## Listener 4125, *Elementary* Number Theory: A Setter’s Blog by Oyler

Posted by Listen With Others on 16 March 2011

On some occasions I’ve set puzzles just to make use of a title, for example *The Times They are A Changin’?* [Magpie Issue 58]. I’d wanted to set a puzzle with the title *Elementary Number Theory* for some time. After all it is one of the many areas that Euler made significant contributions to, as well as providing the title for numerous textbooks, the contents of which are anything but! I also wanted to make use of the word ‘elementary’ in a chemical sense as, although I’m a maths teacher at Madras College in St Andrews, my degrees are in chemistry.

The first ideas started to take shape at about the same time as *Digimix* appeared last year. I was using a resource based on one of Tony Gardiner’s excellent books, namely *Maths Challenge I* with my superb S1 class 1XP or, as I knew them, Windows!! Quite simply they were the best S1 class I’ve ever taught as they were incredibly receptive and insightful. The resource in question was Define a Number, in which there were 26 statements about numbers lettered A to Z. A – it is prime, B – it is perfect square, etc, and I’d get them to find me a number less than 100 say that satisfied certain statement letters or which statements did the number 42 satisfy or which number satisfied most or least of the statements as part of a class competition. It was great fun and maths teachers reading this should give it a go if they haven’t done so already!!

At this point I was reminded of Piccadilly’s *Properties of Numbers* puzzle and my own *Euler’s Spoilers* that used number definitions, and realised that I could use this resource as the basis for a puzzle. As the puzzle was based on number definitions I felt that I could at last make use of the title *Elementary Number Theory* as it described what the puzzle was about albeit in a fairly loose sense.

My first thoughts were to have two unclued 5-digit entries crossing in the middle of a 7×7 grid and, in order to find the statements that applied to them, solvers would have to decode two 3-digit entries to get the statement letters. But there was a twist: using the conventional A=1, B=2 would lead to nonsense, so instead the individual digits of those entries referred to the atomic numbers of the elements so H=1, B=5, C=6, N=7, O=8 and F=9. I set the puzzle and sent it off to The Magpie who quickly rejected it on the grounds that they hadn’t twigged the code and felt that solvers wouldn’t either. In addition they couldn’t suggest any way the puzzle could be saved.

I went back to the drawing board and came up with the idea of reversing the process, namely having the unclued entries fully checked apart from the middle square, and having solvers find the statement letters that applied to them and then rearrange the letters to give the name of an element.

I informed The Magpie that I had found a way to save the idea and would be in touch. There were a number of things that I had to consider – size of grid, what statements to use and which element. As I was going to send the puzzle to The Magpie and as this was at the same time as *Digimix* I decided to use the 1 – 9 digits theme and revisit two of my previous Magpie puzzles *Quadratum* and *Quadratum II* which were, as I never tire of saying, fully sudoku compliant and published well before The Times published their first sudoku. *Quadratum* had three 9-digit squares and *Quadratum II* had three sums of the type 391 + 284 = 675 arranged in the 3×3 blocks with a further hidden one using the centre squares of each 3×3 block. So I wondered if I could combine the two. I retrieved the list I’d used for Q and QII and set to work. I wanted to use two 9-digit squares crossing in the middle and have sums at the top left and bottom right along with the hidden variety. I pored over the lists of squares (there are only 30 of them) and selected two then went to the sums lists (there are 336 of these if you count A + B = C and B + A = C as different) to achieve a fit. This was of course done by hand and took about an hour. No doubt other setters would write a computer program to get the desired fit but I don’t do that.

That done I had 37 out of the 81 digits placed in the grid and all I had to do was to hope that the sudoku had a solution, which took about 20 minutes to get by hand. The next stage was to bar off the grid so that there were no unchecked cells apart from the middle one, and that each entry was unique and no more than 4 digits long. After all solvers would have to deal with a couple of 9-digit numbers and given the feedback I’ve received on puzzles from solvers bemoaning the fact that their calculators only have an 8-digit display I decided to be kind and have short entries. (Goodness only knows how they coped with Elap’s 11-digit triangular numbers which I personally felt was a step too far.)

The next stage was the prime factorisation of all the entries and this was achieved using my new calculator a Casio fx-85GT+ that cost about £7 from Amazon and has a 10 digit display and a prime factorisation function.

I had to think about the statements that I was going to use and which element. The element would have to have two letters the same, be fairly short and preferably have letters that weren’t too far on in the alphabet. A quick scour of a list of elements soon revealed Iodine as a candidate and as it had two Is, the statement I was now fixed as “It is a perfect square” – and what was this? Iodine has atomic number 53 which would be my age on my next birthday.

I had followed with interest the discussion on the Answer Bank about *Digimix* and the relevance of the number 73194 which was just a check sum. Some speculated that it was my birthday using the American date method. So I was either very young or very old. Here was a way of answering that speculation in the best way possible with a puzzle. At this point I decided that the puzzle would go to The Listener and not The Magpie as it is only Listener puzzles that are talked about on the Answer Bank as The Magpie has their own on site discussion forum.

I studied the prime factorisations of the entries as well as the entries themselves for some time and needed to have statements that would yield the correct letters for the 9-digit squares. Of course I couldn’t just add any old statements as I had to check that once I had found the ones for the 9-digit squares that any other statements added wouldn’t apply to the 9-digit squares.

Eventually I had 16 statements and writing the clues was very easy!! Then came the most important part of any numerical puzzle – the cold solve or the logical pathway through the puzzle. It is very easy to set a numerical puzzle (honest). You could start with a completed sudoku for instance then bar it off and write clues!! The important bit though is that it can be solved without recourse to exhaustive trial and error and computational power. I recognise that many solvers don’t want to spend hours sitting at their computers using spreadsheets and writing programs but prefer to sit in an armchair relaxing with their favourite tipple armed with a calculator and a few lists of numbers.

It was during the cold solve that I realised I would have to have more statements as there were too many loose ends, so I added one more and that seemed to rectify the situation. I started off again but realised I had made an assumption, namely that there were no repeated digits in any entry during my first cold solve. So I added statement R which said “it has a repeated digit” which of course would never appear in the clues. That done I finished up with 4 cells that had multiple possibilities and of course the unchecked centre cell. I had always intended that solvers would have to study their final grid in order to resolve the central cell but hadn’t thought about any others. I decided to leave things as they were in that there were enough completed rows and columns which satisfied the Latin criterion as well as completed 3×3 blocks which would hopefully force solvers into realising they had a sudoku. And there you have it!!

In retrospect I should perhaps have used 26 statements so that solvers may have thought that the word to be found was sudoku. But maybe that would have been just a tad too clever.

Xenon for next year perhaps?

PS. As Simon Long and Arden have pointed out, there was only one ambiguous cell apart from the centre one and that was the middle digit of 41 across. The other three could be found by logic and I should have found that if only I hadn’t been so incredibly thick.

**Oyler.**

## John Pooler said

7 of the results were Euler’s numeri idonei numbers. This could have been another clue statement. IDONEI is another answer for the thematic word. Why did you reject it?

Best wishes.

## Oyler said

I didn’t notice that. All I was interested in was the play on words with the title.

## Oyler said

Also John I don’t think the editors would have allowed such an obscure set of numbers as a statement. They would have to have been explained. I don’t set puzzles that require solvers to spend time trawling the internet or writing computer programs to aid solution. I’m a bit old fashioned in that way!

## john pooler said

Thank you for setting the puzzle. I enjoyed solving it (with a tipple handy) but I did opt for IDONEI in preference to IODINE.

Best wishes.