Re: Re: List-Selection

• To: mathgroup at smc.vnet.net
• Subject: [mg20157] Re: [mg20060] Re: [mg20021] List-Selection
• From: kewjoi at hixnet.co.za (Kew Joinery)
• Date: Sat, 2 Oct 1999 03:05:12 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```Hello Everyone ,
I would like to thank Mr. Kozlowski  for spending more then 2 hours for this
problem. (I spent much more- Human  is curious about unknown).
The example illustrate the ability of Mathematica to be weak in the most important
spectrum of Discrete Mathematics Searching (Sorting is handily controlled). There
is no build  in function Search (probably is impossible to build  in one). The
best custom implementation is Combinatoricas Backtrack, because is using bit vector
(incrementing) and one could be able to search longer  without run out of memory.
Mr. Hayess backtrack is faster but is memory hungry .So Mathematica needs one
build-in function just to perform at least exhaustive search over a candidate
solutions. Better one would be to involve pruning possibilities like cut off
k-subsets from rank r1 to rankr2.
Consider in this case custom implementation  construct all 7-subsets from 32 set
(which is 3365856) and map them over the matrix, then perform exhaustive search.
Thanks to Mr. Pratt for the nice pruning in the solution space. So Mr.Kozlowski was
patient to perform complete search and to find all 320 solutions.
Now I ask you  how to approach the next instance?
This is steal small instance: n=6 so the desirable matrix is m(64x7)
Note: to construct matrix whit property NO number appears twice in the same column
or the same row + that all numbers are equally distributed   the matrix should be
long prime.
*** the task is: Find JUST ONE solution satisfying the condition:
Union[Flatten[ some 12 rows of m ]] = = Range[64]

Here is the modest matrix for n=6 >m( 2^6=64  x  n+1=7)

{{1,2,4,8,16,32,64},{2,1,3,7,15,31,63},{3,4,2,6,14,30,62},{4,3,1,5,13,29,61},{
5,6,8,4,12,28,60},{6,5,7,3,11,27,59},{7,8,6,2,10,26,58},{8,7,5,1,9,25,
57},{9,10,12,16,8,24,56},{10,9,11,15,7,23,55},{11,12,10,14,6,22,54},{12,
11,9,13,5,21,53},{13,14,16,12,4,20,52},{14,13,15,11,3,19,51},{15,16,14,10,
2,18,50},{16,15,13,9,1,17,49},{17,18,20,24,32,16,48},{18,17,19,23,31,15,
47},{19,20,18,22,30,14,46},{20,19,17,21,29,13,45},{21,22,24,20,28,12,44},{
22,21,23,19,27,11,43},{23,24,22,18,26,10,42},{24,23,21,17,25,9,41},{25,26,
28,32,24,8,40},{26,25,27,31,23,7,39},{27,28,26,30,22,6,38},{28,27,25,29,
21,5,37},{29,30,32,28,20,4,36},{30,29,31,27,19,3,35},{31,32,30,26,18,2,
34},{32,31,29,25,17,1,33},{33,34,36,40,48,64,32},{34,33,35,39,47,63,31},{
35,36,34,38,46,62,30},{36,35,33,37,45,61,29},{37,38,40,36,44,60,28},{38,
37,39,35,43,59,27},{39,40,38,34,42,58,26},{40,39,37,33,41,57,25},{41,42,
44,48,40,56,24},{42,41,43,47,39,55,23},{43,44,42,46,38,54,22},{44,43,41,
45,37,53,21},{45,46,48,44,36,52,20},{46,45,47,43,35,51,19},{47,48,46,42,
34,50,18},{48,47,45,41,33,49,17},{49,50,52,56,64,48,16},{50,49,51,55,63,
47,15},{51,52,50,54,62,46,14},{52,51,49,53,61,45,13},{53,54,56,52,60,44,
12},{54,53,55,51,59,43,11},{55,56,54,50,58,42,10},{56,55,53,49,57,41,9},{
57,58,60,64,56,40,8},{58,57,59,63,55,39,7},{59,60,58,62,54,38,6},{60,59,
57,61,53,37,5},{61,62,64,60,52,36,4},{62,61,63,59,51,35,3},{63,64,62,58,
50,34,2},{64,63,61,57,49,33,1}}

Note: use ColumnForm to see some nice features of m.

Thank you for your attention and any suggestions.
Eugene

```

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