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Topic: Advanced Combinations
Replies: 2   Last Post: Jul 13, 2014 11:20 PM

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Einstein

Posts: 35
Registered: 11/22/07
Advanced Combinations
Posted: Jul 6, 2014 8:20 AM
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The idea of Advanced Combinations is simple, where I have a laymans
description and the math definition.

The laymans argument is one of cubes (though the pending patent allows
for any method, this is the easier challenge). In a given dimension
space a certain number of single cubes, or of larger shapes made up of
cubes joined together, can fit, as well as represented empty spaces.

This creates a condition where it would seem possible, in a new manner,
to encode more data via the act of creating Combinations, than the
binary would normally allow. Note I said seem, were we have not yet
advanced far enough to test if this is true.

-----------------------------------------------------------------
definition: legal object
-----------------------------------------------------------------

Let S be the set of solid unit cubes in R^3 with all vertices at
lattice points.

A legal object is a subset X of R^3 such that

(1) X is a nonempty, finite union of elements of S.

(2) The interior of X is connected.

Congruent objects are considered the same.

_______________________

Empty spaces are allowed, which will increase the total possible
outcomes.
_______________________

Now we were able to do a variety of lower level efforts, solving the
exact count for a 2x2x2 container, and maximum shapes, a 3x3x3
container for maximum shapes, and a 3x3x4 container for maximum shapes.
The exact results are below

A 2x2x2 container has 9472 total outcomes from 14 possible shapes.
A 3x3x3 container has 1,585,580 total shapes with an estimated
quadrillion plus possible outcomes.
A 3x3x4 container has 1,828,003,418 possible shapes, and defies an
estimation of possible outcomes (most certainly exceeds the quadrillion
by a factor or two)



2x2x2 = 9472 total combinations
For a 2x2x2 container, the number of distinct legal objects is 14.

Exact Shapes count for a 3x3x3 array
Units     Shapes
  0:         0
  1:         1
  2:         1
  3:         2
  4:         7
  5:        25
  6:       111
  7:       485
  8:      1,844
  9:      6,134
 10:     17,322
 11:     41,998
 12:     86,803
 13:    152,959
 14:    226,410
 15:    277,767
 16:    277,390
 17:    223,802
 18:    145,803
 19:     77,251
 20:     33,413
 21:     11,772
 22:      3,356
 23:       769
 24:       138
 25:        22
 26:         4
 27:         1


Exact Shapes Count for a 4x3x3 Array.
#Bits     #Shapes
  1:         1
  2:         1
  3:         2
  4:        10
  5:        34
  6:       181
  7:       959
  8:      5,383
  9:     27,582
 10:    124,741
 11:    486,690
 12:   1,655,982
 13:   4,934,805
 14:  12,969,641
 15:  30,109,904
 16:  61,740,472
 17: 111,353,404
 18: 175,542,991
 19: 239,819,535
 20: 281,599,067
 21: 282,326,036
 22: 241,011,786
 23: 175,295,801
 24: 109,002,113
 25:  58,119,504
 26:  26,620,643
 27:  10,455,818
 28:   3,509,584
 29:    997,606
 30:    238,069
 31:     46,638
 32:      7,435
 33:       901
 34:        92
 35:         6
 36:         1
Total: 1,828,003,418



So the question is, can anyone here make better software for this? Or
perhaps someone here has access to a cluster and can run the
computations faster than our systems?

Here is links to the existing software

http://pat7.com/js/3shapes/1cellcount.c
http://pat7.com/js/3shapes/4cuber.c
http://pat7.com/js/3shapes/4cuber-inc.c
http://pat7.com/js/3shapes/3cuber.py
http://pat7.com/js/3shapes/rotate24x8.py

It is my observation that the combinations rapidly spiral out of
control, where 100x100x100 would be computationally impossible without
a reasonable system to shortcut from checking all possibilities and
instead build a full binary tree from the results.


One issue we are having is poor software and poor computational
capabilities. I would like to see all possible outcomes of a 3x3x3
array enumerated, but our efforts cannot do this. We also are unable to
count the shapes in a 3x4x4 sized array. I suspect that a 5x5x5 array
could make feasible a Pending Patent of mine using this math theory.



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