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From: Erik Carlson on 6 May 2010 02:32
I'm working on a project that depends on solving this puzzle.
Any help would be greatly appreciated, even perhaps help in rephrasing
this problem in a less clumsy way:

I'm looking to create a string of letters (or integers) with certain
properties.

The main property involves looking at consecutive groups of 4 letters
within the larger string
(i.e. the 1st through 4th letters, the 2nd through 5th letters, the
3rd through 6th letters, etc).

For example, in the string:

A B C D D E B C D

the first 4-letter groups is: ABCD
the second group is: BCDD
the third group is: CDDE
etc.


The goal is for every combination of 1-4 letters from a set of N
letters to be represented by these 4-element collections once and only
once.

For example, if there is a set of 5 letters {A, B, C, D, E} being used
Then the possible 1-4 letter combinations are:
A, B, C, D, E,
AB, AC, AD, AE, BC, BD, BE, CD, CE, DE,
ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE,
ABCD, ABCE, ABDE, ACDE, BCDE

If in a 4-letter group, a single letter is repeated, it counts only
one time.
For example, the 4-letter group AABC would represent the combination
ABC.


Here is a working example that uses a set of 5 letters {A, B, C, D, E}
with the use of 3 blank spaces to start.

_ _ _ D E C B B B B A E C C C C D A E E E E B D A A A A C B D D B

The 1st through 4th elements form the combination: D
The 2nd through 5th elements form the combination: DE
The 3rd through 6th elements form the combination: CDE
The 4th through 7th elements form the combination: BCDE
The 5th through 8th elements form the combination: BCE
The 6th through 9th elements form the combination: BC
The 7th through 10th elements form the combination: B
etc.

This could also be represent by a string integers (A=1, B=2, C=3, D=4,
E=5, _=0):

0 0 0 4 5 3 2 2 2 2 1 5 3 3 3 3 4 1 5 5 5 5 2 4 1 1 1 1 3 2 4 4 2

I am hoping to find a working example of a string that uses 12
different letters (a total of 793 4-letter groups) starting with 3
blank spaces.


Though less desirable, blank spaces are allowed in the middle of the
string.
As at the beginning, they do not add to the resulting combination.

Here is an working example that uses a set of 6 letters (56 4-letter
groups) with blanks:

_ _ _ A F D E C A _ A D C F _ C A B E F E D B _ _ E _ D A B F F F _ E
A _ B _ B C E F C A B D E C C _ _ D _ F B C D B

or, with integers:

0 0 0 1 6 4 5 3 1 0 1 4 3 6 0 3 1 2 5 6 5 4 2 0 0 5 0 4 1 2 6 6 6 0 5
1 0 2 0 2 3 5 6 3 1 2 4 5 3 3 0 0 4 0 6 2 3 4 2

From: rossum on 6 May 2010 06:40
On Wed, 5 May 2010 23:32:06 -0700 (PDT), Erik Carlson
<eriksbike(a)yahoo.com> wrote:

>Here is a working example that uses a set of 5 letters {A, B, C, D, E}
>with the use of 3 blank spaces to start.
>
>_ _ _ D E C B B B B A E C C C C D A E E E E B D A A A A C B D D B
Why do you need the three blanks to start? You can get the same
result with three D's in their place:

D D D D E C B B B B A E C C C C D A E E E E B D A A A A C B D D B

rossum

From: Chip Eastham on 6 May 2010 08:31
On May 6, 6:40 am, rossum <rossu...(a)coldmail.com> wrote:
> On Wed, 5 May 2010 23:32:06 -0700 (PDT), Erik Carlson
>
> <eriksb...(a)yahoo.com> wrote:
> >Here is a working example that uses a set of 5 letters {A, B, C, D, E}
> >with the use of 3 blank spaces to start.
>
> >_ _ _ D E C B B B B A E C C C C D A E E E E B D A A A A C B D D B
>
> Why do you need the three blanks to start?  You can get the same
> result with three D's in their place:
>
>   D D D D E C B B B B A E C C C C D A E E E E B D A A A A C B D D B
>
> rossum

I think the use of blanks matters (to the OP) only in that it
minimizes
the count of letters used (because blanks, even internal ones, aren't
counted).

regards, chip
From: Chip Eastham on 6 May 2010 09:14
On May 6, 2:32 am, Erik Carlson <eriksb...(a)yahoo.com> wrote:
> I'm working on a project that depends on solving this puzzle.
> Any help would be greatly appreciated, even perhaps help in rephrasing
> this problem in a less clumsy way:
>
> I'm looking to create a string of letters (or integers) with certain
> properties.
>
> The main property involves looking at consecutive groups of 4 letters
> within the larger string
> (i.e. the 1st through 4th letters, the 2nd through 5th letters, the
> 3rd through 6th letters, etc).
>
> For example, in the string:
>
> A B C D D E B C D
>
> the first 4-letter groups is: ABCD
> the second group is: BCDD
> the third group is: CDDE
> etc.
>
> The goal is for every combination of 1-4 letters from a set of N
> letters to be represented by these 4-element collections once and only
> once.
>
> For example, if there is a set of 5 letters {A, B, C, D, E} being used
> Then the possible 1-4 letter combinations are:
> A, B, C, D, E,
> AB, AC, AD, AE, BC, BD, BE, CD, CE, DE,
> ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE,
> ABCD, ABCE, ABDE, ACDE, BCDE
>
> If in a 4-letter group, a single letter is repeated, it counts only
> one time.
> For example, the 4-letter group AABC would represent the combination
> ABC.

[snip]

> I am hoping to find a working example of a string that uses 12
> different letters (a total of 793 4-letter groups) starting with 3
> blank spaces.

Perhaps it will help to begin by investigating a strictly easier
problem of finding Hamiltonian paths on the graph of 793 nodes
representing the subsets of 12 symbols of size 1,2,3,4.

Given such a path one can then try to "fill in the blanks" by
choosing consecutive representations of the the {1,2,3,4}-sets
by 4-place strings (including blanks).

For example, if we had only two symbols, then a Hamiltonian path
might be {A} -> {A,B} -> {B}, which could be realized by the
minimal string _ _ _ A B _ _ _.

regards, chip

From: Chip Eastham on 6 May 2010 14:03
On May 6, 9:14 am, Chip Eastham <hardm...(a)gmail.com> wrote:
> On May 6, 2:32 am, Erik Carlson <eriksb...(a)yahoo.com> wrote:
>
>
>
> > I'm working on a project that depends on solving this puzzle.
> > Any help would be greatly appreciated, even perhaps help in rephrasing
> > this problem in a less clumsy way:
>
> > I'm looking to create a string of letters (or integers) with certain
> > properties.
>
> > The main property involves looking at consecutive groups of 4 letters
> > within the larger string
> > (i.e. the 1st through 4th letters, the 2nd through 5th letters, the
> > 3rd through 6th letters, etc).
>
> > For example, in the string:
>
> > A B C D D E B C D
>
> > the first 4-letter groups is: ABCD
> > the second group is: BCDD
> > the third group is: CDDE
> > etc.
>
> > The goal is for every combination of 1-4 letters from a set of N
> > letters to be represented by these 4-element collections once and only
> > once.
>
> > For example, if there is a set of 5 letters {A, B, C, D, E} being used
> > Then the possible 1-4 letter combinations are:
> > A, B, C, D, E,
> > AB, AC, AD, AE, BC, BD, BE, CD, CE, DE,
> > ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE,
> > ABCD, ABCE, ABDE, ACDE, BCDE
>
> > If in a 4-letter group, a single letter is repeated, it counts only
> > one time.
> > For example, the 4-letter group AABC would represent the combination
> > ABC.
>
> [snip]
>
> > I am hoping to find a working example of a string that uses 12
> > different letters (a total of 793 4-letter groups) starting with 3
> > blank spaces.
>
> Perhaps it will help to begin by investigating a strictly easier
> problem of finding Hamiltonian paths on the graph of 793 nodes
> representing the subsets of 12 symbols of size 1,2,3,4.
>
> Given such a path one can then try to "fill in the blanks" by
> choosing consecutive representations of the the {1,2,3,4}-sets
> by 4-place strings (including blanks).
>
> For example, if we had only two symbols, then a Hamiltonian path
> might be {A} -> {A,B} -> {B}, which could be realized by the
> minimal string _ _ _ A B _ _ _.
>
> regards, chip

Actually a shorter solution, although it
doesn't start with three blanks, is:
_ A _ _ B _ .

--c
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