A Little Math Puzzle to Ponder

My good friend Dave Eldan sends me interesting tidbits on a regular basis. More often than not, they are pulled from obituaries. Everyone needs a hobby, I guess.

I found his latest missive very interesting. It is from an obituary by Morton White for the great philosopher and mathematician Willard Van Orman Quine. (Unless I am dreaming, I actually had dinner once with Quine at the Harvard Society of Fellows. He was almost 90 at the time and quite a wit. I remember when the topic of Thomas Kuhn‘s work on scientific revolutions came up, Quine shook his head ruefully and said, “Paradigm Lost.”)

White tells the following story about Quine: Upon hearing that White’s son Steve was not taught multiplication properly due to a switch in school districts, Quine sent him a letter with the following passage:

Since Stevie misses multiplication, he may enjoy gorging on this one. Well, so you have these two numbers, see, and you want to multiply one of them by the other. O. K., so you write them down more or less side by side, as potential headings of two potentially parallel potential columns, roughly thus:

19 27

Then you go to work on the left one, cutting it in half. Write the half underneath. No fractions, though. If it was odd, just forget the fraction. If it was 58 .. 537, just put down 29 .. 268 as its half. That’s near enough. Then, under that in turn, put its half (ignoring, again, the fraction if any); and so on, until you get down to 1. That completes your left-hand column.

19 27

Then go to work on the right-hand number, making a column under it by exactly the opposite method: doubling each time. This could go on forever, but don’t let it. Keep your right-hand column lined up with your left-hand column, entry by entry, and stop as soon as you are opposite the bottom of your left-hand column.

19 27
9 54
4 108
2 216
1 432

O. K., so now you have the two columns side by side. The next thing to do is to start in on the right-hand column and cross out a lot of it. Cross out all the entries which have even numbers opposite them in the left-hand column. Keep only those entries in the right-hand column which have odd numbers opposite them in the left-hand column. All right, now add up the right-hand column, what’s left of it after all the crossing out. The result, unless I have made a mistake somewhere, is the answer to your original multiplication problem.

19 27
9 54
4 108 [XXXed out]
2 216 [XXXed out]
1 432
513 [19 * 27 = 513]

It may be that this information reaches Stevie a bit too late to be altogether useful. If I had tipped him off earlier, he would never had [sic] had to learn the multiplication table.

The algorithm actually works, which you will see if you play around with it. And indeed, although certain aspects of the storytelling are played up to make it seem like magic, after a little study it becomes clear (even to someone as mathematically handicapped as I am) why it works. If you like this sort of thing, it is fun to figure out.

Do any readers know who first invented this trick? I doubt it was Quine, but it’s possible.


It is essential using a binary system instead of decimal. The left column is the 2 to the power of 0 . . .1 . . 2. . 3, etc. You XX out the even, because it is like the zero's in the decimal system.


I have an old "odd facts" book at home (at least 25 years old) that has this trick in it. It attributed its development to Ethiopia. I'm at work right now, so I don't have the book's name handy.


I'm not sure who invented that specific calculation, but I know that Mike Byster teaches young students a number of "tricks" for how to solve complex math equations in their heads quickly. He was featured in an ABC 20/20 segment a couple of years ago.



I don't know who first came up with this algorithm, but my guess is that it was a Greek about 250 B.C.


Someday we'll get visited by Aliens and they'll go, "what the heck are doing not using a binary based numbering system"?!!!


Yawwwn. Old news.

This is known as "Egyptian Multiplication" and more recently as "Russian Peasant" multiplication. Wikipedia has a really nice explanation of how it works.


This is commonly known as Russian Peasant Multiplication, see http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml
for a tentative history and related discussion.


How can you mention Quine without mentioning the Duhem-Quine hypothesis? It pinpoints a big problem for analysis of the Freakonomics type.


RobertSeattle: It could well be that the aliens would expect us to us the base 12 system since it has a lot of virtues in the practical world. There are good reasons why we have 12 inches in a foot, beer in six-packs, cases of many products are 12 or 24 units, etc.


I had to write a basic program using this method for multiplication. It was attributed to Russian peasants who could add and subtract.


This is binary multiplication.

Dividing the left number by 2 (without remainder) and crossing out the even numbers locates the '1's in the binary representation of the left number (because multiplying by 0 is 0 so you don't need those). Doubling the number on the right is like how you add an extra '0' for each row when doing multiplication out long hand in decimal (doubling in binary is the same as multiplying by 10 in decimal). Adding them is the final step in decimal multiplication as well.


It's certainly cute, but it can't have been from Russian peasants who could only add and subtract. Perhaps Russian peasants who could add, and multiply by 2, and divide by 2 and round down.

Using multiplication and division in order to achieve multiplication is the kind of thing you call "academic" and mean it badly. :)


Jason: Yeah, but multiplying and dividing by two isn't the same difficulty as multiplying by 27.

Halfing and doubling numbers is a lot easier to do by eye than multiplying or dividing by larger numbers.

Or in particular, if you're losing the 1/2 part if you divide an odd number by two, all you're doing is bit-shifting... or if you're doing it by hand, at least you're working with a much smaller set of numbers to remember.

So, yeah: It may not be as straightforward as the "algorithmic method" we all learned in second grade or whenever that was, but it is reducing it to a different problem--if not easier, at least not a harder one.

Katie S.

For those who found Quine's description hard to follow, here is an implementation of his technique in haskell:

emult x y = sum [ j | (i,j) 0) (iterate (`div` 2) x)
y2 = iterate (*2) y


My guess (without having read the wikipedia article) would be that this has arabic origins. I've heard of many neat tricks like this, most of them far too cryptic to figure out so easily. I don't know if the ancient mathematicians even had proofs of all the methods, it's bizarre stuff like how to find the third root of a number close to 5000.

I don't see why binary is necessary to understand this trick though. Basically you're just playing with factors. Dividing one factor by 2 (or 3 or 42, whatever) and multiplying the other by the same number will keep the product intact due to the associative property of multiplication. When you drop a remainder, what you're really doing is subtracting 1 from the odd number ABOVE in the left column, therefore your final answer (432) is missing one group of each right column number associated with a left column odd (i.e. 54 and 27).

I hope that's clear, thats the way I see it without appealing to base-2.



Sorry Levitt, but I don't believe for a second that you're "mathematically handicapped."

Rita: Lovely Meter Maid





(collapses to floor, sobbing bitterly)...


"Someday we'll get visited by Aliens and they'll go, “what the heck are doing not using a binary based numbering system”?!!!"

I am guessing that those aliens will have 2 fingers? See, we have 10 - which is why we use decimal. There is really nothing magic about binary other than it is easy for transistors and, apparently, russian peasants to deal with.


Has anyone seen the method that involves drawing intersecting lines and counting the intersections? I may be behind that times, but I found that recently and was amazed. Why did't anyone teach us these types of tricks in school?


why use paper.. be a little economic and green.. use your head..

19x21 is 20^2-1^2 e.g in human language 400 less 1

then we lack 19x6 which is 20x6 less 6 = 120 less 6

so our brain tells us it's 400+120 less 1 and less 6.. 513...


The algorithm does not work for 29 x 31.