Feynman vs. the Abacus (1985) | Hacker News Reader

Feynman vs. the Abacus (1985)

ee.ryerson.ca

54 pointsmarcodiego38 comments

"The number was 1729.03. I happened to know that a cubic foot contains 1728 cubic inches, so the answer is a tiny bit more than 12."

There are people lauding and panning Feynman in this thread, but this story illustrates a very important lesson that I've seen a lot of very smart people fail to learn.

Whenever you're doing a calculation, especially if you're using a computer or calculator, make an approximate estimate of what the result should be. Don't just assume that whatever method you use will produce a correct answer.

A man with an abacus (or computer) is probably going to be faster than someone like Feynman most of the time, but he's also going to make mistakes. Big ones. He might not make them frequently, but mistakes always happen sooner or later. The trick is to catch them when they do happen. That's hard to do if you have absolutely no sense of what the answer should be. If, however, you start from a rough estimate using anything your brain can come up with, as Feynman does in his anecdote, you're a lot less likely to produce a badly wrong answer without realizing it.

If this contest had gone on long enough, Feynman would have been beaten badly in several rounds if he messed up a calculation and had to start over. The thing is, Feynman would know when this happened because the outcome wouldn't agree with his initial estimate. The man with the abacus would, eventually, have produced answers off by several orders of magnitude without realizing.

Rough estimates are important.

Back in the slide rule era, we had to learn this lesson. The slide rule would give you the (rather approximate) digits, but you had to know where to put the decimal point.

It feels like, in every one of Feynman’s arithmetic stories I’ve read, he always says how lucky he was that someone used a specific number.

At some point it stops being luck.

Or, when you enough special cases, you're almost always lucky to hit one of them.

Which is another way of saying what you're saying, I think.

Feynman actually touches on this in the same chapter, a few pages before the anecdote quoted in the article:

> I had a lot of fun trying to do arithmetic fast, by tricks, with Hans [Bethe] [...] He was nearly always able to get the answer to any problem within a percent. It was easy for him—every number was near something he knew.

“In the fields of observation chance favors only the prepared mind.”

Murray Gell-Mann did say once that he was annoyed by Feynman's tendency to invent anecdotes about himself. He wasn't lucky in the sense that he was a brilliant scientist but he also had a huge ego which he needed to validate publicly.

It may be annoying, but I'll grant he earned it. The world is full of gigantic egos with little of substance to say.

If I remember correctly, according to Leonard Mlodinow's Feynman's Rainbow, Gell-Mann wrote that in Feynman's obituary for the Physics Review. It "raised quite a few eyebrows".

He says the same in this interview too https://www.youtube.com/watch?v=rnMsgxIIQEE&t=109s

I don't doubt the claim, but this story is so specific ("the cube root of 1729.03") that it seems plausible there may be at least some truth to it. And it starts off with him being soundly beaten in adding and multiplying, so it's not totally egotistical. (I get that it's still pretty egotistical, since the parable is that he understands the actual numbers while the salesman is just using a rote, mechanical process without fully understanding the underlying theory.)

I could definitely see him totally making this up by working in reverse (decide to teach a lesson about mechanical vs. fundamental understanding, start with a tough operation, pick a mentally tractable number like a bit over 1728, tack on the simpler arithmetic contests before it to contrast and build tension), so I'm not saying it's a good argument for it being real. Just that it's tough to say one way or another.

That’s my favorite thing about reading Feynman’s autobiographical works or anecdotes about him. There’s a feeling that he must be embellishing, but the stories are never entirely outside the realm of plausibility.

it's like the math equivalent of being fielded softball questions

When I read his book I got the distinct impression that he grooms anecdotes. That is, he will subtly manipulate a situation (or the recounting of it) in order to make a better story. His stories are highly entertaining as a result.

A lot of us do this to some extent, don’t we? I mean, I’m no Feynman, but I have done some things thinking, if this works out, it’s going to be a great story. Sometimes there really is no other good reason for doing them.

I like, and respect t Feynman, but feel most of his stories are a humble brag.

I can only take so much of his writing. On a psychological level, I have thought about it over the years, and come up empty. I probally don't know enough about the man.

What got me thinking about it was one of his stories about the kid who could tell you what's wrong with your radio with his hearing.

I understand needing to protect your ego later in life when you didn't get the respect you deserved. In the stories, the trait started very young.

And maybe I'm completely mistaken? It just might be his way writing that has me wondering?

I get the opposite impression. The facts of many of these anecdotes would force you to believe that he must have unfathomable genius. But he honestly seems to believe that luck played a huge role; sometimes, the only role. I remember one story where someone puts a huge blueprint in front of him of some kind of plumbing installation of immense complexity. It’s not working right. He feels overwhelmed, so, just to have something to say, he points at random at a valve and asks what it does. That turned out to be the key that let them solve the issue. He thinks this is just blind luck, because he had no idea what anything did in the diagram. I think it’s more likely that his genius subconscious saw something.

I agree with you except for that last sentence—but in that anecdote he didn’t even know if the symbol he was pointing at was supposed to be a valve or a window. I think a much more likely explanation is just selection bias: if he’d asked “what happened if this valve gets stuck?” and they’d replied “oh, this pressure release over here will activate” the anecdote would have simply been too boring to make it into the book. Take the “best of” anecdotes from a busy lifetime and add a flair for storytelling and of course you’ll get an inflated view; the alternative would be a lot less likely to produce discussions on HN.

They said "When you grow up, you won't always have an abacus in your pocket!" And yet here we are.

And if anyone saw that coming it was Feynman. This isn't quite prophetic for 1959 but it's close:

> I don't know how to do this on a small scale in a practical way, but I do know that computing machines are very large; they fill rooms. Why can't we make them very small, make them of little wires, little elements – and by little, I mean little. For instance, the wires should be 10 or 100 atoms in diameter, and the circuits should be a few thousand angstroms across.

Quite a coincidence... Robert Noyce invented the first monolithic integrated circuit chip at Fairchild Semiconductor in 1959.

The you won't always have a calculator justification was always a poor one, even ignoring that we all carry computers these days.

You don't study mathematics just to improve your mental arithmetic. If mental arithmetic were the point, you'd just practice mental arithmetic for your whole mathematical education, rather than progressing to more advanced topics.

Flash Anzan at the All-Japan National Soroban Championship 2012: [1]

Background on how students train to do this using the Soroban (the Japanese Abacus): [2]

Another demo: [3]

Two nine-year-old girls play shiritori[4] ("a Japanese word game in which the players are required to say a word which begins with the final kana of the previous word") while adding 30 three-digit numbers flashed in 20 seconds: [5]

[1] - https://www.youtube.com/watch?v=7ktpme4xcoQ

[2] - https://www.youtube.com/watch?v=Px_hvzYS3_Y

[3] - https://www.youtube.com/watch?v=JawF0cv50Lk

[4] - https://en.wikipedia.org/wiki/Shiritori

[5] - https://www.youtube.com/watch?v=_vGMsVirYKs

One past thread:

Feynman vs. The Abacus - https://news.ycombinator.com/item?id=5849665 - June 2013 (33 comments)

I was always wondering what method the Abacus guy used for calculating the cubic root

Suppose you have a number:

  x = a0 10^1 + a1 10^0 + a2 10^-1 + ...

If you write down x * x * x, you get something like:

  x^3 = a0^3 10^3 + 3 a0^2 a1 10^2 + (3 a0 a1^2 + 3 a0^2 a2) 10^1 + ...

Now equate this term by term to your goal, which is:

  x^3 = 1 10^3 + 7 10^2 + 2 10^1 + ...

From the first term, you get:

  1 = a0^3

So a0 = 1, giving us an answer of 10 so far. Plug that in to the second term and you get:

  7 = 3 a1

which gives a1 = 2---you always round down. The answer is 12 so far. We have a carry of 1, which we need to add to the next one. That gives us:

  12 = 3 * 1 * 2^2 + 3 * 1^2 * a2 = 12 + 3 a2

Which leaves a2 = 0. So the answer is 12.0 so far.

As you go further, there are more and more terms hence the "scaling" phenomenon you see. Every time you are solving a polynomial equation in one variable, where the solution is an integer 0 through 9; my guess is that on an abacus you do binary search instead of root finding. On paper this sounds easy, but on the abacus it sounds impossible—each of those adds and multiplies is its own crazy sequence of steps.

1729 was also Ramanujan’s taxi number

https://mathworld.wolfram.com/Hardy-RamanujanNumber.html

it's because 12^3 differs from 1729 by just 1, making the error very small when using series

This is how I remember that 1728 = 12³. I remember getting puzzled looks from classmates in my elliptic curves class first when I was able to provide 12³ without a moment's delay and then again when I explained why I knew that.

feynman is always smart in his autobiography.

If you're gonna try and convince people Feynman wasn't smart, try a bible group.

Low-precision mathematics saves lives.

(Allowing wider margins of error and larger tolerances means better reliability and stability.)

I'll have to pull this one out next time I hear somebody complaining about the imperial system of measures being useless

nah, that's clearly an argument in favour of full blown duodecimal system.

1729 decimal is 1001 duodecimal, Every kid raised in a duodecimal system that had to learn how to convert into decimal in order to interact with us barbarians knows this fact.

I love the chaos of the imperial system. It has the taste of life itself.

The clock is base 12 because it has more rational fractions, those Babylonians!

Their number system was base 60, actually (one prime factor more).

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