The triangle paradox

This seems to be turning into Mathematics week at The “paradox” shown in the latter part of this article was presented to the author and some of his work colleagues at a recent seminar. It kept company with some well-know trompe l’œil such as:

Old or young woman?




Parallel lines?

However the final item presented was rather more worrying as it seemed to be less related to the human eye’s (or perhaps more accurately the human brain’s) ability to discern shape from minimal cues and more to do with mathematical fallacy. The person presenting these images (actually they were slightly different ones, I have simplified the problem) claimed that they themselves had no idea about the solution.

Consider the following two triangles:

Spot the difference...

The upper one has been decomposed into two smaller triangles – one red, one green – a blue rectangle and a series of purple squares.

These shapes have then been rearranged to form the lower triangle. But something is going wrong here. Where has the additional white square come from?

Without even making recourse to Gödel, surely this result stabs at the heart of Mathematics. What is going on?

After a bit of thought and going down at least one blind alley, I managed to work this one out (and thereby save Mathematics single-handedly). I’ll publish the solution in a later article. Until then, any suggestions are welcome.

For those who don’t want to think about this too much, the solution has now been posted here.

Half full, or half empty?

Glass half, er...

Someone being described as a “glass half-full” or “glass half-empty” sort of person is something that one hears increasingly frequently. I was recently discussing this with a friend and we both agreed that the analogy was unhelpful. First it supports a drastically simplistic and binary view of people having fixed attitudes and behaviours in all circumstances. Day-to-day observation suggests on the contrary that a person my be an avid optimist one day about one thing and a manic pessimist the next day about another thing. This rather shallow type of characterisation rather reminds me of some of the subjects I touched on in The Big Picture and Pigeonholing – A tragedy some time ago.

However, there is a more fundamental consideration; wilful inaccuracy. A glass that is half empty is also half full; that’s the definition of a half. Either description is 100% valid and therefore logically can tell you nothing about the person’s mindset.

Instead what might be more apposite is to adopt a different way to divide sheep from goats. This is still rather too binary for my taste, but at least it has the merit of a greater degree of rigour. I propose dividing people according to how they view a glass that is three quarters empty:

  • I still have some left: optimist
  • There isn’t very much left: pessimist

I think that all of our lives would be much the better for adopting this simple principle.

The International Organisation for stamping out sloppiness in spoken speech

Accordingly, I am going to submit this recommendation to the International Standards Organisation for their urgent consideration. I’ll make sure that I keep readers up-to-date with how my submission progresses.

Medical malpractice

8 plus 7 equals 15, carry one, er...

I was listening to a discussion with two medical practitioners on the radio today while driving home from work. I’ll remove the context of the diseases they were debating as the point I want to make is not specifically to do with this aspect and dropping it removes a degree of emotion from the conversation. The bone of contention between the two antagonists was the mortality rate from a certain set of diseases in the UK and whether this was to do with the competency of general practitioners (GPs, or “family doctors” for any US readers) and the diagnostic procedures they use, or to do with some other factor.

In defending her colleagues from the accusations of the first interviewee, the general practitioner said that the rate of mortality for sufferers of these diseases in other European countries (she specifically cited Belgium and France) was greater than in the UK. I should probably pause at this point to note that this comment seemed the complete opposite of every other European health survey I have read in recent years, but we will let that pass and instead focus on the second part of her argument. This was that that better diagnoses would be made if the UK hired more doctors (like her), thereby allowing them to spend more time with each patient. She backed up this assertion by then saying that France has many more doctors per 1,000 people than the UK (the figures I found were 3.7 per 1,000 for France and 2.2 per 1,000 for the UK; these were totally different to the figures she quoted, but again I’ll let that pass as she did seem to at least have the relation between the figures in each country the right way round this time).

What the GP seemed to be saying is summarised in the following chart:

Vive la difference

I have no background in medicine, but to me the lady in question made the opposite point to the one she seemed to want to. If there are fewer doctors per capita in the UK than in France, but UK mortality rates are better, it might be more plausible to argue that less doctors implies better survival rates; this is what the above chart suggests. Of course this assertion is open to challenge and – as with most statistical phenomena – there are undoubtedly many other factors. There is also of course the old chestnut of correlation not implying causality (not that the above chart even establishes correlation). However, at the very least, the “facts” as presented did not seem to be a prima facie case for hiring more UK doctors.

Sadly for both the GP in question and for inhabitants of the UK, I think that the actual graph is more like:

This exhibit could perhaps suggest that the second doctor had a potential point, but such simplistic observations, much as we may love to make them, do not always stand up to rigorous statistical analysis. Statistical findings can be as counter-intuitive as many other mathematical results.

Speaking of statistics, when challenged on whether she had the relative mortality rates for France and the UK the right way round, the same GP said, “well you can prove anything with statistics.” We hear this phrase so often that I guess many of us come to believe it. In fact it might be more accurate to say, “selection bias is all pervasive”, or perhaps even “innumeracy will generally lead to erroneous conclusions being drawn.”

When physicians are happy to appear on national radio and exhibit what is at best a tenuous grasp of figures, one can but wonder about the risk of numerically-based medical decisions sometimes going awry. With doctors also increasingly involved in public affairs (either as expert advisers or – in the UK at least – often as members of parliament), perhaps these worries should also be extended into areas of policy making.

Even more fundamentally (but then as an ex-Mathematician I would say this), perhaps the UK needs to reassess how it teaches mathematics. Also maybe UK medical schools need to examine numeric proficiency again just before students graduate as well as many years earlier when candidates apply; just in case something in the process of producing new doctors has squeezed their previous mathematical ability out of them.

Before I begin to be seen as an opponent of the medical profession, I should close by asking a couple of questions that are perhaps closer to home for some readers. How many of the business decisions that are taken using information lovingly crafted by information professionals such as you and me are marred by an incomplete understanding of numbers on the part of [hopefully] a small subsection of users? As IT professionals, what should we be doing to minimise the likelihood of such an occurrence in our organisations?

No-fooling: A new blog-tagging meme – by Curt Monash

Software Memories - a Curt Monash blog

By way of [very necessary] explanation, this post is a response to an idea started on the blog of Curt Monash (@CurtMonash), doyen of software industry analysts. You can read the full article here. This is intended as an early April Fools celebration.

A summary:

[…] the Rules of the No-Fooling Meme are:

Rule 1: Post on your blog 1 or more surprisingly true things about you,* plus their explanations. I’m starting off with 10, but it’s OK to be a lot less wordy than I’m being. I suggest the following format:

  • A noteworthy capsule sentence. (Example: “I was not of mortal woman born.”)
  • A perfectly reasonable explanation. (Example: “I was untimely ripped from my mother’s womb. In modern parlance, she had a C-section.”)

Rule 2: Link back to this post. That explains what you’re doing.
Rule 3: Drop a link to your post into the comment thread. That will let people who check here know that you’ve contributed too.
Rule 4: Ping 1 or more other people encouraging them to join in the meme with posts of their own.

*If you want to relax the “about you” part, that’s fine too.

I won’t be as dramatic as Curt, nor will I drop any names (they have been changed to protect the guilty). I also think that my list is closer to a “things you didn’t know about me” than Curt’s original intention, but hopefully it is in the spirit of his original post. I have relaxed the “about me” part for one fact as well, but claim extenuating circumstances.

My “no-fooling” facts are, in (broadly) reverse chronological order:

  1. I have recently corrected a Physics paper in Science – and please bear in mind that I was a Mathematician not a Physicist; I’m not linking to the paper as the error was Science’s fault not the scientists’ and the lead author was very nice about it.
  2. My partner is shortly going to be working with one of last year’s Nobel Laureates at one of the world’s premier research institues – I’m proud, so sue me!
  3. My partner, my eldest son and I have all attended (or are attending) the same University – though separated by over 20 years.
  4. The same University awarded me 120% in my MSc. Number Theory exam – the irony of this appeals to me to this day; I was taught Number Theory by a Fields Medalist; by way of contrast, I got a gamma minus in second year Applied Mathematics.
  5. Not only did I used to own a fan-site for a computer game character, I co-administered a universal bulletin board (yes I am that old) dedicated to the same character – even more amazingly, there were female members!
  6. As far as I can tell, my code is still part of the core of software that is used rather widely in the UK and elsewhere – though I suspect that a high percentage of it has succumbed to evolutionary pressures.
  7. I have recorded an eagle playing golf – despite not being very good at it and not playing at all now.
  8. I have played cricket against the national teams of both Zimbabwe (in less traumatic times) and the Netherlands – Under 15s and Under 19s respectively; I have also played both with and against an England cricketer and against a West Indies cricketer (who also got me out), but I said that I wasn’t going to name drop.
  9. [Unlike Curt] I only competed in one chess tournament – I came fourth, but only after being threatened with expulsion over an argument to do with whether I had let go of a bishop for a nanosecond; I think I was 11 at the time.
  10. At least allegedly, one of my antecedents was one of the last hangmen in England – I’m not sure how you would go about substantiating this fact as they were meant to be sworn to secrecy; equally I’m not sure that I would want to substantiate it.
  11. And a bonus fact (which could also be seen as oneupmanship vis à vis Curt):

  12. One of the articles that I wrote for the UK climbing press has had substantially more unique views than any of my business-related articles on here (save for the home page itself) – sad, but true, if you don’t believe me, the proof is here.


Other Monash-related posts on this site:



Alice consulting with an industry expert

  `As I was saying, that seems to be done right — though I haven’t time to look it over thoroughly just now — and that shows that there are three hundred and sixty-four days when you might get un-birthday presents –‘

`Certainly,’ said Alice.

`And only one for birthday presents, you know. There’s glory for you!’

`I don’t know what you mean by “glory”,’ Alice said.

Humpty Dumpty smiled contemptuously. `Of course you don’t — till I tell you. I meant “there’s a nice knock-down argument for you!”‘

`But “glory” doesn’t mean “a nice knock-down argument”,’ Alice objected.

`When I use a word,’ Humpty Dumpty said, in rather a scornful tone, `it means just what I choose it to mean — neither more nor less.’

`The question is,’ said Alice, `whether you can make words mean so many different things.’

`The question is,’ said Humpty Dumpty, `which is to be master — that’s all.’

Through the Looking-Glass and what Alice found there, by Lewis Carroll


Yesterday I was moved to post the above section from one of my favourite books on the Organisational Change Practitioners forum. The precise thread was entitled, Commitment during Change (as ever you need to be a member of and thr group to access the preceding links). The context was an increasingly intricate discussion about what constituted a “burning platform”; if this was a good thing to be standing on, or not; and whether such a situation was likely to lead to a positive or negative reaction on behalf of those standing on it.

My first contribution to this section of the discussion was as follows (with some light editing):

A burning platform tends to suggest panic and an imperative to do something (anything) right now. Think about it; the burning bit and… well… being on a platform. I am not sure that this is the best metaphor for instilling commitment.

Commitment may be passionate, but it is more rational, more of an active choice as opposed to, “what do I have to do to get out of here, my toes are getting hot?”

Telling someone that they are on a burning platform will certainly get their attention – they may also be willing to listen to you if you have some suggestion that might help, but this does not sound like instilling commitment in them to me; more like instilling fear.

Commitment tends to suggest a belief on behalf of the committed that what they are being asked to do is right for them and necessary for the broader organisation (despite it potentially being difficult and/or painful).

Commitment is not fleeing a burning platform – that’s just a survival instinct. Instead commitment might be exhibited by a person deciding to return to a burning platform to rescue some one.

The Alice quote came after I had posted the above thoughts, but before the post that I wanted to focus on in this article. This was about professional jargon and was as follows (again lightly edited):

When I was studying Mathematics, the use of words to mean something other than their ordinary meaning became second nature. The uninitiated would never have guessed the recherché meanings we ascribed to everyday words such as “ring”, “field” or “group”.

Early in my IT career I went over to the dark side, quoting impenetrable acronyms with the best of them. However as my role became more part of the business, I had something of an epiphany. I realised that people were not really that impressed by jargon, that they were more likely to assume (sometimes correctly) that the jargon-user was trying to cover something up or sound clever, and that maybe there was a better way.

Nowadays I am sometimes guilty of using complicated English, but I hope that it is mostly just English (as opposed to English 2.0 – now with even more terminology and even less meaning). I will crave your indulgence about the bit of French above of course :-o.

I think that jargon is both useful and inescapable when communicating efficiently with fellow professionals in a field (no not the Maths meaning of “field”); in all other cases it is mostly a hindrance to being understood.

Now I am sure that an assidious reader would have no problem whatsoever in finding counter-examples to my call for plain-speaking about IT; they are probably sprinkled liberally throughout this blog. Maybe this is a case of doing what I say, rather than what I do. However, it is interesting the number of commentators who have suggested that it would help IT professionals to increase their standing with their colleagues if they dropped the technical jargon and learned to speak more like a business person (e.g. see my comments on Ilya Bogorad’s article about Talking Business).

While getting business people to terminate their love affair with their own version of jargon might be wishful thinking, it is pleasing to go beyond Ilya’s recommendation and contemplate a world where a spade is called a spade and not a terrain relocation appliance.

Sadly it is all too often the case that the number of words used in a business context is inversely proportional to the quantum of meaning that they convey. Perhaps it is time for professionals in all walks of life to take a lead from Humpty Dumpty and begin to better assert their mastery of vocabulary.