I was going to write this blog as a discussion on symmetry and time travel as a means of procrastinating from things that I have to do, but I suddenly realised that the ‘things’ I need to do could actually be quite fun to write about….so symmetry and time travel will have to wait for an….earlier date?
In my “general maths” lecture (so called by me as it is a broad summary of some interesting topics in mathematics, primarily aimed at non-mathematicians) we were given the following question: “if you write the numbers from 1-8 in a circle in any order, prove that there will always be a set of 3 consecutive numbers whose sum is at least 14.”
Some guy came up with a clever but messy proof, using logic to deduce that if you start with 8 and start making groups of three that are all below 13, the numbers you’re left with add up to over 14.
As I said, it was clever….but….it was also an ungeneral proof. Unelegant.in other words, everything that maths is NOT.
So…..onto the true beauty….
Our lecturer asked us to use the ‘pigeon hole’ idea to prove it, so called because….I actually have no idea why they used pigeons as an example. Maybe google knows.
But the idea is basic, and is as follows: if you have n pigeon holes, and n+1 pigeons, then there will be at least 1 hole that contains more than one pigeon….or you need to start searching for your missing pigeon…
A simple and obvious idea, but one that results in some useful results when it comes to counting and efficiency…
Now, on to the question…
Oh, on a side note, I can’t be bothered writing the numbers in circles, so I’m just going to do it a line. It would be much appreciated if you switched your imagination on, or, lacking coffee, use pen and paper.
Ok, so we start off with 8 numbers, their sum being 36, and we’re trying to prove that no matter what order we use, there will always be a set of 3 numbers whose sum is greater than 14. Let’s imagine that we have the simple order 1 2 3 4 5 6 7 8. From this formation, we can form the following groups: 123; 234; 345; 456; 567; 678; 781; 812.
Things to notice about these groups….
There are 8 groups.
Each number appears 3 times.
I had brilliant teacher in high school. He used to say that proving things was very simply a case of writing down what you knew, deducing obvious results, then seeing the “AHA!” stage, and then writing down the total proof. This here is the “aha!” stage…
As each number appears 3 times, the total sum of all the numbers is actually 3 *36 = 108.
As there are 8 different, groups, this total of 108 has to be shared amongst them all…so,108/8=13.5…
As we are only using integers, this means that at least one group has to have a sum of fourteen or more.
If you didn’t follow all of that, think of the pigeon holes. Imagine that there are 108 pigeons, 8 separate structures, each with 13 pigeon holes. If you’re still not getting it, leave a comment, and I’ll get back to you on it….
Now, a small rant, appropriate for a first blog!
Some people seem to get a great kick out of asking (in beautifully derogative tones) “Maths? Where do you hope to get with that?” read this link: Lockharts lament
The above example can be generalised to a random sequence of numbers ( I did have it floating around on my computer somewhere, if anyones interested, a small donation of fresh Columbian coffee in a bottomless mug will be sufficient for me to recreate it), which in turn could have some significance somewhere. At the moment I know not where. For me, the joy in mathematics comes not from the use of maths, but from doing the maths. I hope to introduce to some of you this joy that I find, and to others, I hope to increase your joy.
and it is on that note that i finish this, my first blog. All comments welcome!