# 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
9
4
2
1

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.

#### Katie

In fact, it does not work for the number 29 at all. Why is that?

#### ThePolynomial

I think it does work for 29 x 31, Katie.
29 31
14 62[XXX]
7 124
3 248
1 496

31 + 124 + 248 + 496 = 899
29 x 31 = 899

Is that not right?

#### Matthew P

It seems to work for me with 29 x 31. I got 868 initially, then realized I should add the original 31 (since 29, from the left hand column, is odd). 868+31=699. 29x31=899.

#### Jacob Tomaw

I remember once reading in an algarithms book this method was devised by Muhammad ibn Musa al-Khwarizmi for whom algebra is named.

#### Matt

I tried it with 41 x 10 and it didn't work because of the elimination of the fraction when dividing 41. Am I doing something wrong? Or does this algorithm have holes?

#### Simon

Seems to work for me Matt

41 10
20 20 X
10 40 X
5 80
2 160 X
1 320

10 + 80 + 320 = 410
41x10 = 410

#### frankenduf

I tried it with 76376486374 x 277469471897 and it didn't work because of the elimination of the fraction when dividing 277469471897. Am I doing something wrong? Or does this algorithm have holes?

#### Katie

Ah-ha! So you have to add the original 31! That's where I was making my mistake.

#### Jacob Silj

Did Quine hate Morton White and/or his son Steve? This seems much more complicated that helping Steve to learn to multiply "normally."

#### tim

And then Quine made his son work on large karnaugh maps. Not half as bad as how McCluskey treated his kids!
... my two bits.

#### Mango

Jacob Tomaw: A brief look at the phonetics involved should make it questionable whether 'Algebra' was named after Al-Khwarizmi.

'Algorithm' is derived from his name. 'Algebra' is derived from the Arabic title of a book that he wrote.

I don't know where the method arrived from, but it's probably pretty old.

As it happens, computer science majors should need no convincing that this method works. A compsci geek reading the description would realize very quickly that it is essentially doing shift-and-add multiplication: this is how computers multiply numbers.

#### Greg

19 x 27? Come on, that's too easy without all this nonsense. 20 x 27 minus 1 x 27. 540 - 27 = 513.
But I was interested to learn this for the tougher numbers like 517 x 231.
517 231
258 462 x
129 924
64 1848 x
32 3696 x
16 7392 x
8 14784 x
4 29568 x
2 59136 x
1 118272
So I'll try to remember it to dazzle my son when he gets a little older.

#### McDuck

Had to try the performance difference with the standard integer multiplication in C:

int GetMul(int a, int b)
{
int res = (a != ((a >> 1) 1;)
{
a >>= 1;
b > 1)

#### Ero Carrera

To frankenduf. It works:

{{76376486374,277469471897}}
{{38188243187,554938943794}}
{{19094121593,1109877887588}}
{{9547060796,2219755775176}}
{{4773530398,4439511550352}}
{{2386765199,8879023100704}}
{{1193382599,17758046201408}}
{{596691299,35516092402816}}
{{298345649,71032184805632}}
{{149172824,142064369611264}}
{{74586412,284128739222528}}
{{37293206,568257478445056}}
{{18646603,1136514956890112}}
{{9323301,2273029913780224}}
{{4661650,4546059827560448}}
{{2330825,9092119655120896}}
{{1165412,18184239310241792}}
{{582706,36368478620483584}}
{{291353,72736957240967168}}
{{145676,145473914481934336}}
{{72838,290947828963868672}}
{{36419,581895657927737344}}
{{18209,1163791315855474688}}
{{9104,2327582631710949376}}
{{4552,4655165263421898752}}
{{2276,9310330526843797504}}
{{1138,18620661053687595008}}
{{569,37241322107375190016}}
{{284,74482644214750380032}}
{{142,148965288429500760064}}
{{71,297930576859001520128}}
{{35,595861153718003040256}}
{{17,1191722307436006080512}}
{{8,2383444614872012161024}}
{{4,4766889229744024322048}}
{{2,9533778459488048644096}}
{{1,19067556918976097288192}}

Sum of all right-values with an odd left-value = 21192143339542196431478

76376486374 * 277469471897 = 21192143339542196431478

#### Glenn Gillen

I remember in elementary school being amazed at learning a little trick if you were stuck on your 9 times tables. If you were stuck, you put your hands out in front of you, counted across whatever number you were trying to multiply from the left, and then turned that finger down. The number of fingers up to the left represented the number of 10's, the fingers to the right being the single digits (term??). So for 3*9 with both palms facing down:

count across 3 fingers to get to your middle finger, turn it down. Leaving pinky and ring up (2) and seven others (7) = 27.

Obviously only works for a very very narrow sample of numbers, but it was incredibly useful at that junior age when you got stuck

#### Gerald Deutsch

Being a fan of references to the "Duhem-Quine" thesis, and the ubiquitous dark murmurs that it would pose grave problems for, as posted by pvandervaart in a comment, in this special case for analysis of the freakonomics type (whatever that may mean, imo there is no analysis of the freakonomics type, Levitt just uses well known tools from statistics and economics on unusual data or asking unusual questions. Freakonomics ist just that combined with a well developed sense of how to cause sizeable uproars in the conventional wisdom camp, but I digress).

What is, not surprisingly also in this comment, usually missing is twofold. First and formost, the reference to Duhem might be omitted as he, other than Quine, limited the realm of his thesis to theoretical physics. It is only Quines thesis that might apply to economics. Interestingly neither Quine nor one of his followers ever bothered to supply conclusive evidence to the claim that all human knowledge suffer from being underdetermined. It is presumably precisely this lack of intellectual serenity that made Quine (not on his own free will, to credit him) so popular in post modernist circles.

I might want to conclude that, instead only claiming that thesis 'this-n-that' was introducing grave problems to a certain way of working and/or reasoning providing some evidence (or references to such) would probably be in order. Merely sounding plausible is not enough...

#### Gary

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.

#### David

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.

#### Jeremy

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.