Math Is Fun Forum / Is this cool with you?

bobbym · 2011-10-31 12:58:35

Answers do not get posted. The problems are to be solved. There are 3 or 4 heavy hitters in here who are capable of knocking that problem off.

reconsideryouranswer · 2011-10-31 15:41:55

bobbym · 2011-10-31 15:58:04

Hi;

I do not see what you are driving at. I am the poser of the problems. All answers meet my criterion for a successful solution. If I do not understand a solution I ask, anyone else can do the same.

reconsideryouranswer · 2011-11-04 10:24:18

bobbym wrote:
You are welcome and nice to see you are not rusty.

New problem:

Prove:

When n>0

Last edited by reconsideryouranswer (2011-11-04 10:29:38)

bobbym · 2011-11-04 15:30:56

Hi;

bobbym · 2011-11-11 22:30:55

What is the number of solutions to

a + b + c + d + e + f = 50 with a,b,c,d,e,f being positive integers and
a>b>c>d>e>f ?

A says) No solutions!
B says) 1057
C says) That is correct!
D says) No is isn't. The answer is 1058 by direct count.
E says) How did you get that B?

What is the correct answer?

phrontister · 2011-11-16 01:58:59

Hi Bobby,

EDIT: Tweaked the program a bit...reduced run time by about 35%

Last edited by phrontister (2011-11-17 01:20:08)

bobbym · 2011-11-16 02:10:38

Hi phrontister;

phrontister · 2011-11-16 09:07:36

Hi Bobby,

No complaints, thanks, Bobby. Health is good (at least I think so!), and life is too. Very busy with work as usual, so not much time for MIF.

I've spent a bit more time here lately. Bob resurrected my old Callous Count Crewell puzzle and solved most of it - which got some discussion going - and now I've tried this puzzle of yours. Nice one...I enjoyed wiping the dust off LB and trotting it out again.

Btw, what's the 'best' way to solve your puzzle? I thought it might be something to do with combinations and went down that road first - until my head started spinning! Then I tried it by hand with Excel's help - which was taking too long - and, anyway, it was unsatisfying because I knew there'd have to be an easier way. So I turned to LB. Came up with the code fairly quickly, and it runs in under 2 seconds...so I'm happy.

The last couple of days I've had a go at doing a few of ganesh's 10-second questions (that must be ganesh's typing time). They're good and make me think...especially #3385 because I didn't realise what a diagonal of a cube was and based my initial calculations on a single hypothenuse. I posted that as my answer but while at work I got to wondering if "diagonal" could mean something else, and couldn't wait to get home to find out. Googled it, discovered the right meaning and reworked my answer. Learning all the time...albeit slowly.

Last edited by phrontister (2011-11-16 15:34:28)

bobbym · 2011-11-16 15:44:06

Hi phrontister;

The best solution to that problem is a matter or taste. Programming will almost always solve a combinatorics or probability problem. There is another method by using generating functions.

Glad everything is well for you and yes the ganesh problems are nice.

phrontister wrote:
Learning all the time...albeit slowly.

Is there any other way?

phrontister · 2011-11-16 16:06:27

Hi Bobby,

generating functions

A term I'd never come across before I joined MIF. I've seen you and others use it, but I'll abstain...for now at least.

Is there any other way?

Yes...quickly. Often after a fall one very soon becomes much wiser!

bobbym · 2011-11-16 16:09:39

Hi;

A term I'd never come across before I joined MIF. I've seen you and others use it, but I'll abstain...for now at least.

Still remember how to use 8? Then you could do it even faster than LB!

I remember seeing that in a movie or was it a documentary? Doc Emmet Brown fell in his bathroom and hit his head against the sink. He immediately envisioned the flux capacitor which makes time travel possible.

phrontister · 2011-11-16 16:52:51

Hi Bobby,

Yes...I might try that (7) for fun. I might even learn something!

bobbym · 2011-11-16 19:11:10

Hi;

Yes, that is from a trilogy of "Back to the Future," movies. Pretty good entertainment!

phrontister · 2011-11-16 19:24:24

Hi Bobby,

Yes...I've seen it before, and it's terrific value! Excellent humour right the way through at every corner, and great storyline too.

I watched the third of the trilogy a few weeks back when it came on tv here and planned to watch the other two soon...but with one thing or another on the go at the time I promptly forgot! Thanks for getting me back on track!

bobbym · 2011-11-16 20:01:10

Hi;

Actually I think the second one is the best of them but my favorite is the third one. Excellent plot and good solid performances in all three.

phrontister · 2011-11-17 23:11:36

Hi Bobby,

I tried 7 but I can see it's going to take longer for me to work out than I'll have time for.

I'm also lacking inclination somewhat as LB does it well enough for me: if I only get it to print the number of solutions and not the solutions themselves, running time shrinks from 1844ms to only 30ms.

I also shortened my code a bit and neatened it up.

That'll do me, I reckon.

Last edited by phrontister (2011-11-17 23:13:40)

bobbym · 2011-11-17 23:13:52

Hi phrontister;

I understand what you are saying, but I meant a whole different method of solution. On that uses generating functions. Give me a chance to work it out and I will post it.

phrontister · 2011-11-17 23:18:45

Ok...that'd be good. I'd like to see how it can be done that way, and maybe I can then use it on other problems down the track.

Thanks.

bobbym · 2011-11-17 23:28:37

HI;

Enter this just as you see it.

Expand[Sum[x^k,{k,1,50}]Sum[x^k,{k,2,50,2}]Sum[x^k,{k,3,50,3}]Sum[x^k,{k,4,50,4}]Sum[x^k,{k,5,50,5}]Sum[x^k,{k,6,50,6}]]

You will get a big polynomial. Check the coefficient of x^50.

phrontister · 2011-11-17 23:40:14

Thanks, Bobby.

Yes...that works. But how or what it does I wouldn't have any idea about right now. It took 15ms, and I got it to play a pretty alert via EmitSound[Sound[SoundNote["BellTree"]]].

bobbym · 2011-11-17 23:46:35

Hi phrontister;

That is really a very brutish way of handling that problem. basically that is the generating polynomial that solves the equation:

For a + b + c + d + e + f = 50 with a>b>c>d>e>f>0

phrontister · 2011-11-18 00:01:48

Hi Bobby,

I've just read up a bit on generating functions on the www, but it all goes whoooosh waaay over my head. I just don't know enough to even begin to try to fathom any of it out.

Is this the sort of stuff you learn at uni...or later?

All I can say is that Abraham de Moivre was a very clever boy!

bobbym · 2011-11-18 00:10:31

Hi phrontister;

I've just read up a bit on generating functions on the www, but it all goes whoooosh waaay over my head. I just don't know enough to even begin to try to fathom any of it out.

Yes, you do. Mathematics has two kinds of people. One kind knows that most math is easy and explains it like that. They are called applied mathematicians. The other type have heads so large that they need a can of 10w-60w pennzoil to squeeze their heads through a door. They are called pure mathematicians.

Generating functions are nothing but the multiplication of polynomials!

Is this the sort of stuff you learn at uni...or later?

They teach it to graduate students but like everything else in the world that is backwards.

Abraham de Moivre and Newton are my heroes and they were pals.

phrontister · 2011-11-18 00:32:07

You'd better put me into the 'applied mathematician' category...even though I don't know enough to make me dangerous.

Generating functions are nothing but the multiplication of polynomials!

I'd better not ask what a polynomial is, then!

What was putting me off in one article about generating functions were terms like:
- formal power series in one indeterminate;
- coefficients encode information about a sequence of numbers;
- indexed by the natural numbers;
- ordinary generating functions, exponential generating functions;
- Lambert series, Bell series, Dirichlet series;
- closed form;
- operations defined for formal power series;
- differentiation with respect to x.

MIF doesn't seem to cover it.

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#1126 2011-10-31 12:58:35

Re: Is this cool with you?

#1127 2011-10-31 15:41:55

Re: Is this cool with you?

#1128 2011-10-31 15:58:04

Re: Is this cool with you?

#1129 2011-11-04 10:24:18

Re: Is this cool with you?

bobbym wrote:

#1130 2011-11-04 15:30:56

Re: Is this cool with you?

#1131 2011-11-11 22:30:55

Re: Is this cool with you?

#1132 2011-11-16 01:58:59

Re: Is this cool with you?

#1133 2011-11-16 02:10:38

Re: Is this cool with you?

#1134 2011-11-16 09:07:36

Re: Is this cool with you?

#1135 2011-11-16 15:44:06

Re: Is this cool with you?

phrontister wrote:

#1136 2011-11-16 16:06:27

Re: Is this cool with you?

#1137 2011-11-16 16:09:39

Re: Is this cool with you?

#1138 2011-11-16 16:52:51

Re: Is this cool with you?

#1139 2011-11-16 19:11:10

Re: Is this cool with you?

#1140 2011-11-16 19:24:24

Re: Is this cool with you?

#1141 2011-11-16 20:01:10

Re: Is this cool with you?

#1142 2011-11-17 23:11:36

Re: Is this cool with you?

#1143 2011-11-17 23:13:52

Re: Is this cool with you?

#1144 2011-11-17 23:18:45

Re: Is this cool with you?

#1145 2011-11-17 23:28:37

Re: Is this cool with you?

#1146 2011-11-17 23:40:14

Re: Is this cool with you?

#1147 2011-11-17 23:46:35

Re: Is this cool with you?

#1148 2011-11-18 00:01:48

Re: Is this cool with you?

#1149 2011-11-18 00:10:31

Re: Is this cool with you?

#1150 2011-11-18 00:32:07

Re: Is this cool with you?

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