## 4088, Digimix: A Setter’s Blog by Oyler

Posted by Listen With Others on 19 June 2010

I tend to set puzzles of the same type in batches using what I call the Jethro Tull method which involves making a small change to get a different puzzle although it is effectively the same. This puzzle was no exception and was set alongside a couple of puzzles for The Magpie ( Pandigital Products and Mind Your Ps and Qs ) and one for the Scottish Maths Council’s Journal and one which appears on the earlier incarnation of this site.

The puzzle had an old fashioned feel as I was in a Rhombus clone phase at the time and it wouldn’t have looked out of place 30 or 40 years ago so a denouement would not be required. The satisfaction for the solver is in finding the solutions used, using just the clues given, to an interesting digital relationship. I found the solutions to the problem on the internet at The World of Numbers website and knew full well that solvers could do likewise. However, if I could give solvers enough feedback in the grid and they applied some basic mathematical facts that would all be unnecessary. In fact it is worth remembering that in these types of puzzle if the only way of solving it is to have the solutions in front of you then it won’t be published as it would be unfair! I checked the 78 solutions by writing a bog standard BASIC program and was delighted to find that it agreed with those on the site.

The next stage was the tedious but necessary task of finding the prime factors for all of the solutions. That done I scoured the lists looking for potential candidates for the clues. The sort of thing I was looking for were

• A pf or multiple thereof repeated in the X, Y or Z.

• The reverse of a pf as puzzles of this type tend to make use of this.

• The same P or Q.

There were a number of candidates for these and I chose the ( 9168, 23475 ),

( 2436, 18975 ) and ( 6471, 23589 ) P, Q pairs as the start point and these would provide the way in for solvers.

Then came the grid. I toyed, but only briefly, with the idea of a 9×9 Latin square grid. The reason for rejecting that was based on my experiences setting the Quadratum puzzles ( Sudoku based ) for The Magpie as the Latin criterion very quickly takes over. In fact I recall solving Arden’s Domino Squared puzzle in The Magpie which was a 7×7 Latin square with extra bits sticking out using only a fraction of the clues! So I decided on a rectangular one as Rhombus used them and wanted to have no more than 23 across or down clues as using I, L and O causes confusion in this sort of puzzle and Greek letters are a pain. Also I wanted to have some unches but not many, as I wanted to give solvers as much feedback from the grid as possible as this is a CROSS number puzzle after all and so, in a way, circumvent the need for some solvers to calculate the solutions or surf the net looking for them. It only took a matter of minutes by hand to come up with a suitable grid.

Having the grid already barred and lettered makes setting this type of puzzle much easier. In the past when I started setting I would have started with a blank grid and barred off as I put in entries but not any more.

Further study of the list of prime factors was called for and the neural pathways needed to be cleared so a good dose of keyboard driven 70s prog rock was called for to aid the setting process. As I listened to Emerson, Lindh, Fritz, van der Linden, van Leer, Premoli and Lord batter hell out of their Hammonds I decided that I would leave out parts of some clues if I couldn’t fit them into the grid and get solvers to add them up as a sort of denouement which was something I wouldn’t have dreamed about or dared to do when I started setting in 1993! In those days I would have doggedly obtained clues for all the parts regardless of how clumsy they were. I noticed that 8796 for P appeared twice and 8976 once and decided to use the latter and one of the others for Qs that would be unclued just to make things a bit more awkward especially for those who would be sitting with all 78 solutions in front of them. Those who were solving the puzzle without the 78 solutions in front of them and just using the clues ( solving it properly and in the spirit that the puzzle was set!!! ) would be blissfully unaware of any such problems and so in a sense rewarded for this.

The puzzle was completed in a couple of days and all that was left was the logical pathway through it which made use of the h, 5h and 6h start point along with basic number facts learned at school especially the terminal digits of square numbers. For example if the last digits of P and Q are known then so too is the last digit of Z of course. But if the last digits of P and Z or Q and Z are known then you can limit the possibilities for the last digit of Q and P respectively down to at most 2 digits. For example if P ends in 4 and Z ends in 7 then Q2 ends in 1 so Q ends in 1 or 9. The fact that each digit appears only once in a P, Q pair and X, Y, Z triple as well as almost 90% of the cells being checked cuts down a lot of trial and error and aids the solving process enormously as does the realisation that Q must start with a 1 or a 2.

As for 73194 it is a number of no real significance which would probably get some solvers wasting their time trying to find some. But why? To expect PDMs and denouements in all puzzles is unrealistic and not having one would be a nice change. Simples as a meerkat would say!

PS It appears that 73194 does have some relevance to the puzzle after all and thanks go to Mr Magoo who spotted that 731942 + 58622 = 5391724680 and I thought that I was a saddo when it came to numbers!!! It was not, as some on the ‘ answer bank ‘ surmised my date of birth in an American calendar – I’m not 16 or 116!! The 31st of July 1894 though is memorable as it was the day of the first powered flight by Sir Hiram Stevens Maxim!!

PPS I leave it as a pleasant exercise for readers to discover which groups the keyboard wizards played for.

PPPS As a thank you for all the comments about the puzzle I give you a true pandigital experience that The Magpie rejected as far too simples!!

## erwinch said

Thank you for this insight into numerical puzzle setting. So, you must first find the prime factors of solutions.

I thought that you might have been more dismayed at solvers who threw their computers and especially the Internet (despite your change of title) at the puzzle as if it was merely a chore to be disposed of as quickly as possible rather than something to be savoured. You will have noted that several of these solvers still came a cropper so perhaps the spirit of Rhombus is still alive! For many of his puzzles, solvers would not even have had the luxury of an electronic calculator although they would have been expected to have access to lists of prime numbers, squares, etc and, as you say, would be familiar with the effect of various functions on especially the endings of numbers. I was still using a slide rule for my Chemistry degree in the early seventies. The labs were each equipped with a single large electronic calculator that was an object of universal wonder.

We have heard in the past that certain puzzles were tedious trawls through long lists of numbers but I can only think that not enough thought was given to eliminating some of these numbers from the outset. I didn’t take full note but the longest list that I remember checking here was 32 for the value of

Qin the first clue.Finally, I shall concede that you were genuinely not aware of the pandigital possibilities of 73194 at the time of setting. Of course, a check was required in this case since there was otherwise no need to determine the three asterisked values in the final clue.