## Listener No. 4399, *Square Time Sums*: Setter’s Blog by Oyler

Posted by Listen With Others on 11 Jun 2016

### 9 x 9 grid + digits 1 to 9 + Oyler = Sudoku?

Judging from the comments, it seems that solvers appreciated the fairly gentle solve. I presume it was because the puzzle appeared on Cup Final day both north and south of the border and they were able to polish it off well in time. There was also the matter of a Test Match at Headingly with Jimmy’s ten-for to savour.

However that didn’t stop solvers from speculating that they’d felt they’d missed something and indeed most solvers did!! The danger with sites like answerbank and crossword forum is that you can easily be fooled into thinking that there is more to something than there actually is or indeed the opposite. Yes the highlighted rows/columns were all zero-less pandigital ( ZPD ) and yes they were also square numbers to boot. So when you see the first post saying that the highlighting was obvious only for the same person to post 45 minutes later saying ‘Now I see it. Very clever‘, you start to wonder if you’ve missed something. That poster may just have spotted the ZPD with their first post and then the fact that they were squares with the second. Or they may have spotted the ZPD squares straightaway with their first post and then spotted something else. Panic!!! Now you may or may not have spotted that they were squares at the start so it gets you thinking. More panic!!

Many solvers were concerned with the word time in the title and came up with numerous ideas all of which were wrong. The word time was used as in play time or bed time that’s all. No the word they should have focussed on was sums as that can have two meanings. It can be used to describe arithmetic in general. Just ask a primary school kid or secondary school kid for that matter what they did at school that day and once they’ve torn themselves away from some electronic device may reply sums. The puzzle certainly was to do with arithmetic however sums can mean to add up. Some solvers may have wondered about the 2×2 square that the highlighting fully enclosed and contained the digits 1, 4, 7 and 9. Now I had intended that solvers go further and had an extra sentence in the preamble which read, ‘The region enclosed by these rows and columns contains further thematic information and solvers must circle the digit therein that is most pertinent to the theme.’ You may care to pause and think about that before you read on.

There is a nice fact about square numbers that I read many years ago and can be found on p140 along with a proof in Beiler’s book Recreations in the Theory of Numbers and involves the reduced digit sum or digital root. To find the digital root of a number you add the individual digits of the number together and continue until you arrive at a single digit. In other words you are casting out nines – now where have I heard that before!? If you do this for any square number the result is – yes you’ve guessed it – always 1, 4, 7 or 9. The digit to be circled would have been the 9 since the digital root of the highlighted rows and columns was 9. Of course solvers may have guessed that because it was a 9×9 grid and we were using the digits 1 to 9. Others may have circled 4 since a square has 4 sides and corners and I think that common time , 4/4 time, is also known as square time. Some may have gone for 1 in that it was one puzzle and others thinking of a bluff may have gone for 7 in that it was the only one that wasn’t itself a square number! This fact about the digital root of square numbers is far more obscure than I’d realised and when I submitted the puzzle I told the editors that the last part could be removed if it proved to be too much of a quantum leap or specialised. And so it proved to be.

Now to the setting process.

One of my previous Listener puzzles L4125 Elementary Number Theory was fully Sudoku compliant as were four puzzles in The Magpie. Of course if solvers twig that then they can short cut the solving process by applying Sudoku solving techniques. I have set quite a number of puzzles that involve using sets of numbers that between them contain all of the digits from 1 to 9 and I wondered if I could lull solvers into a false sense of security by setting a puzzle that at first appeared to be a Sudoku but wasn’t. Essentially it’s a puzzle by Oyler, it’s a 9×9 grid and we’re using all of the digits from to 1 to 9 so it’s going to be a Sudoku and hope that solvers would get themselves into a bit of a fankle by making that assumption.

I decided that it was time to use the fact about the digital root of square numbers and have them in a 2×2 square surrounded by ZPD squares. I looked out my file that contains the 30 possibilities for the squares and I wanted to choose four that I hadn’t used before. I had more leeway than usual as the grid wasn’t going to be a Sudoku and I chose four that would make the bottom right hand 3×3 block contain all of the digits from 1 to 9. Having said that I still chose 4 that would give the illusion of a Sudoku so there wasn’t much leeway at all! By chance I had chosen two that contained 2-digit squares, one at the start and one at the end and realised that they could appear in the clues. You’ve no idea how frustrating it is trying to fit ZPD squares into a grid to make it fully Sudoku or Latin Square compliant.

I entered the four squares in place followed by the 1, 4, 7 and 9 in the 2×2 block. What next? How was I going to clue the puzzle?

There’s only set of three 3-digit squares that contain all of the digits from 1 to 9 so I looked at sets which had 2-digit, 3-digit and 4-digit squares that contained the all of the digits from 1 to 9. There are eight of them and I suspect many solvers started off by working out all the possible sets. The solution on the Listener site avoids this and only requires solvers to know 2-digit and 3-digit squares.

The sets are 16/784/5329, 25/784/1369, 25/784/1936, 25/841/7396, 36/729/5184, 81/324/7569, 81/576/3249 and 81/729/4356, and don’t take that long to find by hand. There’s a nice rule that links the last and second last digit of square numbers and helps to speed things up a bit so you don’t have to look at all the possibilities.

Last digit | Second last digit |

0 | 0 |

1 | must be even |

4 | must be even |

5 | 2 |

6 | must be odd |

9 | must be even |

For example if you take 16 with 289 then the remaining digits for the 4-digit square are 3457. If it’s to end in a 5 we need to have a 2 and we don’t. If it’s to end in a 4 then we need to have another even number and we don’t. So the 16/289 is impossible and we didn’t even have to look up any 4-digit squares.

I could use a maximum of four triples as all the entries were going to be different. Obviously there was no choice for the triples involving 16 and 36 but there was for the 25 and 81 triples. Unfortunately the triple for 16 also contains 784 and meant that I couldn’t use those that had 784 for the 25. Then solvers would have had two possibilities for the 4-digit square 1369 and 1936 to consider. In retrospect perhaps I should just have had two or three triples and made use of that ambiguity.

The golden rule when setting is to provide a way in and as I wanted to suggest that it might be a Sudoku I decided to have a triple appear in the top left hand corner. The 16/784/5329 looked good as that gave 175 going down which was 7 x 25 and 25 was part of another triple. I calculated 7 times the remaining 2-digit squares and was delighted to find that all bar 81 gave a repeated digit. Even better 81 could be eliminated as there’s no 2-digit square that starts with a 5. So the way in was set. I noticed that next to 175 was 683 and a quick check revealed that that was a prime number. Thus the puzzle was going to be a bit of a hybrid combining letter/number assignments along with number definition and would hopefully appeal to both camps.

I stuck a 9 into the 3rd cell in the top row to continue the illusion of it being a Sudoku. I wonder if any solvers did this automatically and perhaps having to back track when they found out it wasn’t a Sudoku only to discover that the 9 was correct all along. I barred off the grid and there were going to be a lot of clues, 55 to be precise and 15 unches. I placed the remaining squares then started on the remainder of the grid.

I had to remember that the remaining rows and columns would have to have a repeated digit in them so that those to be highlighted would be obvious. [ Some who commented remarked that they’d have put in a spoof ZPD number that wasn’t a square and I did consider it however some solvers still use calculators that can’t handle 9-digit numbers so I decided against it and was being kind ].

Some number definition type puzzles make use of factors and multiples and have clues that state the obvious and can be quite annoying for solvers. For example 3ac is a multiple of 15dn whilst the clue for 15dn is factor of 3ac. A pair like that just had to be used!

With the grid complete the cold solved loomed and this was straightforward with a lot of feedback being gleaned from the grid coupled with a minimal amount of computation.

Of course some solvers would spot that only two rows and two columns were ZPD and highlight them without knowing they were perfect squares. However you have to remember that I had intended to have solvers do a bit more.

Some fans of The Listener numerical puzzles may not know that The Magpie publishes one per month and that they have an excellent archive from which individual or batches of puzzles can be purchased at reasonable rates. So if you want to improve you know what to do!!

## clanca1234 said

A very interesting read, Alastair. Quite how I’ve got this far through life without knowing the rule about whether the penultimate digit of a square is even or odd, I have no idea. I need to expand my reading!

Seriously though, a very enjoyable puzzle.

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http://www.iisantonellomessina.itListener No. 4399, Square Time Sums: Setter’s Blog by Oyler « Listen With Others