## Listener No 4451, *Numerical Playfair*: A Setter’s Blog by Zag

Posted by Listen With Others on 11 Jun 2017

I enjoy solving both ordinary and numerical puzzles but tend to focus on numerical setting. Sometimes the devices employed in normal crosswords suggest a parallel numerical treatment and I’ll earmark that for a future puzzle. The Playfair puzzle idea arose from my earlier addiction to AZED. When his Playfair puzzles appeared I always approached them with a certain amount of trepidation but I did recognise the numerical potential of the idea. For a long time I never took it any further, possibly because of my mixed feelings about the format. I then had a spell when I produced a series of puzzles involving different aspects of coding and decided that the time had come to overcome my doubts and tackle a Playfair based challenge.

As those of you who are familiar with my puzzles know, I like small grids, using number palettes that are readily available in tables and requiring only a calculator to achieve the solution. Sometimes the theme may force a departure from the ideal, but that is always the aim. The compactness means there are only a limited number of clues and so the challenge is to find a suitable start that supports the theme without being too obvious.

AZED’s application of Playfair usually limited it to the 6-letter entries present in the grid (it only works with an even number of letters to encode). The natural numerical equivalent would be 2-digit or 4-digit numerical entries and with my desire to avoid larger numbers I chose to use 2-digit entries. With 10 digits, the equivalent to the 5×5 Playfair letter square is a 3×3 Playfair code square which could then operate in exactly the same way. Whereas I/J are often used to do double duty in a normal Playfair there is no obvious double digit combination. The simplest option, which I adopted, is using the digits 1-9.

A traditional approach would have been to create a grid fill establishing some of the 2-digit entries. Then, comparing these entries with the corresponding clue answers, the code square could be derived and any outstanding 2-digit entries encoded. That was my original plan until, when considering a starting logic for the puzzle, I came up with the idea of using a 2-digit Playfair code number as one of the entries. The nature of the code square means that the encoded entry corresponding to its 2-digit code number can only end in 1, 2 or 3. With a crosschecking clue requiring that digit to be the last digit of a square it could only be 1 and the puzzle was underway. Always gratifying to find a neat way of starting a puzzle.

Another intriguing possibility was to use some aspects of the code square as part of the puzzle. I thought it would enhance the Playfair theme, complementing the use of the code number starting point; hence the appearance of a row and a column of the code square in the grid. There was a danger such additional content could prematurely reveal too much about the code square so it was crucial to be careful how much information I gave away at each stage in the solution path.

With so much of the code square targeted for the grid, I felt the puzzle was best served by only having a small amount of additional structure. I aimed for a compact grid, one with possibly no more than 25 squares. I wanted a balance of about a quarter to a third of the entries being 2-digit and the rest 3-digit. The starting logic provided an L shaped section which naturally could occupy the bottom right part of the grid and, through symmetry, the top left. Experimenting produced the final grid.

The code square row and column that were to be present in the grid should not be discovered too early, nor should they interlink as that gives away too much information. This dictated that the left side of the puzzle should include these two elements since the right hand side provided the logical starting point for entries and progress would be made from there. The finish would be the top left hand corner and to keep options open to the end probably meant that 1d was not the best location for the code row or column. That left 4a&8a as the locations for these entries.

From there, it was a case of playing around with the armoury of numerical tools to allow a gradual development of the solution and progressive determination of the code square. From the start I had hoped this puzzle would be suitable for the Listener and so aimed for an appropriate level of difficulty, hopefully about the average for a numerical whilst offering satisfaction to the solver. Hopefully the challenge will have demonstrated that a Playfair puzzle need not be as intimidating as I and many others had found them to be in the past.

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