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119 competition/games/cube.p




Description

This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

119 competition/games/cube.p


What are some games involving cubes?

competition/games/cube.s

Johan Myrberger's list of 3x3x3 cube puzzles (version 930222)

Comments, corrections and contributions are welcome!

MAIL: myrberger@e.kth.se

Snailmail: Johan Myrberger
Hokens gata 8 B
S-116 46 STOCKHOLM
SWEDEN

A: Block puzzles

A.1 The Soma Cube

 ______                   ______       ______               ______
|\     \                 |\     \     |\     \             |\     \
| \_____\                | \_____\    | \_____\            | \_____\
| |     |____       _____| |     |    | |     |____        | |     |____
|\|     |    \     |\     \|     |    |\|     |    \       |\|     |    \
| *_____|_____\    | \_____*_____|    | *_____|_____\      | *_____|_____\
| |\     \    |    | |\     \    |    | |     |\     \     | |     |     |
 \| \_____\   |     \| \_____\   |     \|     | \_____\     \|     |     |
  * |     |___|      * |     |___|      *_____| |     |      *_____|_____|
   \|     |           \|     |                 \|     |
    *_____|            *_____|                  *_____|
 
 ______                             ______                      ____________
|\     \                           |\     \                    |\     \     \
| \_____\                          | \_____\                   | \_____\_____\
| |     |__________           _____| |     |____          _____| |     |     |
|\|     |    \     \         |\     \|     |    \        |\     \|     |     |
| *_____|_____\_____\        | \_____*_____|_____\       | \_____*_____|_____|
| |     |     |     |        | |     |     |     |       | |     |     |
 \|     |     |     |         \|     |     |     |        \|     |     |
  *_____|_____|_____|          *_____|_____|_____|         *_____|_____|

A.2 Half Hour Puzzle

 ______                        ______            ______
|\     \                      |\     \          |\     \
| \_____\                     | \_____\         | \_____\
| |     |__________      _____| |     |____     | |     |__________
|\|     |    \     \    |\     \|     |    \    |\|     |    \     \
| *_____|_____\_____\   | \_____*_____|_____\   | *_____|_____\_____\
| |     |     |     |   | |     |     |     |   | |     |\     \    |
 \|     |     |     |    \|     |     |     |    \|     | \_____\   |
  *_____|_____|_____|     *_____|_____|_____|     *_____| |     |___|
                                                         \|     |
                                                          *_____|
 
       ______            ______                    ______
      |\     \          |\     \                  |\     \
      | \_____\         | \_____\                 | \_____\
 _____| |     |    _____| |     |                 | |     |
|\     \|     |   |\     \|     |                 |\|     |
| \_____*_____|   | \_____*_____|______        ___|_*_____|______
| |\     \    |   | |     |\     \     \      |\     \     \     \
 \| \_____\   |    \|     | \_____\_____\     | \_____\_____\_____\
  * |     |___|     *_____| |     |     |     | |     |     |     |
   \|     |                \|     |     |      \|     |     |     |
    *_____|                 *_____|_____|       *_____|_____|_____|

A.3 Steinhaus's dissected cube

 ______                            ______          ______
|\     \                          |\     \        |\     \
| \_____\                         | \_____\       | \_____\
| |     |__________          _____| |     |       | |     |____
|\|     |    \     \        |\     \|     |       |\|     |    \
| *_____|_____\_____\       | \_____*_____|       | *_____|_____\
| |     |     |     |       | |\     \    |       | |     |\     \
 \|     |     |     |        \| \_____\   |        \|     | \_____\
  *_____|_____|_____|         * |     |___|         *_____| |     |
                               \|     |                    \|     |
                                *_____|                     *_____|
 
 ____________                          ______                    ______
|\     \     \                        |\     \                  |\     \
| \_____\_____\                       | \_____\                 | \_____\
| |     |     |                       | |     |      ___________| |     |
 \|     |     |                       |\|     |     |\     \     \|     |
  *_____|_____|______        _________|_*_____|     | \_____\_____*_____|
      \ |\     \     \      |\     \     \     \    | |     |\     \    |
       \| \_____\_____\     | \_____\_____\_____\    \|     | \_____\   |
        * |     |     |     | |     |     |     |     *_____| |     |___|
         \|     |     |      \|     |     |     |            \|     |
          *_____|_____|       *_____|_____|_____|             *_____|

A.4

 ______
|\     \
| \_____\
| |     |____              Nine of these make a 3x3x3 cube.
|\|     |    \
| *_____|_____\
| |     |     |
 \|     |     |
  *_____|_____|

A.5

                           ______                    ____________
                          |\     \                  |\     \     \
                          | \_____\                 | \_____\_____\
 ____________             | |     |____             | |     |     |
|\     \     \            |\|     |    \            |\|     |     |
| \_____\_____\           | *_____|_____\           | *_____|_____|
| |     |     |           | |     |     |           | |     |     |
 \|     |     |            \|     |     |            \|     |     |
  *_____|_____|             *_____|_____|             *_____|_____|
 
                           ______                    ______
                          |\     \                  |\     \
                          | \_____\                 | \_____\
 ______      ______       | |     |____             | |     |__________
|\     \    |\     \      |\|     |    \            |\|     |    \     \
| \_____\   | \_____\     | *_____|_____\           | *_____|_____\_____\
| |     |___| |     |     | |     |     |____       | |     |     |     |
|\|     |    \|     |     |\|     |     |    \      |\|     |     |     |
| *_____|_____*_____|     | *_____|_____|_____\     | *_____|_____|_____|
| |     |     |     |     | |     |     |     |     | |     |     |     |
 \|     |     |     |      \|     |     |     |      \|     |     |     |
  *_____|_____|_____|       *_____|_____|_____|       *_____|_____|_____|

A.6

 ______                   ______       ______               ______
|\     \                 |\     \     |\     \             |\     \
| \_____\                | \_____\    | \_____\            | \_____\
| |     |____       _____| |     |    | |     |____        | |     |____
|\|     |    \     |\     \|     |    |\|     |    \       |\|     |    \
| *_____|_____\    | \_____*_____|    | *_____|_____\      | *_____|_____\
| |\     \    |    | |\     \    |    | |     |\     \     | |     |     |
 \| \_____\   |     \| \_____\   |     \|     | \_____\     \|     |     |
  * |     |___|      * |     |___|      *_____| |     |      *_____|_____|
   \|     |           \|     |                 \|     |
    *_____|            *_____|                  *_____|
 
       ______                      ____________               ____________
      |\     \                    |\     \     \             |\     \     \
      | \_____\                   | \_____\_____\            | \_____\_____\
 _____| |     |____          _____| |     |     |       _____| |     |     |
|\     \|     |    \        |\     \|     |     |      |\     \|     |     |
| \_____*_____|_____\       | \_____*_____|_____|      | \_____*_____|_____|
| |     |     |     |       | |     |     |            | |     |     |
 \|     |     |     |        \|     |     |             \|     |     |
  *_____|_____|_____|         *_____|_____|              *_____|_____|

A.7

 ____________
|\     \     \
| \_____\_____\
| |     |     |
|\|     |     |  Six of these and three unit cubes make a 3x3x3 cube.
| *_____|_____|
| |     |     |
 \|     |     |
  *_____|_____|

A.8 Oskar's

       ____________            ______ 
      |\     \     \          |\     \
      | \_____\_____\         | \_____\
 _____| |     |     |         | |     |__________         __________________
|\     \|     |     |         |\|     |    \     \       |\     \     \     \
| \_____*_____|_____|  x 5    | *_____|_____\_____\      | *_____\_____\_____\
| |     |     |               | |     |     |     |      | |     |     |     |
 \|     |     |                \|     |     |     |       \|     |     |     |
  *_____|_____|                 *_____|_____|_____|        *_____|_____|_____|

A.9 Trikub

 ____________         ______                           ______ 
|\     \     \       |\     \                         |\     \
| \_____\_____\      | \_____\                        | \_____\
| |     |     |      | |     |__________         _____| |     |____
|\|     |     |      |\|     |    \     \       |\     \|     |    \
| *_____|_____|      | *_____|_____\_____\      | \_____*_____|_____\
| |     |     |      | |     |     |     |      | |     |     |     |
 \|     |     |       \|     |     |     |       \|     |     |     |
  *_____|_____|        *_____|_____|_____|        *_____|_____|_____|
 
 ______               ______                       ____________
|\     \             |\     \                     |\     \     \
| \_____\            | \_____\                    | \_____\_____\
| |     |____        | |     |____           _____| |     |     |
|\|     |    \       |\|     |    \         |\     \|     |     |
| *_____|_____\      | *_____|_____\        | \_____*_____|_____|
| |\     \    |      | |     |\     \       | |     |     |
 \| \_____\   |       \|     | \_____\       \|     |     |
  * |     |___|        *_____| |     |        *_____|_____|
   \|     |                   \|     |      
    *_____|                    *_____|      

and three single cubes in a different colour.

The object is to build 3x3x3 cubes with the three single cubes in various
positions.

E.g: * * *  as center    * * *  as edge    * *(3)  as          * *(2) as
     * S *               * * *             *(2)*   space       *(2)*  center
     * * *               * * S            (1)* *   diagonal   (2)* *  diagonal

(The other two variations with the single cubes in a row are impossible)

A.10

       ______         ______                     ______
      |\     \       |\     \                   |\     \
      | \_____\      | \_____\                  | \_____\
 _____| |     |      | |     |____              | |     |____
|\     \|     |      |\|     |    \             |\|     |    \
| \_____*_____|      | *_____|_____\         ___|_*_____|_____\
| |\     \    |      | |     |\     \       |\     \     \    |
 \| \_____\   |       \|     | \_____\      | \_____\_____\   |
  * |     |___|        *_____| |     |      | |     |     |___|
   \|     |                   \|     |       \|     |     |
    *_____|                    *_____|        *_____|_____|
 
 
 ______                           ______               ______    
|\     \                         |\     \             |\     \
| \_____\                        | \_____\            | \_____\
| |     |__________         _____| |     |____        | |     |____
|\|     |    \     \       |\     \|     |    \       |\|     |    \
| *_____|_____\_____\      | \_____*_____|_____\      | *_____|_____\______
| |\     \    |     |      | |     |     |     |      | |     |\     \     \
 \| \_____\   |     |       \|     |     |     |       \|     | \_____\_____\
  * |     |___|_____|        *_____|_____|_____|        *_____| |     |     |
   \|     |                                                    \|     |     |
    *_____|                                                     *_____|_____|

B: Coloured blocks puzzles

B.1 Kolor Kraze

Thirteen pieces.
Each subcube is coloured with one of nine colours as shown below.

The object is to form a cube with nine colours on each face.

 ______
|\     \
| \_____\
| |     |   ______     ______     ______     ______     ______     ______  
|\|  1  |  |\     \   |\     \   |\     \   |\     \   |\     \   |\     \  
| *_____|  | \_____\  | \_____\  | \_____\  | \_____\  | \_____\  | \_____\ 
| |     |  | |     |  | |     |  | |     |  | |     |  | |     |  | |     | 
|\|  2  |  |\|  2  |  |\|  2  |  |\|  4  |  |\|  4  |  |\|  7  |  |\|  9  |
| *_____|  | *_____|  | *_____|  | *_____|  | *_____|  | *_____|  | *_____|
| |     |  | |     |  | |     |  | |     |  | |     |  | |     |  | |     |
 \|  3  |   \|  3  |   \|  1  |   \|  1  |   \|  5  |   \|  5  |   \|  5  |
  *_____|    *_____|    *_____|    *_____|    *_____|    *_____|    *_____|
 
 
 ______     ______     ______     ______     ______     ______
|\     \   |\     \   |\     \   |\     \   |\     \   |\     \
| \_____\  | \_____\  | \_____\  | \_____\  | \_____\  | \_____\
| |     |  | |     |  | |     |  | |     |  | |     |  | |     |
|\|  9  |  |\|  9  |  |\|  3  |  |\|  6  |  |\|  6  |  |\|  6  |
| *_____|  | *_____|  | *_____|  | *_____|  | *_____|  | *_____|
| |     |  | |     |  | |     |  | |     |  | |     |  | |     |
 \|  7  |   \|  8  |   \|  8  |   \|  8  |   \|  7  |   \|  4  |  
  *_____|    *_____|    *_____|    *_____|    *_____|    *_____|

B.2

Given nine red, nine blue and nine yellow cubes.

Form a 3x3x3 cube in which all three colours appears in each of the 27
orthogonal rows.

B.3

Given nine red, nine blue and nine yellow cubes.

Form a 3x3x3 cube so that every row of three (the 27 orthogonal rows, the 18
diagonal rows on the nine square cross-sections and the 4 space diagonals)
contains neither three cubes of like colour nor three of three different
colours.

B.4

Nine pieces, three of each type.
Each subcube is coloured with one of three colours as shown below.

The object is to build a 3x3x3 cube in which all three colours appears in each
of the 27 orthogonal rows. (As in B.2)

 ______                     ______                     ______
|\     \                   |\     \                   |\     \
| \_____\                  | \_____\                  | \_____\
| |     |____              | |     |____              | |     |____
|\|  A  |    \   x 3       |\|  B  |    \   x 3       |\|  A  |    \   x 3
| *_____|_____\            | *_____|_____\            | *_____|_____\
| |     |     |            | |     |     |            | |     |     |
 \|  B  |  C  |             \|  A  |  C  |             \|  C  |  B  |
  *_____|_____|              *_____|_____|              *_____|_____|

C: Strings of cubes

C.1 Pululahua's dice

27 cubes are joined by an elastic thread through the centers of the cubes
as shown below.

The object is to fold the structure to a 3x3x3 cube.

 ____________________________________
|\     \     \     \     \     \     \
| \_____\_____\_____\_____\_____\_____\
| |     |     |     |     |     |     |
|\|  :77|77777|77:  |  :77|77777|77:  |
| *__:__|_____|__:__|__:__|_____|__:__|
| |  :  |___| |  :  |  :  |___| |  :  |
|\|  :  |    \|  777|777  |    \|  :  |
| *__:__|_____*_____|_____|_____*__:__|
| |  :  |     |     |___| |     |  :  |____
 \|  777|77777|77:  |    \|  :77|777  |    \
  *_____|_____|__:__|_____*__:__|_____|_____\
            | |  :  |     |  :  |     |     |
            |\|  :  |  +  |  777|77777|77:  |
            | *__:__|__:__|_____|_____|__:__|
            | |  :  |  :  |     |     |  :  |
             \|  +  |  :  |  :77|77777|777  |
              *_____|__:__|__:__|_____|_____|
                  | |  :  |  :  |
                   \|  777|777  |
                    *_____|_____|

C.1.X The C.1 puzzle type exploited by Glenn A. Iba (quoted)

INTRODUCTION

"Chain Cube" Puzzles consist of 27 unit cubies
with a string running sequentially through them. The
string always enters and exits a cubie through the center
of a face. The typical cubie has one entry and one exit
(the ends of the chain only have 1, since the string terminates
there). There are two ways for the string to pass through
any single cubie:
1. The string enters and exits non-adjacent faces
(i.e. passes straight through the cubie)
2. It enters and exits through adjacent faces
(i.e. makes a "right angle" turn through
the cubie)
Thus a chain is defined by its sequence of straight steps and
right angle turns. Reversing the sequence (of course) specifies
the same chain since the chain can be "read" starting from either
end. Before making a turn, it is possible to "pivot" the next
cubie to be placed, so there are (in general) 4 choices of
how to make a "Turn" in 3-space.

The object is to fold the chain into a 3x3x3 cube.

It is possible to prove that any solvable sequence must
have at least 2 straight steps. [The smallest odd-dimensioned
box that can be packed by a chain of all Turns and no Straights
is 3x5x7. Not a 3x3x3 puzzle, but an interesting challenge.
The 5x5x5 can be done too, but its not the smallest in volume].
With the aid of a computer search program I've produced
a catalog of the number of solutions for all (solvable) sequences.

Here is an example sequence that has a unique solution (up to reflections
and rotations):
(2 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1)
the notation is a kind of "run length" coding,
where the chain takes the given number of steps in a straight line,
then make a right-angle turn. Equivalently, replace
1 by Turn,
2 by Straight followed by a Turn.
The above sequence was actually a physical puzzle I saw at
Roy's house, so I recorded the sequence, and verified (by hand and computer)
that the solution is unique.

There are always 26 steps in a chain, so the "sum" of the
1's and 2's in a pattern will always be 26. For purposes
of taxonomizing, the "level" of a string pattern is taken
to be the number of 2's occuring in its specification.

COUNTS OF SOLVABLE AND UNIQUELY SOLVABLE STRING PATTERNS
 
 (recall that Level refers to the number of 2's in the chain spec)
 
        Level           Solvable        Uniquely
                        Patterns        Solvable
 
          0                 0               0
          1                 0               0
          2                24               0
          3               235              15
          4              1037             144
          5              2563             589
          6              3444            1053
          7              2674            1078
          8              1159             556
          9               303             187
         10                46              34
         11                 2               2
         12                 0               0
         13                 0               0
                       _______          ______
 
      Total Patterns:   11487            3658

SOME SAMPLE UNIQUELY SOLVABLE CHAINS

In the following the format is:

( #solutions palindrome? #solutions-by-start-type chain-pattern-as string )

where

#solutions is the total number of solutions up to reflections and rotations

palindrome? is T or NIL according to whether or not the chain is a palindrome

#solutions by-start-type lists the 3 separate counts for the number of
solutions starting the chain of in the 3 distinct possible ways.

chain-pattern-as-string is simply the chain sequence

My intuition is that the lower level chains are harder to solve,
because there are fewer straight steps, and staight steps are generally
more constraining. Another way to view this, is that there are more
choices of pivoting for turns because there are more turns in the chains
at lower levels.

Here are the uniquely solvable chains for level 3:

(note that non-palindrome chains only appear once --
I picked a "canonical" ordering)

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Level 3 ( 3 straight steps) -- 15 uniquely solvable patterns
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
 
(1 NIL (1 0 0) "21121111112111111111111")
(1 NIL (1 0 0) "21121111111111111121111")
(1 NIL (1 0 0) "21111112112111111111111")
(1 NIL (1 0 0) "21111111211111111111112")
(1 NIL (1 0 0) "12121111111111112111111")
(1 NIL (1 0 0) "11211211112111111111111")
(1 NIL (1 0 0) "11112121111111211111111")
(1 NIL (1 0 0) "11112112112111111111111")
(1 NIL (1 0 0) "11112112111111211111111")
(1 NIL (1 0 0) "11112111121121111111111")
(1 NIL (1 0 0) "11112111111211211111111")
(1 NIL (1 0 0) "11112111111112121111111")
(1 NIL (1 0 0) "11111121122111111111111")
(1 NIL (1 0 0) "11111112122111111111111")
(1 NIL (1 0 0) "11111111221121111111111")

C.2 Magic Interlocking Cube

(Glenn A. Iba quoted)

This chain problem is marketed as "Magic Interlocking Cube --
the Ultimate Cube Puzzle". It has colored cubies, each cubie having
6 distinctly colored faces (Red, Orange, Yellow, Green, Blue, and White).
The object is to fold the chain into a 3x3x3 cube with each face
being all one color (like a solved Rubik's cube). The string for
the chain is actually a flexible rubber band, and enters a cubie
through a (straight) slot that cuts across 3 faces, and exits
through another slot that cuts the other 3 faces. Here is a rough
attempt to picture a cubie:

(the x's mark the slots cut for the rubber band to enter/exit)

                       __________
                      /         /|
                    xxxxxxxxxxx  |
                  /         / x  |
                /_________/   x  |
               |          |   x  |
               |          |      |
               |          |      /
               |    x     |    /
               |    x     |  /
               |    x     |/
               -----x-----

Laid out flat the cubie faces would look like this:

                 _________
                |         |
                |         |
                |    x    |
                |    x    |
                |____x____|_________ _________ _________
                |    x    |         |         |         |
                |    x    |         |         |         |
                |    x    |    x x x x x x x x x x x    |
                |    x    |         |         |         |
                |____x____|_________|_________|_________|
                |    x    |
                |    x    |
                |    x    |
                |         |
                |_________|

The structure of the slots gives 3 choices of entry face, and 3 choices
of exit face for each cube.

It's complicated to specify the topology and coloring but here goes:

Imagine the chain stretched out in a straight line from left to right.
Let the rubber band go straight through each cubie, entering and
exiting in the "middle" of each slot.

It turns out that the cubies are colored so that opposite faces are
always colored by the following pairs:
Red-Orange
Yellow-White
Green-Blue
So I will specify only the Top, Front, and Left colors of each
cubie in the chain. Then I'll specify the slot structure.

        Color sequences from left to right (colors are R,O,Y,G,B,and W):
           Top:      RRRRRRRRRRRRRRRRRRRRRRRRRRR
           Front:    YYYYYYYYYYYYWWWYYYYYYYYYYYY
           Left:     BBBBBGBBBGGGGGGGGGBBGGGGBBB

For the slots, all the full cuts are hidden, so only
the "half-slots" appear.
Here is the sequence of "half slots" for the Top (Red) faces:
(again left to right)

           Slots:    ><><><><<><><<<><><>>>>><>>
                Where
                        > = slot goes to left
                        < = slot goes to right
                To be clearer,
                        > (Left):
                                 _______
                                |       |
                                |       |
                                xxxxx   |
                                |       |
                                |_______|
 
                        < (Right):
                                 _______
                                |       |
                                |       |
                                |   xxxxx
                                |       |
                                |_______|

Knowing one slot of a cubie determines all the other slots.

I don't remember whether the solution is unique. In fact I'm not
certain whether I actually ever solved it. I think I did, but I don't
have a clear recollection.

D: Blocks with pins

D.1 Holzwurm (Torsten Sillke quoted)

Inventer: Dieter Matthes
Distribution:
Pyramo-Spiele-Puzzle
Silvia Heinz
Sendbuehl 1
D-8351 Bernried
tel: +49-9905-1613

Pieces: 9 tricubes
Each piece has one hole (H) which goes through the entire cube.
The following puctures show the tricubes from above. The faces
where you see a hole are marked with 'H'. If you see a hole at
the top then there is a hole at the bottom too. Each peace has
a worm (W) one one face. You have to match the holes and the
worms. As a worm fills a hole completely, you can not put two
worms at both ends of the hole of the same cube.

        __H__               _____               _____
       |     |             |     |             |     |
       |     |             |     |W            |     |
       |_____|_____        |_____|_____        |_____|_____
       |     |     |       |     |     |       |     |     |
       |     |     |W      |     |     |H      |  H  |     |W
       |_____|_____|       |_____|_____|       |_____|_____|
 
        __H__               _____               _____
       |     |             |     |             |     |
       |     |             |     |             |  W  |
       |_____|_____        |_____|_____        |_____|_____
       |     |     |       |     |     |       |     |     |
       |     |     |       |  W  |  H  |       |     |  H  |
       |_____|_____|       |_____|_____|       |_____|_____|
          W
 
        __W__               _____               _____
       |     |             |     |             |     |
       |     |            H|     |H            |     |
       |_____|_____        |_____|_____        |_____|_____
       |     |     |       |     |     |       |     |     |
       |     |  H  |       |     |     |      H|     |  W  |
       |_____|_____|       |_____|_____|       |_____|_____|
                              W

   Aim: build a 3*3*3 cube without a worm looking outside.
        take note, it is no matching problem, as
                 |     |
          worm> H|     |H <worm
                 |     |
        is not allowed.

E: Other

E.1 Rubik's cube

E.2 Magic cube

Make a magic cube with the numbers 1 - 27.

 

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