Maths

[jack]

Last night, I decided I definitely *was* pining for maths, and came home and looked out my numbers and sets example sheets. I was pleased that my storage of maths notes, while not anal, was labelled enough that I could find it, though I couldn't find the lecture notes.

(Numbers and Sets was a course in the very first term, taught by Doctor Leader, notorious for requiring no prerequisites, but being a good introduction to how to think about pure maths, and the example sheet questions graduating from "show you can use the theorems taught" to "no-one solves this ever, apart from a few of the supervisors and professors".)

I've gone through the first half. I could do everything fairly easily that I did fairly easily seven years ago.

Amazingly, I can remember the relevant theorems from the first part of the course after all this time and even how to prove them -- no doubt being the first things, and the first interesting things, they sank in more than everything else[1].

I admit, solving the questions would be basically impossible without remembering them -- you'd basically have to deduce them from the whole cloth, which is possible, but requiring a great leap of intuition. This is something to remember for later courses: often becoming lost was due to not grokking that Blah Theorem was the fundamental part of this section of the course.

My logic is, I need some recreational puzzles, and these are perfect for it: based on knowledge I have, and designed to be solvable by a bright person. And that you can work towards solving, requiring a mix of knowledge, perseverance and insight (well, ok, not much so far). Neither requiring application of only a limited set of skills, nor just the right leap of insight (as eg. soduku or riddles do).

If anyone wants to join me, feel free; I can post the questions and the solutions I do work out :) (I would expect someone bright and interested in maths but without a maths background to be able to make a start, though people may disagree in either direction.)

[1] There are only three: (1) That 1,2...(p-1) all have inverses mod p (2) Fermat's Little Theorem, a^p-1=1 mod p and (3) Wilson's Theorem, (p-1)!=(p-1) mod p.

Comments

[drswirly.livejournal.com]

I've supervised Numbers&Sets for nearly ten years now, so if you do get stuck then you can always ask for hints. It's by far the loveliest course I do.

Imre has put some new questions on his sheets in the past two years. Here are a couple of the prettiest.

Let a and b be distinct positive integers, with say a < b. Prove that every block of b consecutive positive integers contains two distinct numbers whose product is a multiple of ab. If a, b and c are distinct positive integers, with say a < b < c, must every block of c consecutive positive integers contain three distinct numbers whose product is a multiple of abc?

Is there a positive integer n for which n⁷-77 is a Fibonacci number?

(The first one doesn't require any particular knowledge from the course. The second is harder and does, sort of.)

[cartesiandaemon.livejournal.com]

Thank you! That's very much appreciated. I wondered if you might say something like that, but supervising n&s for work and then coming home and supervising n&s on LJ seemed like a little much :) I'll let everyone know how I do; I expect questions near the end will be hard enough I need advice or confirmation, but easy enough I don't give up entirely :)

Imre has put some new questions on his sheets in the past two years. Here are a couple of the prettiest.

Ooh, thank you. And of course I remember the "Your series certainly wins the `brilliancy prize' for Q14!" question you were thinking about before :)

[drswirly.livejournal.com]

supervising n&s for work and then coming home and supervising n&s on LJ seemed like a little much

It's Numbers&Sets! How could anyone get bored by it?

Besides, it's been two months since I taught any, and it's two months until I do so again. Perhaps I'm suffering from withdrawal symptoms.

[cartesiandaemon.livejournal.com]

LOL. I guess that's what *I* think, though most people I know DID get bored by numbers and sets; and as often as not switched to physics or compsci :)

Well, I'll let you know. So far I'm nearly word-for-word the same as what I wrote seven years ago, so I'm probably right. (Though sometimes I just naturally go down the same path :))