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Posts Tagged ‘Zag’

Listener 4660 Prime Cuts by Zag

Posted by gillwinchcombe on 12 Jun 2021

Oh my goodness this one took some solving! First, congratulations Zag on your great choice of title – another contender for best title of the year.

Prime Cuts set a record for me – 17 pages of meticulous working and an elapsed time of a week. I came to the Listener via the numericals and I enjoy solving them, but had I not been so keen, I may have been tempted to give up.

In fairness one page (but only one) of the 17 was reworking caused by a certain mathematician telling me that there were 3 pairs of reversible primes, rather than the actual 4. That sorted, it was a steady and rather strenuous uphill climb to reach the gridfill. I confess I was hoping for some sort of a revelation in the 21 letters but with all the vowels absent that was unlikely (I hope I haven’t missed it).

Without the test for triangular numbers (8n+1 gives a square number), kindly supplied by said mathematician in penance for his earlier misinformation, Prime Cuts would have taken me even longer. Still, I’m proud of finishing it and I’m glad I didn’t need to resort to the offer of Mathematika.

Thanks Zag for providing a week on and off of sustained mental workout,


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L4660: ‘Prime Cuts’ by Zag

Posted by Encota on 11 Jun 2021

Thanks Zag for a top-quality numerical puzzle.

There were several features that I liked: 

  • the fact that all 21 two-digit primes made an appearance was neat;
  • that lack of uniqueness in the answers still led to uniqueness in the entries / final grid (I have 16d’s (19)95 / 95(19) in mind here;
  • how the ascending/descending digit clues gave significant information about the entries, as well as helping narrowing down the removed primes themselves.  8d was a perfect example of this, where deriving its removed prime from elsewhere as 61 immediately fixed its entry at 987 – I loved it!

I have a strong suspicion that there’ll be a neater solution path than the one I took though, having said that, I wasn’t having to tabulate huge quantities of options along my route, which was good.  I did resort to Excel to keep track of my workings, which felt a very reasonable thing to do.

As is often the case with me and numericals, I did go down one wrong path, putting the wrong reversed entry in at 9ac.  Its implications soon reduced this to absurdity, so I quickly deleted 81 as 9ac’s entry and replaced it with 62 – and all worked out fine!

Cheers all,

Tim / Encota

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Listener No 4660: Prime Cuts by Zag

Posted by Dave Hennings on 11 Jun 2021

The last Zag Listener was that (thankfully) rare instance of a mathematical Playfair puzzle. This week, we had to deal with numbers from which 2-digit primes needed to be removed before entry in the grid.

The starting point was 15ac where the 5-digit answer was the cube of prime D with prime L being removed to give the square of prime S. A quick examination of D from 23 (lowest 5-digit cube of a prime) to 43 (highest 5-digit cube of a prime) revealed that only D = 23 or 29 worked, with 121(67) and 2(43)89 giving S = 11 or 17. 10dn with ascending digits precluded 121(67) since it could not end in 2.

As usual, I will leave a detailed analysis of this Listener mathematical to my co-bloggers and

The enjoyable feature of this puzzle was that, using Adrian Jenkins’s The Number File, simple scanning of the entries under “Squares of integers to 100”, “The first 200 triangle numbers” and the two pages of primes, together with limited calculator checking was all that was needed. Obviously, those of you anticipating a more taxing mathematical would have been disappointed, but no doubt the next one will appease them!

Mind you, it wasn’t a particularly quick solve and my notes ran to six pages.

Thanks for a gentle quarter, Zag.

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‘Numerical Playfair’ by Zag

Posted by Encota on 9 Jun 2017

It’s that week in the quarter when the Listener Numerical appears.  I suspect that the combination of the words ‘numerical’ and ‘Playfair’ may well have deterred some regular solvers, as they both do seem to be in the Marmite category for some Listener regulars.  However, this was a great puzzle and I hope most people gave it a try…

As a precursor I skimmed the grid and marked every cell that couldn’t be a zero – it might be important later, I thought.  All 2-digit entries, every starting cell and those implied by reversals left only a very few cells that could be 0 (and even most of those soon disappeared once I got started).

Then this was actually quite a gentle puzzle, once a few items were spotted.

  1. The top right square of the Playfair (PF) can by definition only be 1, 2 or 3.  7d is a square and so ends in the usual 0,1,4,9,6,5 choices.  And 9a once converted via the Playfair square must end in the top right square too.  So the end of 9a Entry (9aE) is 1.  First cell filled!  And this significantly limits the options for 6d.
  2. As 3d is the reverse of 2a, then their centre digits must be the same, so 3d must be of the form xxy, i.e. its first two digits are the same.  Where 7a meets 3d, and with 7a defined as a divisor of 3d gives a very limited set of 7a/3d pairs in the form xxy.  3d  can only be one of 558, 882, 992 and 996.
  3. The options for 5d as an anagram of 6d are again very limited, especially when it is clear that the 1 must be its central digit.
  4. After a while it becomes clear that 5d can only be either 316 or 613.  As the 3 and the 6 are both in the centre of a row or column of the PF square, then neither 3 nor 6 can be in a corner of the PF square.  This allows two options for the Playfair square to be created, one based on each value at 5d – remembering that in rows two and three of the PF square then there can only be three possible numbers in each cell.  Using these partially completed squares then the options for 2d can be fully reduced.
  5. The last couple go in at 1a and 1d.  One option can be eliminated as it results in 16 entered at 1a, which isn’t possible as it is already in at 7d.  And the final one uses one of the cells that could be 0, at the end of 1d.

Double-check that all Answers and Entries align and I’m sorted.  I hope!  Feeling good about Numericals – I must soon have a go at the couple in the last Crossnumber Quarterly issue that I haven’t started yet (just in case the feeling wears off!).

One thought: I wonder how easy it is to use Sympathy software as a solver-assist tool for numericals?  I have been creating a lot of custom Sympathy dictionaries (.tsd) recently for various puzzles I’ve been writing, but I have never delved much into using Sympathy for Numericals except as a general purpose editor.  Could I get the Playfair gimmick to work on this puzzle for example?  If anyone reading this has already done similar then I’d love to hear about it!

Cheers all,

Tim / Encota

P.S. Can you provide the answer to the following deep and meaningless question: “What number do you get if you take all digits not used in the grid and place them together in descending order?”

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Listener No 4451: Numerical Playfair by Zag

Posted by Dave Hennings on 9 Jun 2017

Zag’s last puzzle was 4295 Codebreaker. In my blog for it, I wrote that it was a very small puzzle (5×7) with precious few clues (16). This week, we had an even smaller puzzle (5×4) with even fewer clues (12). I suppose the first thing that crossed my mind was “Why has nobody thought of this before?”

All 2-digit answers had to be Playfair-encoded before entry, which I suppose meant that one of the 3-digit entries was the place to start. 2ac intersecting with 3dn Reverse of 2dn which in turn intersected with 7ac Divisor of 3dn seemed as good a place as any.

The first two digits of 3dn had to be the same given its intersection with 2ac, so I listed all the possibilities together with the 7ac equivalent. The middle digit of 7ac was the last digit of 3dn, so that eliminated a lot in the list. In fact, there were only ten such values for 3dn/7ac, from 336/168, 442/221, 558/186 through to 996/166.

Given that 2dn was Sum of the digits of 7ac, values for 7ac such as 221 could be eliminated as their digit-sums were less than 10.

Further entries in the list could be eliminated where 2ac Multiple of another grid entry was prime, eg 775/155 where 2ac was 577. I was left with 336/168 where 2ac 633 (factored by 3 and 211), 558/186 with 2ac 855 (lots of factors) and 996/166 with 2ac 699 (factored by 3 and 233). If 3-digit factors, they had to go at 1dn since they weren’t square (6dn) or an anagram of one (5dn) or had all digits different (4ac and 8ac were rows in the Playfair grid).

You’d think I was home and dry here, but realising that the relevant factor of 2ac was the 19 at 1ac took me quite a long time, much of it thinking that I’d missed something somewhere.

In the end, I had 89 as the code-number and the full code-square:

8 9 1
2 3 4
5 6 7

Great fun, thanks Zag, but I just hope you’re not working on Hexadecimal Playfair!

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