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Posts Tagged ‘Arden’

Dice Nets by Arden

Posted by shirleycurran on 10 Dec 2021

Arden is familiar to us in the Magpie numericals (especially those where we have monks manipulating obscure harvests) and I see that he has produced a number of numerical Listeners – for us, those dreaded tri-monthly events. I tend to hand over to the other Numpty who grumbles each time a careless error sends him back to his pencil and paper calculations – and the paper piles up.

We do wonder how anyone manages these things without a computer. This time he created a couple of mini programmes to work out what the triangular number and square were that had to be additions of 16 to 27 and 31 across.

Yes, there were errors and back-tracking and he was happily in bed and dreaming, with relief, of the next verbal crossword by midnight, leaving me with a full grid and what looked like an entertaining endgame. Of course, the Internet gave me the eleven potential six-shell shapes that could be folded to make a standard cubic die – and my first muddle failed. It was difficult confirming those shapes that were rotated and making sure they appeared only once – until I hit on the fact that there were only 11 2s in the grid and 11 4s, so that if, as I surmised, all 11 shapes were to appear in the 80 cells, all those 2s and 4s must be included. Then it was plain-sailing and quite a pretty grid emerged. What a feat, Arden, to fit them all in, with the helpful bars.

“Bars”! I was just about to say that Arden hadn’t earned his place at the bar at the Setters Dinner in Stirling but it looks as though he has.

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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

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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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A Defter Premier by Arden

Posted by shirleycurran on 7 Sep 2018

There was my usual despondency as I downloaded A Defter Premier by Arden. The dreaded numerical! There was not much hope of Arden’s earning his Listener oenophile qualification with this. Then I saw the title. It had to be a premier cru! “It must be an anagram” said the other Numpty who had his pencil and paper ready to begin (yes, he solves the numerical ones that way!) TEA suggested to me that we were dealing with FREE TRADE PRIME. Well, there’s hope for us all with Brexit looming – maybe those prime wines will escape heavy duties. Numpty disillusioned me with a delighted snort. “Primes indeed, but it spells PIERRE DE FERMAT. it is going to be about his theory of prime numbers”. Cheers anyway, Arden!

Must be a hard problem, a numerical compilation! Absurd ideas, going off on a tangent, rooting around… Log in looking for topics? Prime the pump how? It’s an irrational field, sometimes, and there’s a limit to solvers’ tolerance in seeking a complex solution. How to differentiate between rational options, integrate odd ideas, converge to a real result? This was a splendid example of an approachable numerical, I think, with the corner clues being fairly easy , and strongly hinting at an ‘all prime numbers’ solution. The clues made it possible to decide on the ‘sum’ or ‘product’ variety with relative confidence, and without generating too many options to check, which can be a real ‘off putter’ for those who aren’t experts.

The other Numpty now laboriously worked his way through the clues, happily solving the ones where the clue was ‘the product’ of those two integers but muttering when the clue was the ‘sum of the integers’. Of course, knowing that only primes were being used helped enormously and, with a break for dinner, we had a full grid by mid-evening.

We suspected from the start that the two diagonals were the two symmetrically placed 9-digit numbers that share that feature (prime). I use a really valuable website (is this cheating? I really don’t know whether GCSE maths, which is the ‘supposed’ level of a solver of Listener mathematical puzzles, teaches one how to work out whether a 9-digit number is a prime  http://www.bigprimes.net/primalitytest but this does) and those two numbers were primes. I double checked and found that no other symmetrically placed pair fitted the requirement.

Clearly we needed the squares of two 5-digit numbers to produce those 9-digit primes and we both squared the available ones in the grid but as I despairingly discovered that all but two of the available solutions were way too large, the other Numpty gave his second happy shout. Of course, he had worked out that we were looking for those four diagonal 5-digit primes. What an achievement. I imagine Arden must have formed the skeleton of his grid with the two main diagonals and those four intersecting primes, then built his grid round them. I hope he will tell us how he did it!

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Listener No 4516: A Defter Premier by Arden

Posted by Dave Hennings on 7 Sep 2018

This was Arden’s fourth Listener mathematical puzzle, although it’s been over eight years since his last. Over at Magpie he has set a fair few, with some word puzzles thrown in for good measure. What was worrying was that the mathematicals seemed to be D- or E-graded!

Anyway, not much to say this week from me as a holiday is imminent. Also not much to say because it was a fairly straightforward solve, and probably a B- or C-grade in Magpie terms. Tackling the 2-digit entries first got the puzzle underway, and it was fairly soon that the primeness of entries was revealed. Mind you, the title pretty much gave that away — although the Wiki article on Free-Trade Primes seems to have gone missing!

I don’t know how easy the puzzle would be without knowing that all entries were primes, if indeed that were possible. It certainly helped me, though. To assist, I created a table showing, for each clue, both the limits for the sums of two squares and the factors for the products. For example for 11ac 57 (4): the sums ranged from 29+28 through to 56+1 with the associated entries being in the range 1625 – 3137; the factors are 1.3.19. 11ac turned out to be 10²+47²=2309. I hope I’ve explained that adequately. Suffice it to say, it enabled certain integers to be discarded as generating sums of squares that were too large or too small, or not having factors to produce an entry to fit with digits already in the grid.

It was an ideal mathematical for me: not too difficult but requiring a lot of patience and analysis. The endgame was pretty straightforward as well. Most 9-digit rows and columns were divisible by 3, a couple were even, and the other two wouldn’t have been symmetrical. It was the two diagonals that were the culprits: 617984977=21791²+11964² and 933181037=25331²+17074².

Thanks for a relatively gentle puzzle, Arden.
 

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