# Listen With Others

## Listener No 4751, Instruction: A Setter’s Blog by Oyler

Posted by Listen With Others on 12 Mar 2023

### Solve One Get Another One Free

It’s spring-time 2018 and one of our cats — Tango, he of EV1118 fame, had been put to sleep a few weeks earlier. I was coming to terms of life without him, he was almost 18 and been part of the family since he was about 8 weeks old. So, to keep myself occupied I had launched into a bout of puzzle setting. They were short puzzles and most of them weren’t up to much. I needed something that would be really taxing, time demanding and require a lot of concentration.

I’d always been impressed by the way setters could get a message into a puzzle and as I hadn’t set one like that for some time, I revisited the idea. I decided to use the old mod10 trick whereby a digit can be one of two or three letters. Then came the message — what would it be? There were various possibilities — leaving a pretty pattern with ERASE ALL ODD DIGITS or ERASE ALL EVEN DIGITS or to give John Green a week off, ERASE ALL ENTERED DIGITS.

Eventually I opted for REVERSE ACROSS ENTRIES. At this stage it had never crossed my mind to have the clues still hold true when the entries were reversed. I set to work finding a suitable grid which meant that the zero which would appear for T would have to be an unched cell or a cell which wasn’t the first or last cell of an entry before and after it was reversed. It was easier to do the former. This didn’t take too long and I entered the relevant digits for the message in their correct cells and got to work

One thing I like to try and do is to introduce solvers to a variety of different sets of numbers other than the usual suspects square, prime, triangular, Fibonacci, Lucas etc. To this end I decided to use Happy numbers and Lucky numbers as well as the multiplicative persistence. To make things easier for the solver I decided that the Happy and Lucky sets would be restricted to 2-digit numbers and to list them in the preamble. Calculating the 2-digit and 3-digit Happy numbers as well as the 2-digit Lucky numbers isn’t that hard by hand. It is a different matter entirely though for the 3-digit Lucky numbers as the method is akin to the Sieve of Eratosthenes for finding primes and requires a clear head.

Setting was going well and I had a few entries in place when I took a step back and began to wonder what the comments would be like on the various forums. I reckoned they would be along the lines of “yeah and”, ”so what” or worse still “why?”. It was just a random set of numbers that were being reversed. There was nothing special about them.

I looked at my entries again and noticed that I’d put in 169 for 3ac which when reversed remains a square and there was another entry that when it was reversed had the clue still hold. However, the rest of what was entered didn’t. My subconscience was telling me something. So, I removed those offending entries and continued with the new idea that the clue would still hold when the entries were reversed. This would make it more difficult for me but also more interesting and the end result more pleasing for the solver as well. I took the opportunity of re-clueing some of what was already in place by adding digit sums and digit products to some. This had the effect of cutting down some of the numbers in the sets as well as making my life a bit easier in that the digit sums and products don’t change when they are reversed. I had used Σ for digit sum and Π for digit product however it seems that The Times can’t handle Greek letters in puzzles!

I had another few entries in place when a devious thought struck me — could I have the same holding true for the down entries as well? That would surely have a PDM and/or wow factor.

This would require careful checking and be far more difficult for me to set but as that was what I wanted; I went for it. If I failed then I’d still have a puzzle where all the clues for the across entries would still hold. So, I made two copies of the grid side by side with a clue list underneath and painstakingly went through what was already in place. This necessitated a few changes to what was already in but thankfully nothing major.

After a week I had the puzzle completed and cold-solved. I was concerned that my solution had involved having to find the letters or potential letters of the message about half-way through. However, I decided to go with it as I thought that that is what solvers would try and do anyway and sent it off to my test solvers. Both testers solved and enjoyed the challenge. All that was left was to decide where to send it, Listener or Magpie? I felt that it deserved the widest audience possible and so, with apologies to the bird, this tribute to the sadly missed Tango who I’m sure would have approved of the deviousness of the puzzle, went to The Listener.

Thank you for all the kind comments about the puzzle which are much appreciated. A number commented regarding why they didn’t get the reverse grid instead as their first solution. The answer is that the message would have been gobbledegook. According to one poster there are 3432 different grid-fills that satisfy all the clues! I presume that there’s only one of them that gives the message though.

PS. I note from The Listener site that the reverse puzzle number 1574 appeared in July 1960 with the title Auto-suggestion-II set by Fudge. The grid was car-shaped and each entry was a word from which three consecutive letters had been omitted; the clue was a “number plate” showing the discarded letters, the entry length and sum of the letter-values of the entry. Looks interesting!