Listener 4190: Can You Cube A Cube?
Posted by Listen With Others on 9 June 2012
2×2×2 from the setter’s point of view
When I came into school on the Monday morning following publication, I found this on my desk.
It was from two of my colleagues in the Madras College maths department who are married to each other, and this was the first Listener puzzle they had managed to complete after 3 years of trying. You will note it only took them 2 hours; however they didn’t send in their entry!!
The puzzle is based on the Eight Blocks to Madness which is a subset of the 30 MacMahon Colour Cubes. I first discovered these cubes after reading about them fifteen years ago in Martin Gardner’s book ‘Fractal Music, Hypercards and More‘.
Take a cube and six different colours then follow the rules that each face must be a solid colour and each cube must contain all of the colours. There are 30 different arrangements that split into 15 mirror image pairs.
I was so taken with them that I went to my local DIY shop and purchased a long baton of wood and cut it up into 30 cubes. The wood was in fact long enough to make 3 sets of the cubes!! I spent a few days carefully painting the faces until I had 3 sets of the cubes. One set I took to school to use with classes and the other two stayed at home.
In Gardner’s article, he explained how the problem was solved by the English mathematician John Conway who is based in the USA. Essentially you want to build a 2x2x2 replica of one of the cubes and, in order to identify the cubes you need, you make use of Conway’s matrix. This is a 6×6 matrix/array/square with the leading diagonal empty. The 30 off-diagonal elements are where the cubes are placed so that mirror image cubes are on opposite sides of the diagonal which acts as a mirror. You choose one of the cubes then take the mirror image cube. In other words, if you choose the cube in row 3 column 2 then you want the one in row 2 column 3. Then you take the 4 other cubes in row 2 and the 4 other cubes in column 3 and those are the 8 that will make your enlargement of the cube in row 3 column 2.
I chose one of the cubes and then with the help of Conway’s matrix I identified the 8 cubes that were required to build the 2x2x2 enlargement.
I had intended to have a grid that contained 8 top views of a cube – basically a square with 4 trapezia on each edge so that you saw 5 of the 6 faces as shown below.
I was going to use clues for the colour of the faces, then clues for the arrangement of the colours and finally a clue detailing which cube went where. Solvers would then have to orient the cubes correctly to get the solution. It was quite complicated.
I put the puzzle aside and, over the next 14 years, it would resurface at the top of the pile now and again only to find itself placed firmly at the bottom! I just couldn’t make any headway with it at all.
One night last year whilst sitting bemoaning the fact that I’d failed to solve one of The Magpie numericals correctly, and basking in the post ENT L4125 euphoria – only tainted by my own stupidity for not realising that there was only one ambiguous cell apart from the middle one and not four as I’d thought – that I started wondering what I could do for my next Listener submission. I decided that it ought to be a Listener tribute to recognise the contribution that Martin Gardner had made to the world of recreational mathematics. I had already done one for The Magpie – Tribute to a Horticulturist M104 – that was a clone of my Listener puzzle Dedication L3944. In that puzzle to celebrate Euler’s tercentenary, I had used Laplace’s message imploring people to read Euler, so for the puzzle in The Magpie I implored people to read Gardner instead! That had been a tough set and is recorded in a blog on The Magpie site. I had used the message type of puzzle in that there was too much material for me to choose from his writings and more importantly I knew that another setter – I won’t embarrass him again but he knows who he is – would have made a better job of it than I would!! At this point I remembered the MacMahon cube puzzle and that I really should see it off my books and as I’d first learnt about them from Martin Gardner this would be the perfect tribute. Whilst thinking about this, Googly’s Verity Hill’s Dice Box L3737 popped into my head. In that puzzle solvers had to identify 3 nets of a cube that when folded up made 3 dice and allowed the solver to complete the unclued entries and hence find the words FACT and BARROW. Inspiration at last, and why not copy from the master setter Googly himself? I would use 8 different non-overlapping nets of a cube in the grid.
I went to my study and got out the 1cm squared paper. Now what size of grid? 7×7 for 48 cells with 1 left over – more than likely impossible. So 8×8 then? Yes much better as 8 is a perfect cube and so too is the number of cells 64, so more in keeping with the theme. It only took 15 minutes to come up with a grid that had 8 different non-overlapping nets of a cube within the 8×8 boundary. Six of the eleven nets are trivial to find in that they are just a line of four squares with the remaining two squares appearing one on either side of the line of four. The other five are a bit trickier to find and so I used four of them!
The following evening I dug out all my original workings and notes and was surprised to find on my computer that I’d actually come up with clues for the arrangement of the colours on the cubes. Could I use these and go back to my original idea? No. They were absolutely bowfin’!!
Then came the painstaking chore of transcribing the colours to digits, placing them into the nets and rearranging some of the digits so that adjacent cells contained the same digit as they would have to be in a different net. This took the rest of the evening and I checked that there was a logical path that would lead solvers to the 8 correct nets. There would be no point in writing any clues until that was done.
The puzzle was going to have 3 parts – a grid fill, obtaining the nets and finally assembly of the replica. All of these parts have varying degrees of difficulty which I could control, and I reckoned that the middle part, whilst not difficult, wasn’t trivial either in that if you’re not conversant with the nets of a cube then it would be tricky and visuo-spatial ability is something you’ve either got or not. The last part I felt was hard – after all it is the basis for the puzzle Eight Blocks to Madness. You can buy this puzzle from a company called Pentangle whose web address is pentangle-puzzles.co.uk and search for Eight Colour Cubes.
[ If we go back to the original puzzle which used colours on the faces instead of digits and you were just given 8 cubes and told to build a 2x2x2 enlargement which had each face a solid colour and all the colours had to appear on the outside of the cube as well as the domino condition being satisfied for the touching inside faces then you would probably still be at the puzzle even now as you are not told which cube you are trying to enlarge!! Of course, if you adopted some graph theory then you would undoubtedly find the solution and you’d still have your hair intact!! However graph theory, as far as I’m aware, doesn’t appear in GCSE maths and certainly not in its Scottish equivalent and as such would mean that the puzzle would not be acceptable if that was the only means of solution. ]
I felt therefore that solvers would appreciate some help and so decided that the net below the grid that solvers had to complete to show that they’d found the correct replica would contain a couple of digits to give them a start. Giving just one digit is just plain stupid as is giving five digits and, worse still, all six and would show that I’d lost the plot entirely!! This would also help John Green when it came to checking the entries, as the order of the digits would then be unique in that, without this, solvers could have their digits in different places but it would still fold up to make the correct cube.
Now to the clues. What sort would they be? Letter/number assignment or number definition? The latter are much easier to do but ENT had used this. So with some trepidation I opted for the former remembering my recent experience of setting Tribute To A Horticulturist for The Magpie which used 23 letters and an already filled in grid with the clues all having some horticultural connection that had taken 6 months.
I decided to use a smaller set of letters and reuse the ones that I’d used in my first numerical for The Magpie that appeared in issue 2, which was about the 24 MacMahon Squares – namely A, C, H, M, N and O.
This would give an easy grid fill which is what I wanted so that as many solvers as possible would be able to progress and therefore access the last two harder parts and thus experience the delights and joys that Martin Gardner had brought to a wider audience. There’s nothing worse, in my view, to having spent many hours in obtaining the grid fill to then having to spend the same amount of time afterwards on the dénouement or having a puzzle that has an interesting dénouement that you can’t access because you can’t get the grid fill.
[ I appreciate and acknowledge that some solvers were dissatisfied with the easy grid fill however I have given my reason for doing so above and remember I could have made the clues for this part much harder, so much so, that some of those complaining may not have been able to do that part and thus be denied the final two fun parts! ]
Just to make it a wee bit more challenging I decided to use a different arrangement in the across and down clues. If the letters are written out in alphabetical order then the order of the numbers in the down clues is just the reverse of the across order – a sort of mirror image which again fitted with the theme.
With such a small set of fairly unhelpful letters to use to make words I had to resort to a bit of setters’ licence!! So there were a few extra repeated letters in some words, some foreign words, as well as a liberal sprinkling of oohs and aahs but no Cantonas!! The clues took a couple of days to write and of course I had to include MacMahon as a hint as well, Common Man, was too good an opportunity to miss and appeared several times – a reference to ELP and my extensive prog rock collection that Magpie solvers are occasionally given a glimpse of!!
The cold solve for the letter/number assignments was easy with such a restricted set of letters and I knew that, once that was done, the rest was bound to work in that I’d already checked the logic for finding the nets, and the fitting process had been done 15 years ago with the help of Conway’s matrix so lucidly explained by Martin Gardner.
Now to address some of the comments.
Why some solvers boast of writing a computer program to get the grid fill is totally beyond me. Yes, granted they may find it fun but in this case sledgehammer and nut spring to mind and thankfully some even then had difficulty, which pleased me enormously! Fair enough to use a spreadsheet in its capacity as a glorified pocket calculator but not if its goal seeking functions are used. The Listener numericals have to be able to be solved by using at most a standard scientific calculator, unless the editors have changed the rules and not told anybody!!
I had always thought that the majority of solvers would opt for what’s been called the Blue Peter approach to the final part, namely actually assembling the cubes as opposed to the mathematical graph theory method, and this was the case. Children’s building blocks were put to good use for the last part – just goes to show that you should never throw anything away – as well as polystyrene and even sugar and jelly cubes!!
It appears that a number of solvers had difficulty in finding the 8 correct nets thanks to a couple of 2s in adjacent cells. I did notice this when I did the logic for finding the nets but dismissed it with a cursory wave of the hand as not being a problem, in that you had to find 8 differently shaped nets and the other possibility gave a repeated net. How wrong could I be? In this case – to use part of an early Genesis lyric- ‘( Am I ) Very Wrong’. It was the editors that suggested that the nets be shaded or delineated in some way so I can’t really take the credit for this.
I hope that solvers got as much enjoyment in solving this as I did in setting it and I would urge them to Read Gardner!!
So thank you Martin Gardner for all the hours of pleasure your writings gave, not just to me, but to countless others as well.