Ask YC: Math problems

https://news.ycombinator.com/item?id=202665

by almost • 18 years ago

26 29 18 years ago

news.ycombinator.com

Hi Everyone,

I'm going traveling in se asia for 3 months fairly soon. I'm planning on doing a lot of lazing about and reading. I'd quite like to work on some math problems while I'm there. I've been enjoying working through Concrete Mathematics but it's too heavy to take with (I'm traveling light). Does anyone have an suggestions of a source of good interesting problems and exercises? Either somewhere I can download them or in a book that's small enough to carry with me would be good.

lbrandy4• 18 years ago

Just work on the Collatz Conjecture. It's trivial to understand and it'll keep you busy :P

xirium• 18 years ago

See http://en.wikipedia.org/wiki/Collatz_conjecture

trominos3• 18 years ago

I can't tell if this is a joke or not, but if it's serious it's not a good idea. Working on one impossible problem is a good way to get frustrated as hell.

lbrandy4• 18 years ago

Bah.

I love working on impossible (or rather, impossibly difficult) math problems... especially on airplanes when I get sick of reading. The point isn't to solve them but simply to understand why they are hard. When you get a glimpse into the complexity that is these problems, you begin to see why mathematicians go insane.

Collatz is great for this because anyone can understand it and anyone who understands rigor can try to prove it.

Tichy2• 18 years ago

The path is the goal, or how is that saying translated to english?

While thinking about an impossible problem, you can discover lots of other things. Also, who says that it is impossible?

mark-t2• 18 years ago

Now and then I work through problems on projecteuler.net without anything more than a basic calculator. I've gotten 17 of them so far (1, 2, 5, 6, 8, 9, 15, 17, 18, 19, 24, 52, 63, 76, 164, 190, 191).

Also, there are a bunch of contest problems on John Scholes's website: http://www.kalva.demon.co.uk/ . It seems to be down at the moment, but I'm sure it'll be back soon. Of the contests I'm familiar with:

AIME is for fairly advanced high school students (say, on average, one person per high school might take it, and most won't get any of them right)

USAMO is a step up from AIME -- the top ~150 high school students in the US take it each year

IMO is a step up from USAMO -- basically, don't even bother

Putnam is an exam for US university students -- it's generally considered difficult, since the median score is usually 0, even among those willing to take it in the first place, but A1 and B1 are usually pretty easy on every test; I would say it's between AIME and USAMO

lacker• 18 years ago

USAMO and IMO problems are about the same difficulty. Try some of the easier ones first - they are in order of difficulty, so the first ones are generally easier. In general geometry is more likely to require specialized knowledge; go for the problems that sound "programmy". Putnam problems are about the same average level of difficulty but there is more of a gradient from A1-A6 and B1-B6.

mark-t2• 18 years ago

I strongly disagree. USAMO problems range in difficulty quite a bit. I can solve some but certainly not all. Not so for IMO. And nowadays I generally solve all of the Putnam problems in the time allotted.

By the way, the early years of the IMO were very easy, by today's standards (only a few countries were involved at that point). They would probably be approachable to people comfortable with the AIME problems.

ComputerGuru• 18 years ago

Add WYSE to the list - some really intriguing problems there.

te_platt• 18 years ago

Anything by Martin Gardner.

trominos3• 18 years ago

http://www.artofproblemsolving.com/Forum/resources.php is a good source of problems. I'd just stick with USAMOs (click on the "USA" link), they're varied and of pretty high quality.

Oompa2• 18 years ago

You could print out a couple Project Euler problems. Most of them can be solved with pencil and paper, but most people make programs to help them out. They're quite fun.

http://www.projecteuler.net

kalid• 18 years ago

I second that. It's really satisfying to take a crack at these. The forums are fun too, as there's usually several approaches to a given problem.

Oompa2• 18 years ago

Definitely. That's half the fun for me. Especially when someone has posted a really elegant solution.

gambling8nt• 18 years ago

If you are looking for a physical book, try Problem Solving Strategies (http://www.amazon.com/Problem-Solving-Strategies-Problem-Boo...) or Problem Solving Through Problems (http://www.amazon.com/Problem-Solving-Through-Problems-Larso...).

gms• 18 years ago

This is a nice book: http://www.amazon.com/dp/0486647250

DEinspanjer3• 18 years ago

I suggest something light and something out of the ordinary to counterbalance. You'll never know for what type you are in the mood. Here are two that I love:

How to Solve It -- by G. Polya

  http://www.amazon.com/How-Solve-Aspect-Mathematical-Method/dp/0691023565

Alice in Puzzleland -- by Raymond Smullyan

  http://www.amazon.com/Alice-Puzzle-Land-Raymond-Smullyan/dp/0140070567

philcrissman• 18 years ago

Yes -- anything by Smullyan is a pleasure to read. I think much of it is out of print, but should still be findable. "Lady or the Tiger" and "Satan, Cantor, and Infinity" were both good reads. "Alice," as well.

DEinspanjer3• 18 years ago

Yeah the link I supplied actually had a few people selling used copies. I got a first edition of it a while back and I still love to pull it out and go through the problems to freshen up my brain. :)

lunchbox• 18 years ago

One thing I do when I'm bored is to prove the equations I've been taught in my math classes but have never seen derived (like many people, I often skip past the proofs in the textbook when I'm in a rush to finish my homework). Deriving these is enlightening and gives me a nice feeling of "oh, that's why." Also, understanding the derivation of the fundamental math theorems can actually come in handy, in contrast to some contest questions, which can be contrived and arcane in comparison.

jlhamilton• 18 years ago

Titu Andreescu has some good problem books. You might like 102 Combinatorial Problems, http://www.amazon.com/102-Combinatorial-Problems-Titu-Andree..., if you like Concrete Mathematics, and it's fairly small to carry.

manny• 18 years ago

If you truely want a really great, small, book, purchase Peter J. Eccle's An Introduction to Mathematical Reasoning.

http://books.google.com/books?id=ImCSX_gm40oC&dq=an+intr...

almost• 18 years ago

Thanks to everyone who's made suggestions. I've ordered some books and and I'm checking out the other sources. Should give me enough to work on for a while :p

DEinspanjer3• 18 years ago

But you didn't tell us what you decided on! Those of us who did well have to be able to brag! ;)

gensym• 18 years ago

One of my favorite books is To Mock a Mockingbird, by Raymond Smullyan (http://www.amazon.com/Mock-Mockingbird-Raymond-M-Smullyan/dp...). It presents itself as a puzzle book that teaches combinatory logic along the way. Not only is it lots of fun to work through the puzzles, but at the end, you have a pretty solid understanding of the lambda calculus.

Ask YC: Math problems

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