Vast Search Space
This puzzle introduces
novel solver notations. Along the cylindrical prism's vertical spindle axis we
label coodinates as one of Z = 1, 2, 3, 4, 5 going from the base layer Z
= 1 to the top layer Z = 5. We label the center core piece rings
as a single 'cube' in the origin of the radial axis, going from the
center ring at r = 0 to the surface layer at r = 1. Lastly,
we label the prism's outer surface circumference, a modulo12 rotational
symmetry, with longitudes set at an arbitrary but fixed meridian at L =
1, then proceeding 'east' or counterclockwise around the prism spindle
axis, to L = 12, which is immediately 'west' of L = 1.
Diagram and Labels of the 15
Logiq Tower Pieces and the arbitrary number and letter labels assigned to each by
the manufacturer and used in the
software program and solutions catalog (currently known only to me
and the product designer). 
www.logiqtower.com 

Pieces 0,1,2,3 and 4 are corering pieces
that thread onto the spindle. The core ring is assigned a
single 'cube' at zero radius (r=0). The piece label number indicates the number of gap spaces between two
exterior (r=1) 'ear' pieces.
The remaining exterior (r=1)
pieces are labeled according to their approximate resemblence to
their assigned alphabetic letters. Note these are standard
5cube pentominoes bent in a specific rotation position around
the cylinder allowing but two degrees of freedom (protationalsymmetric).
5layer solution
space requires all
pieces:
L,N,Q,U,Y  97 positions
F,T,W  73 positions
I,0,1,2,3,4  61 positions
S  37 positions
4layer solution space omits 3 pieces each:
L,N,Q,U,Y  73 positions
T,W,F,I,0,1,2,3,4  49 positions
S  25 positions
3layer solution space omits 6 pieces each:
L,N,Q,U,Y  49 positions
I,0,1,2,3,4  37 positions
T,W,F  25 positions
S  13 positions
2layer solution space always omits F,T,W,S and omits 5 more
pieces each:
L,N,Q,U,Y,I,0,1,2,3,4 
25 positions
F,T,W,S  0 positions 
Illustration of solutions of heights 2, 3, 4 and 5 pieces
where for solutions of between 2 and 4 high can omit pieces not used
in the solution, and 5 high uses all pieces.

Tower height Z = 5
Analysis
Of the 15 puzzle pieces,
each are required in the solution; 5 may be placed into 97 possible
unique positiions (positions excluding modulus 12 longitudinalsymmetry
and modulus 2, protationalsymmetry), 3 may be placed into 73
possible orientations, 6 may be placed into 61
possible orientations, and 1 may be placed into 37
possible orientations. This means there are 97^{5 }x 73^{3}
x 61^{6}
x 37^{1}
= 6,368,072,304,369,135,136,956,399,733 (over 6 octillion, or 6 trillion
trillion) possible piece position combinations to try!
Tower height Z = 4
Analysis
Of the 15 puzzle pieces
3 must be omitted from each solution but count as a unique special
notinsolution position. 5 may be placed into 73 possible
positions, 9 may be placed into 49 possible unique positiions (after
removing for modulus 12 longitudinalsymmetry and modulus 2,
protationalsymmetry), and 1 may be placed into 25 possible positions.
73^{5 }x 49^{9} x 25^{1 }= 84,395,449,287,076,899,494,381,425 or 84 trillion
trillion possible position combinations.
Tower height Z = 3
Analysis
Of the 15 puzzle pieces
6 must be omitted from each solution but count as a unique special
notinsolution position. 5 may be placed into 49 possible
unique positiions (after removing for modulus 12 longitudinalsymmetry
and modulus 2, protationalsymmetry), 6 into 37 possible positions, 3
into 25, and 1 into 13. 49^{5} x 37^{6 }x 25^{3 }
x 13^{1} = 147,
215,698,144, 155,639,578,125 or 147 billion trillion possible positions.
Tower height Z = 2
Analysis
Of the 15 puzzle pieces
4 (T, W, F, S) must be omitted from ALL solutions and of the remaining
11 pieces, 5 more must be omitted from each solution but count as a
unique special notinsolution position. All 11 pieces may be
placed into 25 possible unique positions. 25^{11 }= 2,384,
185,791,015,625 or over 2 thousand trillion possible positions.
Search Algorithm
An exhaustive
depthfirst combintorial search algorithm called DLX or Knuth's Dancing
Links was used. The spatial encoding and decoding scheme limiting
the search space's modulus 12 longitudinalsymmetry and modulus 2,
protationalsymmetry, was the primary challenge. My agreement
with the product designer, Marko Pavlovic, precludes my release of the software developed
and licensed to them.
Search Results
Exactly 22,069 unique solutions were found! The Logiq Tower can be
solved with tower heights of between 2 and 5 layers, or Z = 2, 3, 4 or
5. There are 23 of the Z = 2 solutions, 2294 for Z = 3, 15588 for Z = 4
and 6164 for Z = 5. Benchmarked on a 2 GHz Intel CD2 CPU.
Tower height 
Search
time % of Z = 5 time 
solutions 
97 positions 
73 positions 
61 positions 
49 positions 
37 positions 
25 positions 
13 positions 
0 positions
(piece is excluded from all solutions) 
Z = 5 
100% 
6164 
L,N,Q,U,Y 
T,W,F 
I,0,1,2,3,4 

S 



Z = 4 
86.65% 
13588 

L,N,Q,U,Y 

T,W,F
I,0,1,2,3,4 

S 


Z = 3 
2.579% 
2294 



L,N,Q,U,Y 
I,0,1,2,3,4 
T,W,F 
S 

Z = 2 
0.008% 
23 





L,N,Q,U,Y
I,0,1,2,3,4 

S
T,W,F 
Logiq Tower Solutions
My agreement with the product designer precludes my release of solutions for an initial embargo period. Until it lapses, here are the example solutions provided with the product!
Did my Logiq Tower
solutions help? Aw, you know it did! Toss me a bone
here  click the donate button!

Tetris Cube Solver Software
The original software
and source code that began this fun puzzlesolving journey in 2008! Download the program tetriscube.exe, a
simple console application without buttons or windowing and its 700
lines of C language
source code file tetriscube.c, in
TetrisCubeSolved.zip. Run it
yourself! Every 1,000,000 piece order permutations are output to
the console screen. Email me at the address at the bottom of the
page and tell me the kind of computer you used and how long it took to
run. With a few trivial tweaks this code determined all
480 unique solutions of Piet Hein's Soma cube puzzle
in 10 seconds (note: this linked reference cites only 240 unique
solutions),
both unique solutions of Hugo
Steinhaus' cube in 1 second, all 19,186 unique
solutions of the Bedlam cube puzzle, all
14,177 solutions of the Brother Cube, and would work for other 3D boxtiling
puzzles. Note there are copyright
restrictions given in the tetriscube.c source module and
readme.txt files to observe
regarding modification of the source code and/or republishing the code
or output data file.
Background and Credits
I owe these puzzlesolving
adventures to my son Dylan, who years ago had challenged me to find "even one solution,
daddy!" and witnessed my struggle to manually restore his Tetris Cube to
its plastic box. I told him there was a way to use a computer to
find every possible solution, so we encoded the cube coordinate
positions of the 12 pieces on paper, and I later put that into this
software over the span of a handful of days. Thank you, Dylan!
In 1986 I wrote a
software program to exhaustively solve and
catalog all solutions of the
2dimensional pentomino puzzle, to the later
delight of Stanford Professor Emeritus
Donald Knuth, which has
tens of thousands of solutions in various box dimensions (3x20, 4x15,
5x12, 6x10 and 8x8 with several 2x2 hole positions) even after sifting
out reflected and rotated copies. I also have over a
decade of experience creating
and running
supercomputercapacity
research projects so I was fully prepared to organize any "heavy
duty processing power" required to catalog all the
solutions and verify their number as claimed by the Tetris Cube's creators,
but a laptop computer was enough.
