Re: Re: Re: Re: circumference of an ellipse

*To*: mathgroup at smc.vnet.net*Subject*: [mg19541] Re: [mg19501] Re: [mg19447] Re: [mg19390] Re: circumference of an ellipse*From*: "Wolf, Hartmut" <hwolf at debis.com>*Date*: Tue, 31 Aug 1999 00:52:28 -0400*Organization*: debis Systemhaus*References*: <199908250525.BAA24787@smc.vnet.net.> <199908281953.PAA24571@smc.vnet.net.>*Sender*: owner-wri-mathgroup at wolfram.com

Mecit Yaman schrieb: > > Hi there, > > I am trying to solve a problem with Mathematica. You have numbers from 1 to n all > numbers twice , namely. > > 1 1 2 2 3 3 4 4 5 5 for example for n=5 > > I am trying to sort the numbers o that between two 'n's there must be exactly n > numbers. > > For example if n=3 the solution is > 2 3 1 2 1 3 . You see there is 1 number between 1 and 1. and 2 numbers between 2 > and 2, and 3 between 3's. > > I know this forum is not for asking problems. Right! So if you do, you should not camouflage you posting behind an answer to an obviously unrelated question! > ..... But i am learning MAthematica and > wanna see how professinals solve a real problem with Mathematica > But I'm deeply sympathetic with everyone who wants to learn, so -- without pretending to be a professional, in fact I'm only an occasional user, still learning myself -- I will just show you how I tried to attack your problem with _Mathematica_. First thought of course, was to search for a direct solution, but after short I believed that there is no recursive formula. So I resorted to brute force (in hope to find some particular solutions which would give a clue for finding a general solution). Ideas for brute force are always simple, so to do was all this: for given n get that List of dubletts (1), construct all permutations of it (2), test all those with all (4) [patters {___,n,<n Blanks between>,n,___} (3)] and collect (6) those results where all (5) tests are true. For (3) I just wrote In[4]:= f[n_] := {___, n, Blank[]^n, n, ___} and now I had to define what I meant with Blank[]^n In[7]:= Unprotect[Blank] Out[7]= {"Blank"} In[8]:= Blank /: (Literal[Blank][])^1 = Blank[]; In[9]:= Blank /: (Literal[Blank][])^n_ := Sequence[Blank[]^(n - 1), Blank[]] It works! In[10]:= f[2] Out[10]= {___, 2, _, _, 2, ___} With that we define a function: In[78]:= prod[n_] := With[{probe = Permutations[ ReleaseHold[Function[x, Hold[Sequence][x, x]] /@ Range[n]](*1*)](*2*)}, With[{r = Function[x, MatchQ[#, f[x]] & /@ probe] /@ Range[n](*4*)}, Cases[Transpose[{probe, #}], {l_, True} :> l] & @ Fold[Inner[And, #1, #2, List] &, First[r], Rest[r](*5*)](*6*)]] If you try the function In[81]:= prod[2] Out[81]= {} This is ok, we see there is no solution for n=2! In[82]:= prod[3] // Timing Out[82]= {0.191 Second, {{2, 3, 1, 2, 1, 3}, {3, 1, 2, 1, 3, 2}}} We find the solution you specified, and its reverse which is another solution (of course! we might say now). Lets try it for n=4 In[83]:= prod[4] // Timing Out[83]= {7.961 Second, {{2, 3, 4, 2, 1, 3, 1, 4}, {4, 1, 3, 1, 2, 4, 3, 2}}} We easily check that as being solutions. But it took much more time. Since now I was going out for lunch, I ordered for dessert: In[84]:= prod[5] // Timing Out[84]= {529.391 Second, {}} I got none, so I leave the rest of the problem for you. Regards, hw __________________________________________ P.S.: it's not a circumference of an ellipse, a sort neither.

**References**:**Re: Re: circumference of an ellipse***From:*"Andrzej Kozlowski" <andrzej@tuins.ac.jp>

**Re: Re: Re: circumference of an ellipse***From:*Mecit Yaman <mecit@iname.com>

**Re: Simple edit ...**

**Re: Re: Re: circumference of an ellipse**

**Q: Simple? 3D Plotting Problems**