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©1999 by Martin H. Watson

For questions or comments regarding this site, contact Martin H. Watson

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

My puzzle collection - Coming soon!!

Images of some of my puzzles - Coming soon!!

Puzzle shops in the UK - Coming soon!!

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,

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.

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:

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.

In the USA wood unit cubes and pentacubes are available from

The following suggestions will help speed up the process for making sets out of unit cubes. (When making my first set I laboriously glued each face, before ‘automating’ the process!) The second set will only take you a few minutes. Follow the glue manufacturer’s advice, with the following extra tips.

Each of the pentacubes are made from two 2x1 cube sections, connected in varying ways, plus the addition of a fifth single cube. The only exception is the X piece. Initially you need twenty two 2x1 sections. If the adhesive manufacturer advises gluing both surfaces to be joined, form a regular layer of 44 cubes, either 4x11 or 6x6 with a few extras. If only one piece is to be glued, use half these quantities. By holding them in a solid layer, it is possible to cover one face of each cube with glue very rapidly.

Then split the regular layer into two equal areas and hold the two glued areas together. Push firmly, then separate the mass into glued pairs. Then lay out the twenty two pairs to resemble the completed pentominoes. (A small sheet of glass or ceramic tile is good for this - newspaper sticks to the pieces.) Next lay out eleven additional unit cubes. The remaining 5 cubes are for the X pentomino. It helps to position them in ‘alphabetical’ order. (The pieces resemble the (western) letters FILNP TUVWXYZ.) It is then clear which faces require glue. Glue the relevant faces, with a diagram in front of you. Assemble the 12 pieces, and leave to dry.

Most of the pentacube pairs are in contact with each other apart from the squares which were glued. I find that gluing just the one square is quite adequate. It also means that there is a little bit of ‘give’ in the assembled result, which is useful if the constituent cubes don’t quite line up.

Mathpuzzles

I found this on Southwest Missouri State University's Challenge Problem Page. Find the dimensions of the smallest rectangular box which can be completely filled with copies of the "U-pentacube". What about the smallest cubical box? Try it mentally first. I haven’t given this a lot of thought, hence no answer or link yet.

©1999 by Martin H. Watson

For questions or comments regarding this site, contact Martin H. Watson