Listen With Others

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Listener No 4686: Dice Nets by Arden

Posted by Dave Hennings on 10 Dec 2021

Reading Arden’s preamble, it seemed that this puzzle would be fairly straightforward. After all, every entry would only contain the digits 1–6. Moreover, the clues indicated that we were just dealing with primes, triangular numbers, squares and higher powers. What could be simpler?!

Of course, comparing clues with grid, I was worried that we would need to somehow wrestle with the 7-digit entries at 9dn and 13dn, one a triangular number and one a square. Unfortunately, Adrian Jenkins’s book, The Number File, only gives triangular numbers up to 20100 and squares up to 10000. Luckily I had two sturdy calculators, one with buttons and a 12-digit display and the other online courtesy of WolframAlpha, my Mrs Bradford of mathematicals!

(Nets always bring to mind a Charybdis puzzle from seven years ago — no. 4310, Net Book Agreement — which involved netting Lynne Reid Banks’s The L-Shaped Room. I got that wrong!)

As usual with mathematicals, I leave the details either to a fellow blogger or to the Listener web site. Suffice it to say that I would have found it very difficult without WolframAlpha, and I look forward to seeing the detailed walk-through (which no doubt doesn’t use it).

In fact, it wasn’t too difficult a solve although it did take a few hours. I wasn’t looking forward to the endgame which required us to “shade in distinguishing colours as many dice nets as possible”. If only we’d been told exactly how many that was, but I suppose that would have detracted from some of the setter’s sneaky enjoyment.

I started by counting the number of each digit in the 80-cell grid: 1 — 17, 2 — 11, 3 — 14, 4 — 11, 5 — 13, 6 — 14. Well that didn’t really tell me a lot. Of course, it was important to note that a standard die has opposite sides summing to 7: 1/6, 2/5 and 3/4. My first attempt had a puny six shapes scattered around the grid, but I was sure that there must be more. Unfortunately, the only web sites that I found didn’t expand on that number, not even Wiki.

It was as a last attempt that I googled “formula for calculating number of nets for a solid/cube”. Voilà! The following site told me all: https://www.skillsyouneed.com/num/3d-shapes-nets.html. There were 11 different nets for a die. And there were 11 2s and 4s in the grid. Was that a coincidence?

It was important to note that opposite sides, eg 1/6, could not touch in any way, even at corners. A bit of perseverance, starting with the S shape in the top right and trying to use all the 2s and 4s, had the psychedelic grid complete.

What a phenomenal piece of jigsawing, and not too sneaky. Thanks, Arden

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