Cosma’s review of Stephen Wolfram, linked to below, says that:

Wolfram refers incessantly to his “discovery” that simple rules can produce complex results. Now, the word “discovery” here is legitimate, but only in a special sense. When I took pre-calculus in high school, I came up with a method for solving systems of linear equations, independent of my textbook and my teacher: I discovered it. My teacher, more patient than I would be with adolescent arrogance, gently informed me that it was a standard technique, in any book on linear algebra, called “reduction to Jordan normal form”, after the man who discovered it in the 1800s. Wolfram discovered simple rules producing complexity in just the same way that I discovered Jordan normal form.

I’m in no sense of the word a mathematician, but I too made a “discovery” in my teenage years, and found out years later that I wasn’t alone – Samuel Beckett, since we’re already talking about him, describes the technique in *Watt*. In Beckett’s words:

In another place, he said, from another place, he might have told this story to its end, told the true identity of Mr Nackybal (his real name was Tisler and he lived in a room on the canal), told his method of cube-rooting in his head (he merely knew by heart the cubes of one to nine, and even this was not indispensable, and that one gives one, and two eight, and three seven, and four four, and five five, and six six, and seven three, and eight two, and nine nine, and of course nought nought).

In other words, each single digit number has an unique cube, and if you know this cube, and do a bit of memorization (e.g. that the numbers 0 to 9 have cubes between 0 and 729, that 10-20 have cubes between 1000 and 8,000, and so on), you can derive the cube roots of quite large sounding numbers very easily (as long as they’re whole numbers). For example, to figure out the cube root of 103,823, the final digit is a 3, which means that the final digit of the cube root is 7, and since 103,823 is between 64,000 (the cube of 40) and 125,000 (the cube of 50), the cube root has to be 47.

I’m presuming that if this trick occurred to me and Beckett independently, it must be common knowledge, but haven’t seen it written up anywhere else. I’d be curious to know if someone else (Martin Gardner???) has described it.

{ 22 comments }

abbistani 07.25.06 at 9:07 pm

It’s described in a book of such tricks called “Mathemagics” by Arthur Benjamin and Michael Brant Shermer (cover blurb by Martin Gardner :-). They don’t say anything about its origin, but I would guess that its been around for quite a while.

Tim Lambert 07.25.06 at 9:29 pm

Yes, I recall Gardner writing about this. Fifth roots are even easier, because the last digit of fifth powers is the same as that of the original number.

justin 07.25.06 at 9:50 pm

I had a friend in high school chemistry who, missing a crucial page of the homework assignment giving the formulae, derived for himself both Charles’ and Boyle’s Gas Laws using the inputs in problems and the answers in the back of the book.

Jason Kuznicki 07.25.06 at 10:01 pm

All that needs to be known about Wolfram comes from the title of his book

A New Kind of Science: “Kind of” is a wordy locution that can often be discarded. But take it away, and the title is absurdly pretentious.Benjamin Nelson 07.25.06 at 10:33 pm

The worst feeling is not knowing whether or not the ideas you have are truly original or not. At least people who know that they’re reinventing the wheel don’t have to worry about it. If you don’t know who’s done what, then you have this pressure to publish your pet project sooner rather than later, regardless of how comfortable you feel with the ideas and their implications; and that sooner-over-later impulse can lead to hasty work. I guess it’s a balancing act between being like a Onsager or a Wolfram, if you have access to smart people in the field who would know where to start looking for similar research. But those who don’t have access, don’t have even the ghost of a chance, for the same reason that Onsanger didn’t have a chance in his novel bits of natural science: disciplinary boundaries.

Harald Korneliussen 07.26.06 at 1:18 am

I made a discovery myself some years ago, of a cryptographic technique known as key strengthening. It had been invented before, but only by a couple of years! Frustrating, isn’t it? So close!

It should also be said that it isn’t a very useful technique, so it probably wouldn’t have brough me fame and fortune.

Brett Bellmore 07.26.06 at 5:35 am

Welcome to the club; In Jr. High I ended up deriving the transfer elipse equations from basic physics, because I was space obsessed, and couldn’t find a book on celestial mechanics in the school library. Of course, I had the advantage of knowing they existed, which was a great help.

chris y 07.26.06 at 5:49 am

I get the impression that small kids do this a lot. I had several blinding “insights” about numbers when I was 7 or 8 which came down to using pattern recognition to solve simple equations. A number of my friends used to do the same sort of thing.

I wonder if children are innately better at seeing such patterns than most adults

[e.g. “Daddy, if you multiply a number by itself it’s always one more than if you multiply the number below it by the one above it” : n^2 = (n-1) * (n+1)]. Are competent mathematicians the people who don’t lose this ability in later life? And why do the rest of us not keep it?Belle Waring 07.26.06 at 7:16 am

when I was 10 I “discovered” short division because I was tired of writing out long division. my teacher wasn’t very happy about it because I wasn’t showing my work. I’ve completely forgotten now, but I imagine if I had to do 20 long division problems a night for a while it might come back to me.

chris y 07.26.06 at 7:24 am

I imagine if I had to do 20 long division problems a night for a while it might come back to me.Or you could use the calculator on your PC, which is why nobody under 25 can do arithmetic any more.

[Disclaimer: if Belle is in fact in her teens, I apologise fulsomely]

Jacob T. Levy 07.26.06 at 7:30 am

My third-grade pattern discovery was squares and odds:

n^2 + (2n+1) = (n+1)^2

not that I knew the notation at the time, or how to prove it algebraically; I just wrote down lots of square numbers in a sequence down one side of the page, and the sequence of odd numbers down the other side and finding that the gaps between sequential squares were sequential off numbers. I was ver excited, but nothing seemed to follow from it, and there was no one around to show me how to abstract it into the algebraic form.

mollymooly 07.26.06 at 7:41 am

Number 5: “not knowing whether or not the ideas you have are truly original or not”. Wouldn’t it be great if there was a way to find out? A kind of “engine” for “searching” some vast repository of knowledge? Gee, I think I’m on to something here…

coujou 07.26.06 at 7:58 am

Here is a method to calculate cube roots of 9 digit numbers. I didn’t “discover” it myself (except if you count finding the web page as a discovery).

http://www.1729.com/blog/CubeRoots.html

kharris 07.26.06 at 9:54 am

I think Belle has the key to Chris Y’s question. Kids do the same operation over and over. In those repetitions, kids have the opportunity to see patterns if they exist. Later in life, we use intermingle various operations. Less opportunity to recognize patterns. Could Wolfram’s assertion of a “discovery” be “youthful” in a similar sense? Seeing things that other have seen, but not knowing that others have seen them?

Jeffrey 07.26.06 at 10:14 am

Your essential disclaimer “as long as they’re whole numbers” needs to be clearer; namely, that it applies both to the cube root and the “quite large sounding number”.

For example, try applying your method to the number 207,646. Your method gives an answer of 56, whereas the real answer is 59.2162893.

Henry 07.26.06 at 10:20 am

bq. Your essential disclaimer “as long as they’re whole numbers” needs to be clearer; namely, that it applies both to the cube root and the “quite large sounding number”.

But surely any number that has a whole number as a cube root is itself _ipso facto_ a whole number???

pdf23ds 07.26.06 at 11:13 am

The ambiguity is caused be the alternate interpretation that “as long as they’re whole numbers” applies *only* to the cubed number.

J. Ellenberg 07.26.06 at 11:26 am

I once had a calculus student who had thought quite hard about the meaning of infinitesimals, and had in some sense independently arrived at the conclusion that nonstandard analysis could be interesting. He called his infinitesimals “Brian numbers,” after himself. So I set up a meeting between him and the person in our department who knew nonstandard analysis. Unfortunately, once Brian found out that others had walked the path before him and that “Brian numbers” was not going to become standard notation, he lost all interest.

BruceR 07.26.06 at 2:47 pm

#8, you’re missing a “-1”, I believe.

eudoxis 07.26.06 at 7:43 pm

That reminds me of a Vedic Math book I recently picked up. 16 sutras! The problem with the nifty tricks is that they work for a great many numbers, but not all numbers.

It’s an odd phenomenon that when a particular problem has been solved it seems to be easier for subsequent generations to solve. It’s almost as if a new piece of information has been added to the Platonic universe, now accessible to all.

Not to diminish the fun tricks that people come up with, but students of mathematics go through the derivations of ancient mathematical discoveries on their own all the time. Remember geometry and the Euclidian ideas you had to discover on your own? My son is presently at a number theory camp where, on a daily basis, the students are asked to derive proofs based on, often, unfamiliar material. There’s some rumbling of the “I’m not Gauss” sort, but the students do come up with the proofs. Once the 3 minute mile is broken, the 3 minute mile is easy to run.

chris y 07.27.06 at 4:17 am

#19 How true. See, when I was 8 and had never heard of algebra, I wouldn’t have made that error.

Benjamin Nelson 07.28.06 at 9:53 pm

Molly (#12), you could do that, if people named their discoveries plainly. But they don’t, so you can’t.

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