You'd be an asset on the debating team, Bobby!

Surely the algebraic method is better for non-integer solutions, particularly those with multiple decimal places?

For me, the grass is greener over your way...

Hi Bobby,

I'm way out of my depth here and can't comment much...except to say that each method probably has strengths and weaknesses that the other doesn't have, and one might suit a particular application better than the other. In what ratio that is I wouldn't have a clue.

And they can work together, as in a 'crossed ladders' problem I came across many years ago (see diagram). I came up with an equation in terms of x but couldn't solve it, and then used Newton's method of successive approximations to find the value of x.

I went to the library to try to find out how to solve nasty equations and read about Newton's method, but then had to learn up on it and try to understand what 'functions' and 'derivatives' were because I didn't get to that level at school.

Eventually I worked out how to program that into a little hand-held computer I've got (a Sharp PC-1500A), which automated the solving process.

That was all very interesting at the time...but now I've forgotten it all.

I've since found lots of discussion about this problem, and similar, on the internet. My equation doesn't seem to be on the sites I looked at...but maybe it's there, in a disguise I missed.

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Last edited by phrontister (2010-02-03 15:04:05)

Hi Bobby,

My point, we have computers, we have the internet let's use them.

I do. I like using BASIC, Excel and other solvers where I can, but I do spark up if I happen to see a nifty algebraic solution.

I had a bunch of Sharp programmable calculators. Last one had 5k of memory and a larger screen.

My Sharp PC-1500A has an 8kB memory, which I tripled to a whopping-great 24kB by adding a 16kB module to it years ago. I still enjoy using it.

It's amazing how many programs could be squeezed into those old machines. Helped me to develop bad program-writing habits, though, by jamming in the max amount of code per line just to save enough kBs here and there to make room for another program.

I used to record my PC-1500A's programs on cassette via a radio-cassette player. Had to twiddle the knobs and tweak the settings to get them just right for a successful upload...and download!

Btw, my equation for the crossed ladder problem was 1/√(16-x²) + 1/√(36-x²) - 1 = 0...which my Sharp 48SX solves easily with its Solver function.

This one is so easy, I expect 414 answers.

Problem #21:

A basketball player has gone 21 for 47 from the foul line for a poor percentage of .4468. In order not to be cut from the team he must get his foul shooting percentage to 70 % in his next 200 free throw attempts. What is the minimum number of free throws he must he make out of his next 200 attempts to get his average to 70%. Round your answer to the nearest free throw and please hide it.

Why 414?

Because it is so easy.

True. But that doesn't change the fact that 414 is the number of responses I am expecting.

Nice answer. You solved it simply, I solved it in a hideous way on some other forum and they are going to shred me for my ugly solution.

I agree that numerical methods are usually quicker and easier to use, but I still think that it's best to be algebraic whenever possible.

It's just so much nicer to be *absolutely correct* than just very accurate. Numerical methods are fine for any and all practical purposes, but to get a closed-form expression is much more pleasing.

Hi phrontister;

How many total integer solutions are there to the equations?

a + b + c + d + e + f = r

with r = 0,1,2,3,4,...60

f >= e >= d >= c >= b >= a >= 0

a,b,c,d,e,f < 11

phrontister wrote:

I wondered also if it could somehow be done with permutations, but I know nothing about that subject.

There are 3 main ways to solve a combinatorics problem (typified by the phrase how many ways).

1) Split it into smaller sub problems that are easier to solve. In programming this is called top down design.

2) Look at a couple of examples of smaller but similar problems and spot the pattern. Then prove the pattern applies for the larger problem.

3) Bijection - turn the problem into a totally different one that has the same number of ways as the original one and is solvable.

The third method is used here: The above problem is translated from the number of integer solutions of a diophantine equation to a block walking experiment. See the drawing. The question now is how many paths are there from the bottom left corner to the top right corner. If you can only go right or up. I have shown one path.

These are easy to solve both combinatorically. You are trying to go from (0,0) to (6,10) which has

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Hi phrontister;

Correct, Very good!

Hi;

Problem #23:

Moderately tough.

Prove that 5x + 7y + 11z = r has no solutions for r = 13.

Extra credit: Prove that r = 13 is the highest number without solutions.


Hi phrontister;

Thanks for providing your code for the Elvis problem.

Hi Bobby,

Thanks for providing your code for the Elvis problem.

I made some changes to it that reduced the run time down to 15 seconds (previously 1 minute).

There was some duplication and other stuff, such as a lot of unnecessary calculating that I eliminated by repositioning some if/then lines.

I've altered the code in my post #120.

Last edited by phrontister (2010-02-10 03:19:20)

Hi Omaima;

How many permutations of 3 letters can you make out of the word:

That's true for actual words but their are many more answers. The answer does not have to be an actual word. For instance cld and sec and sce are also answers. And many more.

A permutation is a rearrangement of a set of objects or values.

For instance their are six permutations of a,b,c.

a b c
a c b
b a c
b c a
c a b
c b a