The twelve pentominoes are familiar to many solvers through the work of Arden and other geometrically inclined setters; my interest in them started at age five when presented with a plastic set called Maestro Pocket Puzzle, which I still treasure, its great age plain to see from the now unheard-of inscription, ‘MADE IN ENGLAND’.

In the 1980s, I began to think about more challenging tiling possibilities not already described in Solomon Golomb’s book Polyominoes. Golomb made no reference to polygonal tiles based on the half-square triangle, so, after a lot of scribbling, I came up with the thirty pentaboloes. I saw that the pieces could fit into a 15×5 rectangle, but had no idea if they would. Eventually, with the aid of scissors and cardboard, I found several solutions, despite the problem being hugely more difficult than tiling a 12×6 rectangle with pentominoes. Wanting a more pleasing shape to tile, I hit on the idea that a 11×7 rectangle with the corner triangles cut off might also be coverable, and so it proved. It’s also not a bad approximation to a golden rectangle.

Much later I discovered (through a Martin Gardner column) that pentaboloes and other polyaboloes had already been described at least as long ago as 1961 by TH O’Beirne in his New Scientist Puzzles and Paradoxes series. Internet searches revealed several other tilings of the 15×5 rectangle, but I haven’t seen the clipped 11×7 rectangle used. For convenience I had a set of pentaboloes made for me by Kadon Enterprises, whose website is well worth a visit. (See www.gamepuzzles.com, in which polyaboloes are referred to as polytans.)

Having reached the current millennium, I thought pentabolo tiling would make an interesting change from pentominoes as a theme for a Magpie puzzle (where else could it go?). I realised that if the diagonal lines from a rectangle tiled with pentaboloes were removed, something resembling a barred grid would be left behind. It would also look a bit like a pentomino tiling with some lines missing, a possible red herring. Solvers would merely have to recapitulate the original tiling from the skeleton grid provided. However, such a grid would necessarily have several regions where only two letter words could be entered. I therefore opted to include two letters per cell, making no entry less than four letters long. The added advantage was that I could use the letter pairs in some devious way to indicate cell diagonals. At this point I worked out the (much maligned) modulo 3 method, and — to preserve sanity — chose only to indicate diagonals in the first cell of each entry. Since solvers are used to the A=1, …, Z=26 coding, I felt that the preamble word combining would prompt them to add the numerical values involved and, given the three possibilities, to realise that the simplest way to indicate these was to use the remainder on division by three. How wrong I was.

I quickly dismissed the 15×5 format as being too restrictive, and decided to use the clipped 11×7 rectangle. Partly informing the choice was the belief that no amount of internet searching would reward sly solvers with a prototype solution.

Given the extreme obscurity of the theme, even by Magpie standards, I elected to reveal essential information in an acrostic, thus providing an excuse not to present the clues in normal order. In addition, I wanted to explicitly indicate somewhere the keyword PENTABOLO. I chose to hide it in the grid, but also to include the first three letters PEN in the acrostic in an attempt to reinforce the PENTOMINO thought in solvers tempted to jump to conclusions.

The unusual two-letter entry method meant that I was unavoidably committing a capital offence by sprinkling double unches everywhere. To minimise the chance of a dawn raid by the Unch Police, and to make use of the happy numerical coincidence, I generously made sure that the first letters of clues corresponded one-to-one with unches. (In retrospect, I wish I’d indicated them with a long anagram and saved myself an extra demanding constraint on clue construction.)

Surprisingly, there are well over a million (perhaps ten million) distinct solutions of the clipped 11×7 rectangle to choose from. I wrote a program using Donald Knuth’s dancing links algorithm to generate hundreds of solutions stripped of diagonals, and pored through these to find a tiling with maximal crossword-style connectivity in which the bars were spread fairly evenly throughout. I made sure that in the one selected the unique placement of missing diagonals could be deduced by logic alone (in case solvers didn’t work out the modulo 3 trick). Care was taken to avoid possible ambiguities sometimes caused by internal symmetry — e.g., note that the two pieces covering IL/AMNESIAC can be inserted in two different ways. The tiling’s biggest flaw was the weak single-cell connection between the rightmost third and the rest of the puzzle, but that was the best arrangement found, believe me.

Filling the grid with sensible words was something of a trial, but I got there in the end. I couldn’t find a fill that didn’t include the unfamiliar (to me) proper name TAGORE: at least he is in Collins. All the self-imposed constraints made devising the clues tough — finding suitable words to ‘misprint’ was a particular challenge — and last minute changes before submission introduced a few embarrassing errors that the Magpie team had to point out. The editors also helped polish some of the less convincing clues, and improved the preamble. It was an unexpected delight and honour to be given the ‘Magpie puzzle’ accolade.

Many thanks for the generous feedback about the puzzle. I apologise to those of you who didn’t work out the modulo 3 method, but as it wasn’t essential to completion, it seems a forgivable fault. The title was tricky to devise without giving too much away: sadly, ‘Economise’ seems to have been far too cryptic for most solvers and at least one editor.

I did consider setting ‘Economise’ as a numerical puzzle. However, I decided that the pentabolo aspect was sufficiently non-mathematical that a word puzzle was justified. Perhaps I should have found an alternative to the slightly mathematical modulo 3 process, but no-one seems to have complained too much. I promise to bear your responses in mind when I set a puzzle based on the law of quadratic reciprocity.

I am unapologetic about the almost obligatory need for on-line research to solve ‘Economise’. Why should thematic ideas be restricted to those found in the core library solvers are expected to maintain? The internet is accessible to almost all solvers, and has significantly enhanced the range and interest of themes. Long may it continue to do so.

chorybdis@gmail.com

Posted on Monday, November 1st, 2010 at 1:38 pm under Setting the scene.
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