# Posts Tagged ‘Dice Nets’

## L4686: ‘Dice Nets’ by Arden

Posted by Encota on 10 Dec 2021

Another beautifully constructed puzzle by Arden – delightful!

I do like it when a clue gives far more information than it appears to at first sight. Here’s one example: 11d’s “Twice a square”. Those familiar with number puzzles will be very used to pencilling in the unit digit of any square as one of 0,1,4,9,6 or 5. However, it was new to me that doubling all these gives the reduced set of possibilities of the unit’s digit being 0, 2 or 8. Combining that with the constraint in this puzzle that only digits 1 to 6 appear – and that final digit can be written in immediately as a 2. Only 79 cells left to be filled!

As a complete aside, seeing a mention of ‘nets’ brought back good memories of working closely with the setter Ploy, under the pseudonym EP. As some readers will know, we’ve created thematic puzzles for the excellent Magpie magazine. In these, so far at least, the grids were nets of some form of 3-D shape that could then be folded, built and then ‘flexed’ in different ways to display various thematic references, our last one being entitled (perhaps unsurprisingly) Paper Folding. In today’s puzzle from Arden the challenge was to find as many nets of a ‘standard die’ as one can. “Left-handed or right-handed?” I can hear some of you asking!

Unfortunately this week I seem to have forgotten to scan my entry before posting it off to John Green – it’s all been a bit of a rush recently – apologies for that.

At first I tried to work out what all the nets could be. I recalled that there were about ten of them. I found ten, then realised that I had missed one, the pure zigzag. All eleven appeared in Arden’s grid (with puzzles of this quality we would, of course, expect nothing less!) and the remaining unused 14 cells were arranged symmetrically, which added to the puzzle’s all-round neatness. Loved it – thanks Arden!!

Cheers all,

Tim / Encota

## 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