# All n-variable combinations can exist within a word

1. Aug 28, 2004

### Sariaht

All n-variable combinations can exist within a word;
n=1 | a
n=2 | ab
n=3 | cabca

Is the fourth shortest word cabdcabcdabcadbca, and how long is the n:th word?

2. Aug 28, 2004

### Hurkyl

Staff Emeritus

Or bac?

3. Aug 29, 2004

### Sariaht

backwards cab = bac

4. Aug 29, 2004

### matt grime

so backwards is permissible, why isn't any other random ordering also equivalent? (ie, exactly what is the full question you want the answer to, seeing as you've given us incomplete information?)

5. Sep 20, 2004

### Sariaht

I ment all permutations, but i hoped i could avoid that word

6. Sep 20, 2004

### matt grime

that still doesn't make it a well formed question though.

7. Sep 21, 2004

Dude, what the hell are you saying?

8. Sep 21, 2004

### matt grime

well, if that's aimed at me, cab counts as the same string as bac for this question, now what other strings are identified with bac? sariaht's answer is "all combinations", but that means surely that the answer for a,b,c is abc, ssince that contains all strings of length 3 up to all permutations.

9. Sep 21, 2004

### TenaliRaman

the question as i see it .... i will try to rephrase it ....
U have a machine suppose that can read from a strip of letters forward as well as backward but only continuous group of n letters ....
Given letters a,b,c
find the smallest strip of letters such that the machine can get any combination of abc from that?

e.g given by sariaht for 3 letters,
cabca -- length of 5

Question now asks to find the smallest strip length for n letters.

-- AI

10. Sep 28, 2004

### Sariaht

Exactly!

What is the smallest word with all permutations of n variables?

11. Sep 28, 2004

### matt grime

and as we keep pointing out, your examples do not contain all possible permutations

n=2 ab doesn't contain ba, for instance, however you say this doesn't matter since ba is ab backwards, so what other permutations are equivalent to abc, say? only cba? why not acb? what particularly odd an arbitrary rules do you have that you're not telling us

edit: just read two posts back, so fowards and backwards are the only permissible variations? fine.

Last edited: Sep 28, 2004
12. Sep 28, 2004

### TenaliRaman

Sariaht,
i haven't given this much thought but i think i get,
::n!/2 + n - 1::

Actually i have assumed something in my logic here , i will put my logic later if i get the time to actually prove my assumption.

-- AI

13. Sep 28, 2004

### Gokul43201

Staff Emeritus
I think Tenali's right.

That would be my guess too.

Last edited: Sep 28, 2004
14. Sep 28, 2004

### Sariaht

what!!?

Have I misunderstod the meaning of permutations?

Yes I have, sorry!

What is the smallest word with all n letters long permutations of n variables?

15. Sep 28, 2004

### TenaliRaman

LOL!
the proof of the assumption i made was so damn easy ....
man it skipped my visualisation for such a long time ....
So if my answer is right, i won't give it away just now ......

-- AI

16. Sep 29, 2004

### matt grime

Here's how you start it:

write out abcd...e whatever the n letters are. then cycle through by putting the abcd.. after it. pick a permutation you've not got, repeat, count and leave this question alone, please.

17. Sep 30, 2004

### Sariaht

Wonder if you can mathematically calibrate the length of the word.

18. Sep 30, 2004

### matt grime

what does calibrate mean in this context, and what;s wrong with the metric 'number of letters in the word'?

19. Oct 1, 2004

### Sariaht

The number of letters x you need must be a function of the number of variables n.
What is that function f(n) = x

Not that it makes a difference

Last edited: Oct 1, 2004
20. Oct 5, 2004

### Sariaht

Oh, i see: n!/2 + n - 1

Then I know, thanx a lot!