This puzzle is called Bolygok, which means Planets in Hungarian. It consists of a
solid cube-shaped frame in which is embedded at
each corner a ball. These 8 balls can be rotated individually. The balls have 6
coloured dots, so that when they are aligned each of the six faces of the cube
will show 4 coloured dots (one on each ball).
The dots on the balls come in four colours - yellow, blue, red and green. Every
ball has one or two dots of each colour. There are several possible objectives for
the puzzle which will be detailed below, but two obvious ones are: Get the four dots
of each cube face to be the same colour, or get the dots on each face to be all different.
This puzzle was invented and patented by Jozsef Bognar, and patented in Hungary
26 August 1980, HU 1347. It also has an equivalent British patent GB 2,076,663,
granted 9 December 1981. Note that the patent also covers the variation of
Instant Insanity pictured on that page.
Below is the arrangement of colours on the balls.
There are 15 possible objectives outlined in the instruction pamphlet (my thanks
go to Milan Vodicka for sending it to me). It first decribes the 5 possible
configurations each face may have:
- Four colours on a face.
- Two colours on a face, coloured as two adjacent pairs.
- Two colours on a face, coloured like a checkerboard (i.e. diagonal pairs).
- Two colours on a face, three of one colour, one of another colour.
- One colour on a face, i.e. all the same colour.
Some additional constraints can then be added too, such as:
- Each face has the colours in the same order
- Opposite faces of the puzzle are identical
- One colour is not visible.
- Each face uses only combinations of yellow and red or of blue and green.
The fifteen possible puzzles are:
1, 1a,
2, 2b, 2c, 2d,
3, 3b, 3c, 3d,
4, 4b, 4c, 4d,
and finally 5.
Solutions:
All solutions are shown in the orientation shown above, i.e. the Upper Front Left ball
is the one with red dots on opposite sides, and the Bottom Front Left has opposing green
dots. Click on a number below to go directly to the appropriate section.
Solutions 1:
I did a computer search for solutions with all 4 colours on each face, and there
are 59418 solutions. If you insist that each ball shows 3 different colours as well
then there are only 3236 solutions left. Of these, 12 solutions have central
symmetry, and these are shown below. Again the puzzle is assumed to be in the
orientation shown above (i.e. the Upper Front Left ball has its red dots on
opposite sides, and the Bottom Front Left has opposing green dots).
| | Y | R | | | | |
| | B | G | | | | |
| B | R | G | B | Y | G | Y | R |
| Y | G | Y | R | B | R | G | B |
| | R | Y | | | | |
| | G | B | | | | |
|
| | G | Y | | | | |
| | R | B | | | | |
| B | G | B | R | Y | R | G | Y |
| Y | R | G | Y | B | G | B | R |
| | Y | G | | | | |
| | B | R | | | | |
|
| | B | Y | | | | |
| | R | G | | | | |
| R | B | G | B | Y | G | R | Y |
| Y | G | R | Y | R | B | G | B |
| | Y | B | | | | |
| | G | R | | | | |
|
| | R | G | | | | |
| | B | Y | | | | |
| Y | G | R | G | B | R | Y | B |
| B | R | Y | B | Y | G | R | G |
| | G | R | | | | |
| | Y | B | | | | |
|
| | G | R | | | | |
| | B | Y | | | | |
| B | G | R | B | R | Y | G | Y |
| R | Y | G | Y | B | G | R | B |
| | R | G | | | | |
| | Y | B | | | | |
|
| | Y | G | | | | |
| | R | B | | | | |
| G | B | G | R | Y | R | Y | B |
| Y | R | Y | B | G | B | G | R |
| | G | Y | | | | |
| | B | R | | | | |
|
| | R | Y | | | | |
| | G | B | | | | |
| Y | B | R | Y | G | R | G | B |
| G | R | G | B | Y | B | R | Y |
| | Y | Y | | | | |
| | B | G | | | | |
|
| | B | Y | | | | |
| | G | R | | | | |
| Y | R | B | Y | B | G | R | G |
| B | G | R | G | Y | R | B | Y |
| | Y | B | | | | |
| | R | G | | | | |
|
| | Y | R | | | | |
| | G | B | | | | |
| B | R | B | Y | G | Y | G | R |
| G | Y | G | R | B | R | B | Y |
| | R | Y | | | | |
| | B | G | | | | |
|
| | Y | G | | | | |
| | B | R | | | | |
| G | R | G | Y | B | Y | R | B |
| B | Y | R | B | G | R | G | Y |
| | G | Y | | | | |
| | R | B | | | | |
|
| | B | G | | | | |
| | R | Y | | | | |
| R | G | B | G | B | Y | R | Y |
| B | Y | R | Y | R | G | B | G |
| | G | B | | | | |
| | Y | R | | | | |
|
| | B | R | | | | |
| | G | Y | | | | |
| Y | B | R | B | R | G | Y | G |
| R | G | Y | G | Y | B | R | B |
| | R | B | | | | |
| | Y | G | | | | |
|
Solutions 1a:
There are 3 solutions in which each face shows the 4 colours in the same
order.
| | B | Y | | | | |
| | R | G | | | | |
| Y | G | Y | G | Y | G | Y | G |
| B | R | B | R | B | R | B | R |
| | G | R | | | | |
| | Y | B | | | | |
|
| | B | R | | | | |
| | Y | G | | | | |
| Y | B | R | G | Y | B | R | G |
| G | R | B | Y | G | R | B | Y |
| | G | Y | | | | |
| | R | B | | | | |
|
| | R | Y | | | | |
| | G | B | | | | |
| R | Y | R | Y | Y | B | R | Y |
| G | B | G | B | R | G | G | B |
| | B | G | | | | |
| | Y | R | | | | |
|
Solutions 2:
My computer search for solutions with each face half one colour and half another,
showed there are 26174 solutions, all but 1848 cases have some balls which show the
same colour twice.
Solutions 2b:
There are 124 solutions with opposite faces identical. Only 9 of these
cases have no balls with duplicate colours, and these are shown below.
| | Y | G | | | | |
| | Y | G | | | | |
| B | R | G | Y | R | R | Y | G |
| B | R | G | Y | B | B | Y | G |
| | Y | G | | | | |
| | Y | G | | | | |
|
| | Y | Y | | | | |
| | B | B | | | | |
| B | Y | R | R | Y | B | R | R |
| B | Y | G | G | Y | B | G | G |
| | B | B | | | | |
| | Y | Y | | | | |
|
| | B | Y | | | | |
| | B | Y | | | | |
| R | R | Y | B | R | R | B | Y |
| G | G | Y | B | G | G | B | Y |
| | B | Y | | | | |
| | B | Y | | | | |
|
| | R | B | | | | |
| | R | B | | | | |
| Y | G | B | R | Y | Y | R | B |
| Y | G | B | R | G | G | R | B |
| | R | B | | | | |
| | R | B | | | | |
|
| | Y | G | | | | |
| | Y | G | | | | |
| G | G | R | B | Y | Y | R | R |
| Y | Y | R | B | G | G | B | B |
| | G | Y | | | | |
| | G | Y | | | | |
|
| | Y | Y | | | | |
| | B | B | | | | |
| R | R | G | G | R | R | G | G |
| G | G | R | R | G | G | R | R |
| | Y | B | | | | |
| | Y | B | | | | |
|
| | Y | G | | | | |
| | Y | G | | | | |
| R | R | G | Y | R | R | Y | G |
| B | B | G | Y | B | B | Y | G |
| | Y | G | | | | |
| | Y | G | | | | |
|
| | G | R | | | | |
| | G | R | | | | |
| Y | B | R | G | Y | Y | G | R |
| Y | B | R | G | B | B | G | R |
| | G | R | | | | |
| | G | R | | | | |
|
| | G | G | | | | |
| | Y | Y | | | | |
| Y | G | R | R | G | Y | R | R |
| Y | G | B | B | G | Y | B | B |
| | Y | Y | | | | |
| | G | G | | | | |
|
Solutions 2c:
There are 102 solutions with only three colours (no blue), all of which have
one or more balls with duplicate colours. The two solutions that also have opposite
faces the same are shown below.
| | Y | G | | | | |
| | Y | G | | | | |
| R | G | R | Y | R | R | R | R |
| R | G | R | Y | G | G | Y | Y |
| | Y | Y | | | | |
| | G | G | | | | |
|
| | R | R | | | | |
| | Y | Y | | | | |
| Y | G | R | G | Y | G | R | R |
| Y | G | R | G | Y | G | G | G |
| | Y | Y | | | | |
| | R | R | | | | |
|
Solutions 2d:
There are 32 solutions that use red/yellow and blue/green combinations. Only 4 of
them have no balls with duplicate colours, and these are shown below.
| | R | Y | | | | |
| | R | Y | | | | |
| B | G | Y | R | G | B | R | Y |
| B | G | Y | R | G | B | R | Y |
| | B | B | | | | |
| | G | G | | | | |
|
| | R | R | | | | |
| | Y | Y | | | | |
| Y | R | G | B | R | Y | B | G |
| Y | R | G | B | R | Y | B | G |
| | B | G | | | | |
| | B | G | | | | |
|
| | Y | R | | | | |
| | Y | R | | | | |
| R | R | G | G | Y | Y | B | B |
| Y | Y | B | B | R | R | G | G |
| | G | G | | | | |
| | B | B | | | | |
|
| | Y | Y | | | | |
| | R | R | | | | |
| G | G | Y | Y | B | B | R | R |
| B | B | R | R | G | G | Y | Y |
| | G | B | | | | |
| | G | B | | | | |
|
Solutions 3:
My computer search for checkerboard solutions showed there are 1035 solutions,
all but 66 cases have some balls which show the same colour twice.
Solutions 3b:
There are 16 checkerboard solutions with opposite faces identical. All are shown
below. The last 4 cases have balls with duplicate colours.
| | Y | R | | | | |
| | R | Y | | | | |
| B | G | Y | G | B | G | Y | G |
| G | B | G | Y | G | B | G | Y |
| | Y | R | | | | |
| | R | Y | | | | |
|
| | B | Y | | | | |
| | Y | B | | | | |
| G | B | R | Y | G | B | R | Y |
| B | G | Y | R | B | G | Y | R |
| | B | Y | | | | |
| | Y | B | | | | |
|
| | G | Y | | | | |
| | Y | G | | | | |
| Y | B | R | B | Y | B | R | B |
| B | Y | B | R | B | Y | B | R |
| | G | Y | | | | |
| | Y | G | | | | |
|
| | Y | B | | | | |
| | B | Y | | | | |
| B | R | Y | G | B | R | Y | G |
| R | B | G | Y | R | B | G | Y |
| | Y | B | | | | |
| | B | Y | | | | |
|
| | B | Y | | | | |
| | Y | B | | | | |
| G | R | G | Y | G | R | G | Y |
| R | G | Y | G | R | G | Y | G |
| | B | Y | | | | |
| | Y | B | | | | |
|
| | G | R | | | | |
| | R | G | | | | |
| Y | G | Y | B | Y | G | Y | B |
| G | Y | B | Y | G | Y | B | Y |
| | G | R | | | | |
| | R | G | | | | |
|
| | Y | G | | | | |
| | G | Y | | | | |
| B | Y | R | G | B | Y | R | G |
| Y | B | G | R | Y | B | G | R |
| | Y | G | | | | |
| | G | Y | | | | |
|
| | B | G | | | | |
| | G | B | | | | |
| G | Y | R | Y | G | Y | R | Y |
| Y | G | Y | R | Y | G | Y | R |
| | B | G | | | | |
| | G | B | | | | |
|
| | G | B | | | | |
| | B | G | | | | |
| Y | R | Y | B | Y | R | Y | B |
| R | Y | B | Y | R | Y | B | Y |
| | G | B | | | | |
| | B | G | | | | |
|
| | Y | R | | | | |
| | R | Y | | | | |
| B | Y | B | G | B | Y | B | G |
| Y | B | G | B | Y | B | G | B |
| | Y | R | | | | |
| | R | Y | | | | |
|
| | B | R | | | | |
| | R | B | | | | |
| G | Y | B | Y | G | Y | B | Y |
| Y | G | Y | B | Y | G | Y | B |
| | B | R | | | | |
| | R | B | | | | |
|
| | G | Y | | | | |
| | Y | G | | | | |
| Y | R | G | B | Y | R | G | B |
| R | Y | B | G | R | Y | B | G |
| | G | Y | | | | |
| | Y | G | | | | |
|
| | G | Y | | | | |
| | Y | G | | | | |
| B | G | R | Y | G | B | R | Y |
| G | B | Y | R | B | G | Y | R |
| | G | Y | | | | |
| | Y | G | | | | |
|
| | B | G | | | | |
| | G | B | | | | |
| Y | R | Y | G | R | Y | Y | G |
| R | Y | G | Y | Y | R | G | Y |
| | B | G | | | | |
| | G | B | | | | |
|
| | R | G | | | | |
| | G | R | | | | |
| B | Y | R | Y | B | Y | R | Y |
| Y | B | Y | R | Y | B | Y | R |
| | G | R | | | | |
| | R | G | | | | |
|
| | R | B | | | | |
| | B | R | | | | |
| Y | R | Y | G | Y | R | Y | G |
| R | Y | G | Y | R | Y | G | Y |
| | B | R | | | | |
| | R | B | | | | |
|
Solutions 3c:
There are 6 checkerboard solutions which use just three colours. Note that the
missing colour must be blue, since each of the other colours occurs on opposite sides
of a ball and hence must be visible.
| | Y | G | | | | |
| | G | Y | | | | |
| R | Y | R | G | G | Y | R | G |
| Y | R | G | R | Y | G | G | R |
| | Y | G | | | | |
| | G | Y | | | | |
|
| | R | G | | | | |
| | G | R | | | | |
| Y | R | Y | G | G | Y | Y | G |
| R | Y | G | Y | Y | G | G | Y |
| | R | G | | | | |
| | G | R | | | | |
|
| | G | Y | | | | |
| | Y | G | | | | |
| Y | R | G | R | G | R | G | R |
| R | Y | R | G | R | G | R | G |
| | G | Y | | | | |
| | Y | G | | | | |
|
| | Y | G | | | | |
| | G | Y | | | | |
| R | Y | R | G | G | Y | R | G |
| Y | R | G | R | Y | G | G | R |
| | Y | R | | | | |
| | R | Y | | | | |
|
| | R | G | | | | |
| | G | R | | | | |
| Y | R | Y | G | G | R | R | G |
| R | Y | G | Y | R | G | G | R |
| | R | Y | | | | |
| | Y | R | | | | |
|
| | G | Y | | | | |
| | Y | G | | | | |
| Y | R | G | R | G | Y | G | R |
| R | Y | R | G | Y | G | R | G |
| | G | Y | | | | |
| | Y | G | | | | |
|
Solutions 3d:
There are just 3 checkerboard solutions which use only green/blue and red/yellow
combinations.
| | Y | R | | | | |
| | R | Y | | | | |
| R | Y | B | G | Y | R | B | G |
| Y | R | G | B | R | Y | G | B |
| | B | G | | | | |
| | G | B | | | | |
|
| | Y | R | | | | |
| | R | Y | | | | |
| R | Y | G | B | R | Y | G | B |
| Y | R | B | G | Y | R | B | G |
| | G | B | | | | |
| | B | G | | | | |
|
| | Y | R | | | | |
| | R | Y | | | | |
| G | B | Y | R | G | B | Y | R |
| B | G | R | Y | B | G | R | Y |
| | B | G | | | | |
| | G | B | | | | |
|
Solutions 4:
My computer search for "3+1" solutions showed there are 2,664,606 of them, but
there are 125,748 solutions in which each ball shows three distinct colours.
Solutions 4b:
There are 4,236 solutions with opposite faces identical, of which only 456 solutions
are such that each ball shows three distinct colours. One of these is shown below.
| | Y | Y | | | | |
| | G | Y | | | | |
| B | B | R | R | G | B | R | G |
| G | B | G | R | B | B | R | R |
| | Y | Y | | | | |
| | Y | G | | | | |
Solutions 4c:
There are 18,530 solutions without the colour blue, all of which have one or more
balls that show only two colours. Of these, 154 also have opposite faces identical, i.e. are
also solutions to 4b. One of these is shown below.
| | G | R | | | | |
| | R | R | | | | |
| R | Y | G | G | Y | Y | G | Y |
| Y | Y | G | Y | Y | R | G | G |
| | R | R | | | | |
| | R | G | | | | |
Solutions 4d:
Each face uses only combinations of yellow and red or of blue and green. There are 2048
solutions, of which 256 have no balls showing duplicate colours. One of these is shown below.
There is no position that solves both 4b and 4d simultaneously.
| | Y | R | | | | |
| | R | R | | | | |
| B | B | Y | Y | B | G | Y | R |
| G | B | Y | R | G | G | Y | Y |
| | G | B | | | | |
| | B | B | | | | |
Solutions 5:
There are two positions where each face has a single colour.
