Maths named after mathematicians

January 5, 2012 math No comments

A finishing carpenter friend of mine recommended I try to use Pythagoras’ Theorem when I asked him, out of desperation, for advice on a difficult math problem that I am stuck on.

When I reported no progress, he asked what other topics in math are named after people, after all, Pythagoras lived a long time ago.

So here’s a very simple list (for now not even in a good order). Please use the links to learn more about what is being named.

There are many, many more, especially from the last century. For example, I left out the Tutte Graph. Wikipedia has a list of things named after mathematicians, however it is still far from complete.

To finish, I’ll list one of a different kind: the Erdős number is the distance to Paul Erdős in terms of coauthorship. (My Erdős number presently is 4)

Of mice and traps

December 19, 2011 bc, economics, systems No comments

Two or three summers ago, we had problems with mice in our house. We have a live trap, sometimes it works. When there are not too many mice, it is quite all right.

I usually get the task to empty it, which is not much of a big deal. I take it to a sports field that is half a block away. After opening the trap, the mouse jumps out. The first two times, crows snatched up the mouse pretty quickly. So now I usually stay around for a while, and the mouse manages to disappear in the grass.

This time, while I was waiting, a man came by, walking his dog. He asked, so I explained, yes, it’s a mouse trap.

He became really angry. i was shocked, because to me it was all really harmless. He asked whether I knew that dogs use the field. He asked if he should call the police. I said yes, since he was pretty intimidating in his anger. Of course, he didn’t. I didn’t know what he was so angry about, I was just glad when he finally left.

Funny thing, and I did notice something odd during this confrontation, when I returned home with the trap, I thought to check it, and actually, yes, the mouse had returned to the trap.

I took it back right away (Mr. Angry wasn’t around anymore).

Later, thinking this over, the only explanation I came up with was that I never mentioned to him that it was a live trap; he may have thought it is a glue trap, or some kind of poison trap. I guess, in his mind, he would not like his dog to sniffle into one of those. So, at the time, using one of those was quite outside of my horizon. At the same time I think a live trap was completely outside of his horizon. Hence the dispute.


The live trap didn’t work well enough. After a few months we got an exterminator to help.

He was a nice man, and we had a friendly conversation about mice and this and that, so at the end I thought I’d ask him a question that I had been wondering about, especially with the saying in mind, “building a better mouse trap”: What would he think a mouse trap designed by Steve Jobs would look like?

He said, “Oh, the guy who does Microsoft?”

Well, no, that’s not who I meant. I think that would be quite a horrible trap. Spreading viruses comes to mind.


Quite a while later, I told an acquaintance about the exterminator and the Steve Jobs disconnect. She was super excited about the idea of a mouse trap designed by Steve Jobs himself. She said, it would be so nice and elegant, you would want to put one in every room of your home.

The One-Two-Three Man

December 19, 2011 bc No comments

A few years ago, returning from a bar (quite a while after midnight) and smoking a last cigarette on the porch of our home, I noticed a man walking in the middle of the street. Wearing a reflective west, and carrying a hockey stick, he kept saying “One, two, three, four, five, six , seven, eight, nine, ten, Push!” One word per step. This is in the dark, nothing else really going on in the neighbourhood.

I call him the One-Two-Three man. No-one else I asked knew about him, and I never saw him anywhere else.

But one day we were selling something through craigslist. I forgot what it was; it wasn’t anything special. The One-Two-Three man came to buy it!

I asked him about his “walks”. I think he said he couldn’t sleep, or wanted to lose weight.

Saw him yesterday again. (Except, this time there was no bar involved)

Accounting for a better world

December 30, 2010 economics, generic, systems No comments

In another weak moment I was pondering progress and whether in some sense people have become better, or the world has become better. It’s a huge topic of course, look it up through a Google search, “Is the world improving?” It will show you a lot of pessimism. On the other hand, Noam Chomsky, an unusually harsh critic (which is excellent since he knows and understands so much), said “Slowly, over time, the world is becoming more civilized, in general“.

My immediate prompt was seeing all this technology around me, an “iphone” here, SSD memory there, oodles of websites with free photos, people coming up with new stuff like bitcoin, quora,,, some dubstep music. It adds up, and it made me think how fast everything is nowadays, and whether people were really slower in the past. There is no one around coming up with loads of, for example, impressive symphonies or novels anymore, as far as I know. There’s loads of average products that one hears about, like for example, Windows Vista. I think being prolific doesn’t quite mean the same anymore. So it’s not clear cut to me.

How to measure?

Now my pondering led me to think about how could one measure this, how could one find an objective answer, how to do a better job at resolving the question. A connection with my personal ongoing tasks of filing tax returns led to this: as you’re probably aware, accountants are updated about all kinds of goings-on in a business. Wouldn’t it be nice if one could look up the “books” of fifty years ago, 25 years ago, and find “the data”.

I run a small company, The Buckmaster Institute, Inc, and I can well see how accounting is useful, if only in providing one perspective. Every little piece counts, after all. I can understand the accountant’s approach, I have learned from conversations with some professionals. I know about software and how it is used for accounting. I have recommended against the naive use of spreadsheets (still need to write those stories up). I have personal insights into the amount of effort that goes into accounting.

Resource Allocation

Have you heard about economists talk about “resource allocation”? For example, free markets are declared to be good at “allocating resources efficiently” by to some of those economists. Then this improvement in resource allocation should show up in the “books”, no? In as much as a drop in the efficient allocation of resources would be a sign that the world is getting worse, we might find evidence for the world improving in the financial statements of the businesses of today. Compare the figures to the financial statements from decades ago and we would see whether the world is getting better, maybe only in some small area. But it would be a start.

Doesn’t work

I think you’ll agree it’s not going to work. They very likely didn’t record the right numbers and amounts.

Should we call it -1 points for the accountants? Or would you go so far as assessing a full  -10 points?

Or do you disagree? Please elaborate by posting a comment below; that would be exciting to learn about! If you think a mere comment would not do your insights justice, please contact me (@stephanwehner on twitter, and we’ll take it from there)

Related Questions

Other slightly related questions that I have not been able to resolve are:

  • What are the recent breakthroughs in the accounting world?
  • Who are the heroes of accounting (what Steven Hawking is to black holes)?
  • Who do young accountants look up to and draw their inspiration from?
  • What are the present “open problems” in accounting? Which obstacles are hindering the full bloom of the profession?
  • Whither accounting: Is there a roadmap? Or is it all just  happy-go-lucky?


Any hint or pointer would be greatly appreciated! Don’t think your insight might be insignificant. Please don’t think there is a limit on space or time, and we cannot handle all responses. Please help by leaving a comment now. Thanks so much.

Couple of thoughts about evolution and economics

August 19, 2010 economics, systems No comments

I saw a documentary about Charles Darwin a few months ago. I think it was part  of David Suzuki’s The Nature of Things.

Evolution and natural selection are of course endless topics with many different and interesting aspects. So I’m writing here about a few thoughts that occurred to me while watching the documentary. Please leave corrections or pointers to related articles in the comments, if I got something wrong or missed something!

So, I used to think this was a good way to think of natural selection: Let’s say your eating potato chips, then towards the end, the bag has only small chips − the big ones are more likely to be pulled out. I’m sure this corresponds with your personal experience. Pretty intuitive isn’t it?

Well, watching the documentary, I realized that it’s not that easy. The potato chips example is primitive in the way that the chips don’t change over the time. They don’t interact while you’re eating. This simplicity might make it a good example, but it really doesn’t capture enough of the phenomenon.

The curious thing to me about evolution and economics is that I got the picture that people learned about evolution and what marvelous creatures are produced by simply letting natural selection do its work. Just around that time (late 19th century) it became popular to think: Why not build society on this principle? Let the best survive, why bother with the inferior (apply to employees/products/politicians/strategies)

And in as much as the theories that evolved (it might be fair to say this thinking led to neoclassical  economics) may be behind a lot of amazing features and products of modern society, I think overall there is also a sense of profound failure.

So what I wanted to write about here, is that in fact, these economic theories did not actually copy over the findings of evolutionary studies.

You see, there is  this distinction that biologists make between sexual and asexual reproduction. What biologists are telling us is that sexual reproduction can produce better organisms, better in the sense that they are more likely to survive, or better in the sense that they fill the available “space” more completely, are bigger, stronger, and more complex overall. (In general, what makes natural selection so general is that what “better” means is very open, and cannot be preconceived)

(Usually you think of sexual reproduction as it involving a male and a female producing offspring. A simpler variation of asexual reproduction may be that it simply takes two, instead of just one organism, to reproduce.)

Sexual reproduction can be viewed as unfair on an individual level, but also as a bad idea on a species level. After all, its form is: if a pair of individuals have the same sex, then they cannot mate, they cannot have any offspring; no matter what, and there are no exceptions. Who cares how superior they are, and even if they are the two best individuals of the species: tough luck.

So here’s a question: what would be the analogous set up of sexual reproduction in economics?

Here is one way: you divide the economic population (people + organizations) into red and blue people.  Red people can only work for blue people, and vice versa. Red people can only buy from blue people, and vice versa. What on earth would that look like? I really don’t know. At first glance, you might call this a sophisticated variation of racism—horror. However, biologists are hinting this might be a better system. Overall, the population is going to have a stronger economy; individuals get to follow their rational preferences,  maximize their utilites, while companies maximize profits, and everyone just acts independently on the basis of full and relevant information.

However, since I already mentioned neoclassical economics, I think the form of sexual reproduction, with its strong “You cannot mate” can also be found in the big pain point of the corporate world: regulations. After all regulations put constraints on what corporations can do. You have probably often personally heared about complaints from the business world. (In fact do they complain about anything else, besides taxes?) However, what we see in natural selection is that even the most basic, and brutally unfair, rule such as we find in sexual reproduction can actually work out.

Rock, Paper, Scissors with Handicap

May 10, 2010 rps No comments

So I was playing Rock, Paper, Scissors with my daughter the other day. (It was her idea; I didn’t tell her about the game, my previous blog post about Rock, Paper, Scissors, nor about the new site). She won, and I lost a few rounds. Then I noticed: she wasn’t using “Paper”!

So I  thought of this variation on the game: one player is not allowed to use one of the weapons. All other rules stay the same.

To make it more interesting, the other player would not know which weapon is not allowed. How to do that? The handicapped player chooses one card from 3, each card showing a different weapon. The player is not allowed to use that weapon. Only after the winner is determined is the card shown to the other player for verification. If the handicapped player cheated by using the forbidden weapon, then they lost!

I doubt this change gives the non-handicapped player an advantage.  But you never know. The number of draws should decrease, though. Then again, maybe not. Would be nice to figure it out (mathematically)!

Rock Paper Scissors for Three Players

April 5, 2010 generic, rps 3 comments

In the summer of 2009, if not to say, a long long time ago, Sharon Twiss asked on twitter: “can groups of more than two use Rock-Paper-Scissors?

In case you’re not familiar with the game,  Rock, paper, scissors (wikipedia link)  is played by two people. “Rock beats scissors, scissors beats paper, and paper beats rock”.

A basic way to play the game with more than two people

It took me quite a while to find this article about Rock, Paper, Scissors at, which explains how the game can be played by more than two:

The game is easily adaptable to more than just two players. This variant works remarkably well, even for large groups. The rules are the same, with the following exceptions:

  • If all three weapon types are played, or only one type of weapon is played, the round is considered to be a draw. A new round begins.
  • If there are only two different weapon types showing between all of the players, then all of the players showing the losing weapon are eliminated.

A different way for three players

Let us add another “weapon”, called Lizard, like this:

Weapons and winning sequence for 3 player Rock Paper Scissors

Weapons and winning sequence for 3 Player Rock Paper Scissors

(I based the illustration image on another one from wikipedia, by Nojhan )

The game for three players is played like the original version, with this order

  • Lizard beats Scissors
  • Scissors beats Paper
  • Paper beats Rock
  • Rock beats Lizard

Count to three, and on three, each player chooses one of the weapons with their hand.

How to determine the winner with the extra weapon?

What you do is look at the sequence of weapons chosen. For example, here

Blue player: Lizard, Yellow player: Scissors, Purple Player: Paper

Player Blue: Lizard, Player Yellow: Scissors, Player Purple: Paper

Player Blue chose Lizard, Player Yellow chose Scissors, and Player Purple chose Paper. Lizard beats Scissors, Scissors beats Paper. So you would say Player Blue, who chose Lizard, won, right?

What to do when two players choose the same weapon? Let’s say like this:

Blue and Yellow player: Lizard, Purple Player: Paper

Player Blue and Player Yellow: Scissors, Player Purple: Paper

Both Player Blue and Player Yellow chose Scissors, Player Purple chose Paper. Scissors beats Paper. There are two winners, so now Player Blue and Player Yellow play a round of 2-player Rock-Paper-Scissors to determine the winner.

Let’s look at this case,

Blue and Purple player: Paper, Yellow Player: Scissors

Player Blue and Player Purple: Paper, Player Yellow: Scissors

both Player Blue and Player Purple chose Paper, and Player Yellow chose Scissors. Scissors beats Paper, so Player Yellow won. No need for a 2-player round.

Obviously, if all players choose the same weapon, then it is a draw, and the game starts from the top.

Two different versions of this game

Now we come to a case that has no equivalent in the 2-player version:

Player Blue and Player Yellow: Lizard, Player Purple: Paper

Player Blue and Player Yellow: Lizard, Player Purple: Paper

Two player chose the same weapon, and the other player chose a weapon that neither beats the other weapon directly, or is beaten by it. (They are opposite in the circle).

Here I see two options. Either this is declared a draw, or the player who chose the single weapon is declared the winner. I think time will tell which is the better choice.


I’ll try to explain the advantage of adding another weapon in a separate post. Stay tuned!


Thanks go to my friends Sara and Gerry who played this game with me yesterday. Sara won.


All illustrations licensed under Attribution-Share Alike 2.5 Generic

Put a website together that allows you to play the game over the Internet: You can either play with one (standard “Rock, Paper, Scissors”), or two other players (“Rock, Paper, Scissors, Lizard”). Operated by my company, The Buckmaster Institute, Inc.

// This probably still needs some editing, will keep updating //

La Cumparsita: Piano Lesson with Linda Lee Thomas

March 19, 2010 music No comments

Last fall I practiced Nocturne No 9/2 by Chopin on the piano (on youtube: Horowitz, Rubinstein, Shebanova, Yundi Li). I found it pretty difficult, because it’s very delicate, and for me there was a lot more stretching, and the left hand moves a lot farther distances than in the Moonlight Sonata). The right hand needs to play louder than the left hand and the pedal is absolutely necessary.

“Discovering” La Cumparsita

I managed to play it reasonably well by the time we had our piano party, after which I wasn’t sure which piece to practice next. I looked through some score books that I discovered in my bookshelf. I could recognize most titles, and tried them out and nothing seemed appealing, when I got to a piece titled La Cumparsita. I didn’t know it, so I went over to trusty youtube, searched for “La Cumparsita piano“, and came across a lot of nice versions, but I found this rendition by Alberto Dogliotti just amazing. Isn’t that wonderful?

Little bit of background

La Cumparsita is originally not a piano piece. From wikipedia:… it was written by Gerardo Matos Rodríguez, an Uruguayan musician, in 1917. It is among the most famous and recognizable tango songs of all time. The title translates as “The little parade” and the original lyrics begin: “The little parade of endless miseries…” It was named cultural and popular anthem of Uruguay. (There are some more links further down)

Starting to practice

So I tried going through the scores that I had. They didn’t come out anything like what I found on youtube. I went to a music store, to buy scores, but they had none (to my surprise). So I searched the internet for other scores, and found this one. I practiced that for a while but didn’t find it too satisfying.

Pointed to Linda Lee Thomas

I mentioned my latest ambition to my friend Chris Startup, a professional Jazz Saxophone player. He recommended I should ask Linda Lee Thomas for help.I called her up and left a message and got no reply. So I thought of looking her up on the Internet. Oh my, she is the principal pianist for the Vancouver Symphony Orchestra. I mean I’m really not that good of a player. I called a subscriber of the Vancouver tango mailing list, Gabriela Rojo, if she knows someone and she also recommended Linda. Ok well, I called again. She was very encouraging! I told her I am kind of stuck because I have no scores, and she said that’s right, I need to “make something up”. She would be away to Argentina, the heart of Tango, until the beginning of March. We could meet after she returns.

“Making something up”

So ok, I tried to make something up. I simply changed the left hand from the scores that I had: just repeat the whole chord + the octave (in the scores the chord is broken up) in quarter notes, and at the last 8th play a half note down. I thought that sounds like tango, and it took me quite a while to complete that for the whole piece. I also modulated the volume a little bit (in fact it gets quite loud). Sorry, I don’t have a recording. Its really quite simple. Still it is enjoyable; for example, I heared my family humming the tune after my playing. On the other hand it is far from what I found on Youtube;  they do a lot more with the left hand, but I couldn’t figure out what it is.

The Lesson

So I thought that was a good basis for a lesson with Linda; I called her up at the beginning of the week, and we arranged to meet today! My hope was to find out what happens in the left hand. From what I could tell it was playing a few bass notes, then contributing to the melody, breaking up the chords into much less notes than my scores had, and how I was playing it.

So now let me explain what I learned at this lesson.

I started to play what I had put together. I didn’t make too many mistakes, still she praised me for getting this far by myself. She said I must have a good ear. I guess I cannot tell, but maybe that helps. However, she said the way I played it is the “American” way, as opposed to the “Argentine” way, the American way being associated with Ballroom dancing, which is looked down upon in the Latin tango world. Hey, that makes sense. I guess I was after that Argentinian way.

So she sat down and played the Argentinian way: yes that is great, that is what I want to learn. How to get there? I really need to practice a phrase over and over again, so on the spot I find it difficult to duplicate. So she reduced it to a lot less notes in the left hand. Let’s just break up the chord, play its notes as individual quarter notes, but then syncopate the first note – play it a little earlier.

Ok, I think I got that. Now we made it even more simpler: play the root note of the chord, and then the same note an octave higher. With the melody in the right hand that already sounds a lot better!

As far as I understood the right hand then plays the chord notes added below the melody note (which is pretty standard for the piano I guess), plus maybe an octave lower.

Well, that is as much as I could reproduce on the spot. She scolded me for using what she referred to as the American phrase, which is the last-but-one bar in the scores that I have. This is associated with the “American” tango. Instead, what she plays at the end of part A (the piece comes in parts A-B-A-C-A), is d-g, possibly completed to chords, possibly the chords broken up into very fast notes. It really sounds very nice!

Now I also asked her about how she starts the piece. While getting to understand the syncopation, she was playing the right hand (and me the left), but she would always have a nice introductory phrase. I wanted to know how that works – I’m not quite able to grasp whole chords and play them. So this phrase has two parts, the first one of which is just F-F#-G which she pointed out as an important element. After that follow three chords, G-minor, F-major, E-flat-major. Then with the D-minor chord the tune starts.

I asked about using the pedal. With my version, I couldn’t make out any use for it. But when playing just a few notes in the base, and the melody with higher notes, it makes sense to use the pedal to make the bass notes last.

She pointed out that Tango likes to take up the full range of the piano, all high and low notes, “just play it higher, and then lower”.

As I mentioned earlier, I couldn’t immediately play that much of what she showed me right on the spot. I hope that I now can better hear what is going on. She gave me a list of names to find artists  for inspiration:

I think what is not in the scores is what you can only learn from teachers, and really at the heart of learning music; here there are no scores available. So thanks a lot to you, Linda, for sharing your knowledge!!

Related Links

(Some of these also appear above) alive again

March 2, 2010 internet, programming No comments

My website is alive again. It’s about a nice math problem that I worked on 10 – 18 years ago.

webcrawlers desperate for content

January 27, 2010 internet, programming No comments

I recently found this in the web server logs of one of the websites I look after: - - [26/Jan/2010:05:01:44 -0800] "GET /application/json HTTP/1.1" 404 763 "-" "" - - [26/Jan/2010:05:01:47 -0800] "GET /following-sibling::* HTTP/1.1" 404 763 "-" "" - - [26/Jan/2010:05:01:55 -0800] "GET /AppleWebKit/ HTTP/1.1" 404 763 "-" "" - - [26/Jan/2010:05:01:58 -0800] "GET /following-sibling::* HTTP/1.1" 404 763 "-" ""

In case you are not familiar with web server log files, these line mean is that someone/something from IP address requested the pages named after “GET” on the website, for example, a page named “following-sibling::*” etc.

Does it need to be said that no such pages exist (that’s what the “404” indicates)?

When I saw this I was rather puzzled; and looked up (the last item on each line). Their home page says they provide some kind of vertical search service, whatever that is. On their FAQ page, I found this:

Why is your web crawler trying to access pages that don’t exist on my website?

Our web crawler attempts to extract links to valid web pages from javascript and other scripting languages. The crawler may misinterpret the information in these scripts and request a page that does not actually exist. These requests are attempts to retrieve valid web content, and are not an attempt to circumvent your webserver security.

(Emphasis mine) Oh ok. They are looking into javascript files on the web site and attempting to extract names of pages that might have content for the “vertical search”. But not successful in this case. As a web developer, I can tell you that javascript files very rarely contain interesting links to web pages.

Looks like a pretty competitive business when people start pulling at straws like this. Also I take it bandwidth is easier to come by than crawling software that avoids such silly attempts.