arael

This has been an unusually productive time in recreational mathematics.

1. It appears that the 40th Mersenne Prime has been discovered:

http://www.mersenne.org/prime.htm

2. And we now know that there are no magic knight's tours of a chessboard, a
150-year old problem:

http://mathworld.wolfram.com/news/20...06/magictours/

3. And the world's oldest math problem,*the Loculus of Archimedes*, has been solved:

http://www.maa.org/editorial/mathgam..._11_17_03.html

Pragma

Very interesting that within a fairly recent time span three fairly significant discoveries have been made. Just goes to show that the field of mathematics still has lots of life left in it, in terms of discovering new things and proving old theorems.

Also shows how heavily we rely on computers to aid us in mathematics - each of those 3 were solved with computers.

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phukraut

math will never run out of things to study. the only thing that would kill math is if people stopped caring about it.

and the avid recreational math fan shouldn't despair thinking that math is only composed of mammoth old theorems and metasystems that need a PhD to even read about. there's LOTS of small and accessible problems that can be worked on. and if there isn't one you like, make up your own.

cartmen34

Wow. I was never good at even intermediate algebra, so I'm impressed.

What amazes me is that someone thought this crap up....a long time ago. I can't even begin to imagine how to solve it, much less able to think the original problem up!

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kel

Well number two isn't really that big a deal is it? Anyone could have enumerated the combinations and checked each one. It didn't even require that much computing power. It could have been done with technology available many years ago.
I would say the same for number 3 also. The only real challenge seems to be number 1.

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Woody182

This is very interesting, do any of you know of any websites with interesting problems designed for the high school/collegiate level?

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cheerios

and yet, the first one is being solved just the same as the last two were...

arael

well, if pc's weren't so popular it wouldn't have happened. just like many great discoveries, it sometimes happens by accident. Just because it's accidental or technology driven does not make it less signaficant

John Henry

In a way it does. Using a computer to solve a problem by an exhaustive search is donkey work. They could at least have tried some sort of heuristic search. The best solutions of these sort of problems are when someone comes up with an elegant formal proof, because if it's something that hasn't been solved for ages, it's more likely to require the development of some new mathematical technique. New techniques can have a far broader-ranging impact than just solving the puzzle itself, thus making such proofs far more significant.

Interesting to see anyways.

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arael

ya.. no one has been able to find any formal proves regarding prime and perfect numbers. we are all waiting for it... but a development of a whole new branch/system or langauge of number theory would be neccessary to do it. Since Godel already proved there is not system powerful enough to encompass it all... we need someoen specific enough to prove it... but within the power of discovered systems. i mean, Euclid did prove tehre are infinite primes, but the problem is no one can prove an equation that gives you. Furthermore, no one has yet proved there are infinite perfect numbers or the existance of odd perfect number.... ahhh who needs science when there is so much to explore in math? :P

kel

You can't compare the methods in the last two to the first one.
There is a couple orders of magnitude difference in computing power between the problems.

The first one could NOT (unless you have the secret to uber long living) have been solved by the technology available a few decades ago.

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John Henry

Yes, but that still means that the problem has only been solvable because of technological progress, it has not, in itself, driven any kind of progress or discovery

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phukraut

sure it has, algorithm efficiency plays a big role in cutting down the number of cases to check. there is math involved in finding out the best way to approach a brute force attack. this does open up new possibilities.

John Henry

you got me there.

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kel

There were no new algorithms developed to solve these problems.

Any mathemetician or programmer knows how it is improbable (really impossible) to find large prime numbers efficiently. If an algorithm had been developed it would be some big ass news because modern cryptography would be ruined.

Number 2:"Bill Cutler used a computer program to enumerate all solutions."

Number 3:"This longstanding open problem has now been settled in the negative by an exhaustive computer enumeration of all possibilities."

Enumeration isn't an efficient way of doing things and anyone could have done it years ago, they just would have taken forever to get the results.

In the case of prime numbers you would have to wait a few hundred (thousand?) years on a 486.

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