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, being American, 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.
Right.
Now what?????
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….
the next step is to form a general theorem of this question (for reasons why, see after).
-please note that I will be using more formal formatting here, as it makes it easier to follow.
let the elements in your circle be defined to be X, where Xi:=i, i Є [a,b] i, a, b Є Z(integers, i can’t get the proper symbol…) (a and b constants of your choice).
now, the different groups that can be formed will be (Xa,Xa+1,Xa+2)……..(Xb-1,Xb,Xa) (Xb,Xa,Xa+1).
this will result in b-a+1 groups.
yet again, each number will appear 3 times, so the total sum of the elements will be 3*(b+a)*b/2 or (3b^2 +3ab)/2.
therefore the lowest possible maximum in the series is (3b^2+3ab)/(2b-2a+2), rounded up.
note: if you choose groupings of numbers other than 3, merely replace the 3 with the new number.
assume it is c, then
(c/2)*((b^2+ab)/(b-a+1)) rounded up is the lowest possible maximum for any of the groups.
Some people seem to get an amazing orgasm out of asking (in beautifully derogative tones) “Maths? Where do you hope to get with that?”go to this link here: http://www.maa.org/devlin/LockhartsLament.pdf
maths is not merely about use. lots of things that rely on mathematics in today’s society use old thereoms that were created merely out of interest. the question above may not be immediately useful, but use is not why i do maths…and nor is it why i share it with you. i share it primarily because i find it beautiful, and even simple things like this leave me feeling awed at the amount of patterns that appear in the world. making up silly exercises merely to find a real life situation where it is necesary takes a lot away from the beauty of the maths.
and it is on that note that i finish this, my first blog. i welcome all comments, however, if you happen to be a mathematician and find my format to be far too coloquial….get stuffed 