Pent-2-Dec Puzzles
Pent-2-Dec Puzzles take the familiar 12 pentominoes into 3D (pentacubes) and
then join them in pairs in specific face-to-face combinations. You then have to
assemble these new 10-cube pieces into a given form. I designed these puzzles
to be very difficult, and to meet other criteria detailed below. I implore you
to make the pieces if you have even the slightest interest in polyomino or
assembly puzzles. You will not be disappointed. If there is sufficient
interest, I have several more available… I would very much like your comments,
and I will shortly start a section containing comments, suggestions, pleas for
mercy etc., etc.
The
XXX puzzle - assemble six very mean
pieces into a 5x6x2 block with no holes or protuberances.
The LX puzzle - very, very, very much harder. Assemble 12 wicked
pieces into a 8x8x2 layer with a central 2x2x2 hole.
The idea
The Solutions - Not that you’ll find them here of course!!
How I
designed the puzzles
Pent-2-Dec Puzzles
The idea
To try and come up with something new for my web pages I wanted to document the
processes involved in designing and making a new mechanical puzzle, and
designing one myself seemed a good way to go. I think that what I have
developed is far better than I would have thought possible. I regret that I
have been unable to find the imagination to come up with a successor of Rubik’s
Cube for the next millennium. I hope to have started the rolling of a grain of
sand which may gather momentum and become perhaps a large pebble…
What I wanted to do was design a puzzle that,
is
very difficult;
has only one solution;
would be difficult to be solved by writing a computer program, owing to the
complexity;
that has a pleasing solution;
has pieces that are nice to look at and touch;
be simple in concept.
I think I have achieved that with Pent-2-Dec, a pentomino-style puzzle for the
next millennium. The name Pent-2-Dec just came to me one day, suggesting the
step up from a puzzle with pieces made of 5 unit cubes to one of 10 such cubes.
Also the solutions are ‘two-deck’ solutions. The name just seemed right!
Despite working in information technology, I have little grasp of software that
has a spatial function, so I cannot confidently say that writing a program to
go through every orientation of 12 very-irregularly-shaped pieces, and to
examine the interconnection of these 12 pieces would be a great challenge. I do
not know if any existing software is functional enough to be able to handle the
requirements here. If it doesn’t already exist, I hope I will give someone a
new challenge.
Something niggles me about using a computer to solve puzzles, whereas using a
computer to design puzzles seems like a Good Thing. I can accept that a program
to generate the best solution to aim for manually is OK, but merely solving a
puzzle using a computer is too close to looking at the answer slip.
Nevertheless if anyone does solve these puzzles by writing software, I would be
interested to know how long the program took to find the solution, and also
whether my solutions are unique.
It seems fitting that I used pentominoes, (or more correctly pentacubes - the
3D version of pentominoes), popularised by Solomon Golomb and Martin Gardner,
as a basis for the puzzle. Pentominoes and Gardner’s books were my earliest
contacts with recreational mathematics, as a child in the late 1960s.
The initial two puzzles here require pentacubes, these being all the different
shapes 5 cubes can make when joined whole face to whole face in a plane. The easier
Pent-2-Dec puzzle which I developed as a stepping-stone, XXX, has six pieces,
and requires one full set of 12 planar pentacubes, while the full LX Pent-2-Dec
puzzle needs two full sets. Most toy shops sell some sort of set for a few
pounds,
Ideally all the pieces of both sets should be of identical colours. The ardent
puzzler wouldn’t want to make things too easy. Alternatively a cheaper option
is to make a set from wooden or plastic unit cubes. The section below on tips
for making your own set contains addresses of suppliers etc.
I have assembled the XXX Pent-2-Dec puzzle perhaps 20 times while testing, and
it is never easy. It assembles into a 5x6x2 regular block. (Two layers of 30,
hence XXX). Once assembled only one individual 10-cube piece can be removed,
all the others must initially be removed as part of a larger group.
The LX Pent-2-Dec puzzle was very difficult to assemble, even when the pieces
of each of the two sets were made up from cubes of differing colours and I had
the (uncoloured) solution in front of me. With two sets, the number of ways of
selecting a full set of 12 different pentacubes ‘face up’ is about forty
million! There are about five dozen ways of assembling an 8x8 area with a
central 2x2 hole. (A layer of 60, hence LX). The LX Pent-2-Dec puzzle requires
the simultaneous solving of two such interlocking layers, forming a ‘square
doughnut’. I think I have selected two solutions which, when combined, form a
pleasing puzzle. Only two individual 10-cube pieces can be removed from the
completed 8x8x2 assembly, all the others must initially be removed as part of a
larger group. Hunting for (partial-)serial-assembly solutions without using a
computer is no fun… I looked at many, many solutions, initially rejected some
single-layer arrangements, as unsuitable for different reasons. Then when
trying to integrate two single-layer arrangements I came across things I didn’t
like, and then it was back to square (or should that be cube) one. I can’t go
into greater detail with regard to some of the things I tried to achieve
without giving hints so you’ll just have to enjoy the puzzles.
The solutions - Not that you’ll find them here of course!!
There are many ways of cheating. You will know deep down inside if you are
cheating in some way. Indications you should look for are:
Lack of a headache.
Not being nasty and irritable to friends, their friends, your family, small
furry animals, etc.
Managing to do a full day’s work.
Getting a good night’s sleep.
Smiling, except bemused leering.
A smell of burning if you try to use any high-intensity heat force on the
puzzle pieces.
Searching the Internet for 8x8 layer solutions with a 2x2 central hole.
How I designed the puzzles
I decided to document the thought processes behind the design of the Pent-2-Dec
puzzles. The idea of the puzzle initially just came into my head. I can't come
up with a specific moment when it came. I had spent some time earlier in the
day looking at pentomino web sites to collect links for my own site, but I can't
say if that helped. It was at this point that I started keeping these notes,
which have only been slightly tidied up. I thought it would be interesting both
for myself and others to follow through the hurdles and pitfalls as they
arrived. I was keen to create a puzzle that was elegant in its simplicity, and
be challenging to those trying it and also would challenge not only those who
enjoy creating software to solve mechanical puzzles, but also to try and
challenge the actual computers and the software. I think I have succeeded.
I decided to use 2 layers of 8x8 with a central square of 4 units, this being
the most aesthetic pentomino solution. Also, as I have discovered while looking
for solutions, there are not many websites publishing the 5 dozen or so
possibilities. The first hurdle was whether or not each of the 12 top layer
pentominoes could be uniquely paired with a (ideally different) pentomino from
the bottom layer. Joining each pair by just a single unit cube seemed a nice
idea. Would it be possible? In a 8x8 grid I subsequently thought, before any
trial, that a one-unit link would be impossible. Perhaps a two-unit constant
link would be possible, and still meets the aesthetics of a puzzle. I might
have to consider several different 8x8 solutions to fulfil that need. This was
a goal which I never achieved, although I don’t think it detracts from the
puzzles whatsoever. Actually, 8 pieces are connected by two cube faces, and the
others by 1, 1, 3 and 4.
The next problem I thought of was would there be too many top/bottom pairs with
an aspect which would make them too obviously edge pieces? or corner pieces? or
parts of the centre void edges or corners? Because of the centre hole, would
they almost all be edge pieces, either inside or outside? Only the X piece
cannot be a corner piece of the 8x8 arrangement. There are some edges of some
pieces which cannot be outside edges of the 8x8. Some pieces suggest themselves
more as edge pieces, especially L,I and Y.
At this point I decided that I needed to assemble some unit cubes to make two
full sets of pentacubes. This gave rise to the question should the 2 sets all
be of one colour? or each set a different colour - this latter would make it a
bit easier though. The 8x8 solutions have eluded me so far, and the 2-colour
option would mean the solver would be seeking just one possible specific
solution for one layer, automatically solving the other.
Another option would be a checkerboard pattern with the top layer rotated 90
degrees to keep the pattern. If all the pieces were the same colour this would
be the hardest option as it would be impossible to tell if a piece was ‘upside
down’. Possibly some pieces could have more than a top face and a bottom face
if the pairs of pieces had a 2x2 cross-section. 5 pieces are only two units
wide, I,L,U,P and Y. If they were paired with another such piece from the other
layer, still with a 2x2 cross section, it would increase the complexity,
although would it spoil the elegance if each layer didn’t contain a full set of
pentominoes? If it was possible to join two of these pieces in such a way that
it was impossible to tell from examinination which two pieces were used, that
would also be a nice feature and add complexity. As it turned out only one
piece has a 2x2 cross-section. Only two pieces, I and U couldn't form the two
sides of a corner of the centre 2x2 square. (Later note: For the elegance of
the puzzle, I decided that each layer must contain a complete set. Any
comments?
For the development I decided to make two full sets of pentominoes from cheap
plastic unit cubes glued together. Each set would be a different colour, to
simplify my initial development task. I would then be able to glue pairs of
pentacubes of opposite colours together with just a dab of glue, as inevitably
some pairs would need much trial and error, and that would also make separation
possible. There was a lot of gluing and separation! Ideally I did not want to
duplicate 10-cube pieces, although it is unlikely that that would happen by
chance. In fact I wanted to avoid two 10-cube pieces being made up from the
same lettered pairs. (Later note: This I achieved and it is now clear that it
was essential aesthetically.)
Having assembled two sets of pentominoes based on 1cm plastic cubes I arranged
each set as an 8x8 with a 4 unit square hole in the centre. I placed one set on
top of the other and started trying to pair each piece with a partner from the
lower level. This was very challenging and time-consuming, and I confirmed my
doubts that I wouldn’t find a solution where there is always a similar number
of cubes touching.
Choosing from the 65 solutions was a challenge. Should I use the same solution
set for both layers, with a rotation and/or flip to make the two sets
different? Or go for two totally different solutions? I won’t comment further.
Practising with the pentominoes forming two 5x6 rectangles I discovered that
sequential construction was something I had not thought of. After enjoying the
beauty of having to assemble some groups sequentially, it soon became a
requirement that as much as possible I would in-build this idea. The 2x5x6 is a
good challlenge, although I have now got to the stage where I am too familar
with the pieces for it to be very hard.
I have now made the 8x8 puzzle, still in 2 separate colous. Some sequential
construction is required, although 2 pieces are free to be removed
individually. The corners are not obvious, and the edges are not too obvious.
Putting it together in two colous is easier then the 2x5x6 single colour
puzzle. If I make a single colour 2x8x8 I think it will be tough even with the
solution.