comics&id=1914#c...

looks similar to the method used by the ancient Romans

http://www.phy6.org/outreach/edu/roman.htm

I've always enjoyed the "Mental Math" section of the "Mentat Wiki": http://www.ludism.org/mentat/

It covers 'tricks' for quickly and accurately calculating squares, cubes, logs and roots (up to fifth root). Worth a look.

iirc in fourier transforms, covolution of signals in time-domain is equivalent of signals in frequency-domain. thus 12 * 23 can be thought of as convolution of two signals (1,2) and (2,3) which gives (6,7,2)...

Yup. And you can then use the FFT to do super-fast multiplication. (I think you omitted something like "pointwise multiplication" after "equivalent of".) This is called "Strassen multiplication"; you can actually do better by doing your Fourier transforms not with complex numbers but with a different algebraic structure; that's called Schönhage-Strassen multiplication. I've seen it claimed that Strassen is actually faster in practice than Schönhage-Strassen.

Your numbers need to be pretty big before the benefits of this sort of algorithm outweigh the overheads, though. For instance, GMP uses naive multiplication up to about 600 bits, various Karatsuba/Toom-Cook schemes (divide-and-conquer) up to about 120,000 bits, and FFT after that.