Posted in category Links

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Having the right mental habits is one of the keys to being a successful problem solver. One of the best lists I’ve seen was recently posted at a math education blog. The list is supposedly for sixth grade (i.e. year six) students, but is much more universal than that. You should go there and read the whole thing, but even the excerpted headings are worth knowing:

**Pattern Sniff**

**Experiment, Guess and Conjecture**

**Organize and Simplify**

**Describe**

**Tinker and Invent**

**Visualize**

**Strategize, Reason and Prove**

**Connect**

**Listen and Collaborate**

**Contextualize, Reflect and Persevere**

A classic (pre 1900) textbook on algebra by Chrystal has recently been scanned and made available electronically. Volume 1 is not likely to be of much interest for training purposes but chapters 23 and 24 (on combinatorics), 32 through 34 (continued fractions), and 35 (number theory), of volume 2 are. However, be warned, the scanned PDF is about 30M, so don’t try this on a slow or expensive connection!

Round about sixth form one learns that every polynomial can be factorized, as a product of linear factors. Why? Well, here’s a polynomial, see. It’s probably a cubic with integer coefficients — after all, most nontrivial polynomials that one encounters are. You play with it until you discover a root, likely by looking at integer factors of the highest and lowest coefficients. Then you polynomial-divide through by the linear factor which that root gives you, and get a quadratic, whose roots there’s a formula for finding. Tada!

Of course, there’s a problem with this algorithm: it depends on figuring out how to break down your polynomial into only linear and quadratic factors.

Posted in category Links, Number Theory

(At least!) a couple of good, comprehensive introductions to elementary number theory are available online. These notes by Jim Hefferon and W. Edwin Clark are nicely written and gently-paced. These ones by Naoki Sato are a bit more Olympiad-focused.

Posted in category Links

Clear mathematical writing often leads to clearer thinking. Of course, in Olympiad problems, it also reduces the risk of losing marks for small errors!

If you want to work on your exposition, I heartily recommend this style guide for writing up solutions to mathematical problems, by John M. Lee. It’s aimed mainly at people writing university mathematics assignments, with lots of time to spare, so some of the details in the last section are more pedantic than Olympiad contestants will need to bother with. But the general instructions in the first half are excellent.

One sometimes-helpful trick (Lee mentions it at the start of the second page) is to try to break down your solution to a question into several sub-arguments. In long complicated questions, clarity, and keeping track of small details, seem to arise much more easily out of a nice sequence of Propositions and Lemmas — each stated clearly, then proved before going on to the next — than out of a single essay-style answer.

Posted in category Links

The site Abstract Math by Charles Wells presents some useful insights on writing and doing proofs. The author’s opinions in some areas are arguable (so you should read and think about them, rather than follow the advice slavishly — the only piece of advice which you should in general follow slavishly!) For instance, I wouldn’t recommend the “Languages of Math” section which is far too detailed at present. However, there are little gems to be found everywhere, and from within that section comes the link to Timothy Gowers article, “The language and grammar of mathematics.”. By the way, the “Tricki” which he (Gowers) promises should be live soon will certainly be a great resource.

Returning to Abstract Math, I particularly recommend the section entitled Doing Math. Just in case you’re too lazy to click a link and explore I will briefly quote one particularly important concept from the “Useful Behaviors” section:

METHOD:To prove that a statement involving a concept is true,start by rewriting the statement using the definition of the concept.

So very simple, and yet so true. But, don’t wait, go and explore Abstract Math now!

*Michael*