Thanks to the clairvoyant genius of Mark and Simon in publishing the Quadratum series of puzzles I have the dubious honour of introducing Magpie solvers to the delights of Sudoku before it was popularised in The Times.

Pandigital and zero-less pandigital (ZPD) curiosities have long since held a fascination for me and I am not alone in this affliction. Madachy and Beiler both devote chunks of their books Mathematical Recreations and Recreations In The Theory Of Numbers respectively to them. In addition the most prolific setter 40 odd years ago was Rhombus who set many puzzles involving them. John Gowland in South Africa also sets puzzles in the Rhombus style and his work can be found in the mathpuzzle.com archive.

Rhombus’s puzzles rarely had the dreaded dénouement which plague mathematical puzzles nowadays relying instead on the satisfaction of having obtained the relevant solution sets used as being more than enough of a reward.

Whilst surfing the WWW I came across a site which had lots of ZPD niceties. I homed in on the problem that was used in the puzzle and I wrote a program to check and found that there were 101 solutions. The contributor Peter Kogel found chains and also a loop of solutions and so I decided to use the loop of 9.

The first stage was to come up with a grid and given that P, Q and R were 3 digit numbers as well as the solutions X, Y and Z I opted for a grid containing only 2 and 3 digit entries. I could have used some entry lengths of 4 or 5 and clued as divisions but I felt it was better with only 2 and 3. Rhombus used rectangular grids quite often and so I chose 9×7 and barred it off. Normally in puzzles like this I have started with a blank grid and barred off as I went along however on this occasion I wanted to try something different and start with a barred off and lettered grid which makes the clue writing much easier as you don’t have to wait until the puzzle is completed to code them.

The next stage was to list the 9 sets used and find their prime factors. I was able to get all of this, grid included, on a single A4 sheet along with 9 rows for the clues.

Flushed by this success I spent a long time studying the prime factorisations and the P, Q, R, X, Y and Zs looking for

· Any that were the same apart from those in the loop.
· Any that were multiples of one another.
· Any that were multiples of 5.
· Any that were reversals of each other.

Eventually I spotted 976 which was 16 x 61 and felt that that could be a good lead in as there couldn’t be too many to test which only give 3 digit numbers with no repeats or zeros. In order to eliminate 23 x 32 a clue which was 5 times a 3 digit number was required so as to fix a 1 at the start and putting all that in one clue solvers would have minimal testing and could enter x and U.

I wanted to make use of deductions which could be made regarding the terminal digit of Z from the sum of 3 cubes and so soon finished off the clue.

Setting continued well and before long the grid was filled. I deliberately left some parts of clues blank in that having bits missing is quite useful especially if you can’t fit them in! Getting solvers to sum those missing bits (something they are quite used to doing!) and writing it under the grid would show that they had obtained all the solutions. I had wanted solvers to write the sum in an appropriate mathematical shape made up of one line to show that they had spotted the loop, basically a circle or any closed loop, but this was rejected by the editors as being too much of a quantum leap. However given the quantum leap dénouement of One Day I have to wonder!!

The editors did come up with an alternative dénouement which used the sum of the numbers that did not form the links in the chain (4807) and its prime factorisation of 11, 19 and 23 which could be found with minor modifications at P, Q and R respectively namely 11, 18 and 33. This was quite an incredible achievement however I vetoed it in that it didn’t involve the missing parts and also gave the game away in that the loop was mentioned in the introductory waffle!

It will be interesting to hear what solvers thought of the puzzle and apparently some have given away their age by commenting on the similarities with Rhombus.

Posted on Sunday, November 2nd, 2008 at 8:07 pm under Setting the scene.
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