Logiq Tower Solved (ALL 22,069 solutions)

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 modulo-12 rotational symmetry, with longitudes set at an arbitrary but fixed meridian at L = 1, then proceeding 'east' or counter-clockwise 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 core-ring 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 5-cube pentominoes bent in a specific rotation position around the cylinder allowing but two degrees of freedom (p-rotational-symmetric).

5-layer 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

4-layer 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

3-layer 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

2-layer 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.

Of the 15 puzzle pieces, each are required in the solution; 5 may be placed into 97 possible unique positiions (positions excluding modulus 12 longitudinal-symmetry and modulus 2, p-rotational-symmetry), 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 97⁵x 73³ x 61⁶ x 37¹ = 6,368,072,304,369,135,136,956,399,733 (over 6 octillion, or 6 trillion trillion) possible piece position combinations to try!

Of the 15 puzzle pieces 3 must be omitted from each solution but count as a unique special not-in-solution position. 5 may be placed into 73 possible positions, 9 may be placed into 49 possible unique positiions (after removing for modulus 12 longitudinal-symmetry and modulus 2, p-rotational-symmetry), and 1 may be placed into 25 possible positions. 73⁵x 49⁹ x 25¹= 84,395,449,287,076,899,494,381,425 or 84 trillion trillion possible position combinations.

Of the 15 puzzle pieces 6 must be omitted from each solution but count as a unique special not-in-solution position. 5 may be placed into 49 possible unique positiions (after removing for modulus 12 longitudinal-symmetry and modulus 2, p-rotational-symmetry), 6 into 37 possible positions, 3 into 25, and 1 into 13. 49⁵ x 37⁶x 25³ x 13¹ = 147, 215,698,144, 155,639,578,125 or 147 billion trillion possible positions.

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 not-in-solution position. All 11 pieces may be placed into 25 possible unique positions. 25¹¹= 2,384, 185,791,015,625 or over 2 thousand trillion possible positions.

An exhaustive depth-first 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 longitudinal-symmetry and modulus 2, p-rotational-symmetry, was the primary challenge. My agreement with the product designer, Marko Pavlovic, precludes my release of the software developed and licensed to them.

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.

The original software and source code that began this fun puzzle-solving 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 box-tiling 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 re-publishing the code or output data file.

I owe these puzzle-solving 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 2-dimensional 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 supercomputer-capacity 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.

Logiq Tower Solved ALL 22,069 Solutions

Logiq Tower Solved
ALL 22,069 Solutions