Enigmatic Variations No. 1213: Pentomino Divisions by Elroy — Setters’ Blog

So who’s Elroy? Well it’s a character in The Jetsons cartoon series however it’s also an anagram of Leroy, Royle or Oyler. As this puzzle was a joint effort by the editor, Samuel, the EL part and Oyler providing the ROY bit I decided on the change of pseudonym in that Chris had considerable input regarding the clues and I’d also promised myself that if I ever did set a crossword I’d use a different pseudonym in order to differentiate between the two. My previous EV puzzle had caused a bit of a stir in that it was a proper mathematical puzzle which didn’t go down too well with some and Oyler is kept for the proper mathematical puzzles.

Not to be put off I was determined to get some mathematical content into a puzzle for EV and as numbers didn’t seem to appeal – or maybe it was the context of t20 cricket match – I turned my attention to shape and looked back through my catalogue of puzzles.

One puzzle in particular seemed to fit the bill quite well – Pentomino Factory which was Listener puzzle 4023. In that puzzle solvers had to find the location of the pentominoes in the grid and colour in the grid so that pentominoes that shared an edge had to be of a different colour – the least number of colours had to be used. I’d assigned each pentomino a number from 0 to 11 inclusive that number appearing five times in each pentomino. The clues were simply the product of the numbers in each row and column. Of course some rows and columns were zero which made it just a bit more difficult. [ Pentominoes first appeared in a puzzle set by Dudeney in The Canterbury Puzzles ( 1907 ) and involved a chessboard that had been broken up into the twelve pentominoes and the square tetromino. They were popularised by Solomon Golomb who used them for a talk he gave to Harvard University maths club in the late 1950s and they’ve never looked back since! ]

Of course the same sort of thing was a non-starter however I came up with the idea of using words so that crossword clues could be written for them and have the letters converted to digits mod5 so that each pentomino would contain each of the digits 0-4. Solvers would have to locate the pentominoes and colour them in as before. After all everyone loves colouring-in! You only have to go into a newsagent or book shop and you’ll be bombarded with numerous colouring-in publications aimed at adults seeking to reduce their stress levels.

I floated the idea to Chris in that I didn’t know if this had appeared before. He replied mentioning a few puzzles that had used pentominoes but not to his knowledge in this way. He ended saying if I needed any help in clue writing then he was happy to help.

The Listener puzzle had used a 5×12 rectangle so I decided on a 6×10 for this puzzle. There are 2339 solutions to this and thankfully these are all helpfully listed on a website [http://isomerdesign.com/Pentomino/6×10/index.html ] So which one to choose? Well I wanted to avoid an arrangement whereby the X pentomino is placed into the U pentomino as that is by far and away the most common way for negating the effect of the X pentomino. I trawled through the arrangements and whilst doing this noticed that some of the arrangement numbers were in blue and other in black. Those in blue were ‘clickable’ and revealed related solutions. By that I mean that once some of the pieces had been placed in certain positions the remaining space could be tiled in more than way by the pentominoes that were left. I decided that I wanted to avoid that and eventually plumped for arrangement 159. I then checked that it could be coloured in using just three colours as opposed to the four that it definitely can be done in ( Four-colour Theorem ) and two colours is impossible.

I had decided to use 12 five-letter words, two in each row, so just 12 clues to write. The down clues were going to be the sum of the digits that the letters converted to which is a crosscheck that’s more than acceptable in a crossnumber puzzle.

The next stage was to assign the digits into the pentominoes making sure that there would be some adjacent cells that would contain the same digit and so would have to be in a different pentomino and that gave real words to boot! This took some time and I made extensive use of Quinapalus. The next stage was the task of finding a logical solution pathway which solvers could take which is something I’m used to do doing in that all mathematical submissions to The Listener and The Magpie have to have this. I started at the bottom left hand corner and noted that there were two possibilities. I deliberately chose the wrong one and was pleased that that quickly led to a dead end after a further few had been placed. I then took the other possibility and after 10 minutes I had the solution and pathway. Now I am certainly no expert in pentominoes but probably know more than the average solver so I was happy with that timing. This really had to be done as there would have been no point in starting to write clues if the last bit didn’t work.

I wrote the clues and with some trepidation sent it to Chris. Whilst waiting for his response I had my concerns regarding the cross checking aspect and hit on the idea of having words and clues for the down entries as well with clashes resolved by the fact that they’d reduce to the same digit. I was just about to email Chris when he responded with the same idea and a completed grid containing 20 words. I had my concerns that the entry lengths weren’t given but Chris assured me it would be fine and he’s the expert.

I wrote some more clues and sent them off to Chris for his comments on them and also the first batch. Chris, painstakingly, ripped them apart, all bar five which were fine. I was gutted that my clue for taiga which was ‘Type of massage given to Silverback in the forest’ was rejected although it would have been fine in The Guardian apparently.

Chris shaped my pathetic attempts at clue writing, probably wishing that he’d never given his offer of help in this in the first place, into the published version!

Thank you Samuel.

Now where did I put my set of hexiamonds again?

Elroy