## Listener No 4686, *Dice Nets*: A Setter’s Blog by Arden

Posted by Listen With Others on 12 Dec 2021

We have a tradition in crossword, and their descendent numerical, puzzles that there’s a grid for solvers to fill. Brain teasers and logic puzzles may require us to think of four 6-letter words that start and end with a B, calculate the range of a ship sailing against a known current, while leaking fuel at a constant and unrepairable rate or say how long it takes two trains to pass one another in a tunnel if one of them deposited its last three carriages in Hemel Hempstead, but this is not the stuff of a CROSSword puzzle. We like to have feedback from a grid, where solving 5 down gives the foothold of the first letter/number in 11 across. This gives a pretty well unbreakable link between solving clues and having some sort of lattice-like array to display the answers. It is therefore inevitable when setters search for ideas for a thematic puzzle they are often drawn to themes that fit neatly, even appropriately, into grids.

It’s less common for number puzzles to have a pretty, final grid with three sensibly placed entries being highlighted to display the theme. In fact, number setters usually just say please work out your answers and write them neatly into the box provided. But, when a theme appropriate to the grid nature is chosen, suddenly 2-D nets of 3-D solids become very popular. More to the point, how can you have a puzzle featuring nets of solids that doesn’t exploit the orthogonal grid nature?

The surprising thing about nets, however, is they have an awkward aspect to them. They are interesting, varied and require some mental manipulation to envisage, they even fit together nicely, what they don’t do is fit into a grid. It’s the corners and edges that are the problem. For their 3-D requirements nets tend to have bits sticking out in all directions, rectangles don’t. It’s probably not very often you attempt to fit a hedgehog into a balloon, but for so many reasons, my advice would be — don’t.

There are 11 different cube nets and the aim was to put them all in as neat an arrangement as could be managed. This rather nebulous requirement had me considering grid sizes and rejecting them for various, slightly arbitrary reasons. Grids were abandoned for being far too big (10×10), or not close enough to square (6×13), 9×9 appealed for a while, but symmetry proved elusive and I settled on 8×10. Clipping away some of the edges and corners allowed the residual 66 cells to be filled with all nets and the grid to remain symmetrical, so that’s how it would be.

The next question was how should the solver identify the various shapes for a unique solution? Obviously with numbered cells dice are very useful, so the nets would not be for cubes, but, more precisely, for dice. Perhaps not the most original idea, it’s probably been used dozens of times, but this is because it’s a good and interesting solve. At least that’s the general idea.

Having drawn up a grid of the final net arrangement I then marked where potential bars could go by selecting only those which matched in traditional 180° symmetry. A small amount of fiddling then tidied the grid into what looked like a usable effort and filling could begin.

In order for the nets to be considered suitable for dice, it was obvious solvers would expect opposite sides to sum 7, 1 opposite 6 etc. But there is also the question of orientation. I decided to be totally fair and give all nets the same orientation, but not expect solvers to have to consider this. So any solver who chose to assume all the nets were the same handedness would be correct, though perhaps a trifle lucky.

After that it was just a case of entering a few digits, which tended to force the entry of rather more, while trying to keep the entry values as “nice” and describable as possible. The fact that all 11 nets were present meant there was very little chance, once the grid was full, there would be different methods to divide the nets up. In fact division of the grid into its nets is relatively easy and something of a separate add-on to the original puzzle.

Overall, it was a fun puzzle to set and, thanks to editorial erudition, I even learnt a new word in ‘chirality’. What could be better than that?

Though, on the subject of chirality, I am aware of a small amount of discussion about the handedness of the dice. With it being reasonably difficult to establish the three-dimensional orientation of a two-dimensional net, folds being flexible, the overall aim was for consistency making all dice the same. There was certainly no intention for the puzzle to have a sinister aspect.

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