# Archive for June, 2016

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

Posted by Listen With Others on 11 June 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!!

Posted in Setting Blogs | 2 Comments »

## Square Time Sums by Oyler

Posted by shirleycurran on 10 June 2016

I know that the editors are keen to publish ‘easy numericals’. For this Numpty, that is an oxymoron. However, if anyone is able to produce a numerical Listener puzzle to order, it has to be Oyler. We approached this ‘Square Time Sums’ with more than the usual trepidation (at least, the other Numpty did) as we were travelling between a son’s wedding and granddaughter’s christening with nothing but pencil and paper. However, I did manage to print out tables of the first 1000 prime numbers and square numbers of two, three and four digits and purchase a £1 calculator which later gave my three-year old grandson immense glee when he found out that you can change a whole row of nines to zeros by simply adding one. I suppose numbers can be fun.

Does Oyler qualify for the Listener Setters’ Toping Club? As I scan his clues and preamble, I realize that he is confirming his seat of honour as a founder member with not just doubles but triples – and even worse, four of them! How on earth did he stay sober enough to compile this! Cheers! Bottoms up.

The other Numpty grumbled and muttered for a few minutes and then announced  “I know I can compute the possible triples in my old favourite GWBASIC if I can convert my French laptop keyboard to an English one to find the odd characters like <, \$ and % and there are at most about 20,000 triplets. The 1-9 appearing once only requirement might cut this down a lot!” Soon  afterwards he happily remarked “Well, there are only seven possible triples'”- and shortly after that he slotted in 1, 9 and 15 across and 2 down and fixed JKL and XYZ

ABC = 36, 729, 5184

JKL = 16, 784, 5329

PQR = 81, 576, 3249

XYZ = 25, 841, 7329

A steady solve followed, with the usual backtrack to find where the fat-fingering on the tiny £1 calculator had led to an error but a couple of hours later the grid was full and a careful check confirmed that it all worked.

What was left to do? Two complete rows and two complete columns “that are related to the theme” had to be highlighted. It didn’t take much nous to see that only four of the rows and columns contained all the digits from 1 to 9 but what was special about those? The answer astonished me – they are all perfect squares! How did Oyler manage that! Thanks to Oyler.

## Listener No. 4399: Square Time Sums by Oyler

Posted by Dave Hennings on 10 June 2016

It had been over two years since Oyler’s last puzzle with its University of St Andrews theme. Before that was 2x2x2 which required us to cut out four cubes and arrange them into a larger cube in order to complete the dice net required for entry. That endgame was certainly tricky, and I hoped this week’s endgame was a bit easier. The fact that it was Oyler’s 13th Listener had me concerned!

Here we had to identify four triples, each consisting of 2-, 3- and 4-digit squares using all the digits 1-9. My first step was to pencil in the last cells of all the squares with 1, 4, 5, 6, 9 and 1, 3, 7, 9 for any entry given as a prime.

After that, 1dn 7 x X (3) seemed a good place to start since X had to be 16-81 and 7 x X provided the first digits of J, K and L so its digits had to be different. Only 175 and 567 filled the bill, but there is no 2-digit square 5?, so 175 got slotted in. 1ac was therefore 16. 2dn was prime, but there is no prime 62? so 2dn was 683 and 15ac was 5329. (J, K, L) was thus (16, 784, 5329).

Unfortunately, the rest of the solve went a bit more slowly and, in some cases, required the teasing out of single digits or the gradual rubbing out of pencilled options. I will leave the detailed solution to other bloggers or to listenercrossword.com, but in summary:

(A, B, C) was (36, 729, 5184)
(J, K, L) was (16, 784, 5329)
(P, Q, R) was (81, 576, 3249)
(X, Y, Z) was (25, 841, 7396)

And so to the endgame which required the highlighting of two rows and two columns which were related to the theme. 13 be damned! The 6th and 9th rows and columns each contained all the digits 1-9 and were also squares.

254871369 = 15963
847159236 = 29106
184231675 = 24807
412739856 = 20316

It wasn’t too difficult a puzzle after all, but enjoyable nonetheless. Thanks, Oyler.

## Boxing Characters by Mountain Ledges

Posted by shirleycurran on 3 June 2016

Mountain Ledges? Now who is that? One solving correspondent spent some time trying to work out who this setter identity might point to or what sort of a clue his crossword title and name might give us. He came up with “There must be scope for some fun anagrams of the Setter’s pseudonym – but if any are relevant, I can’t find any!  Nonetheless, picture the scene, the activity in question being full of champions who ‘dominate lunges’, with it all taking part of course in ‘timeous England’, in full view of statues that could only be described as ‘augmented lions’.  And perhaps they carry on playing ‘until game’s done’.  The champions naturally have ‘untamed legions’ of followers, most of whom, however, are simply ‘demeaning louts’.  [‘Gelatine mounds’ affixed with ‘nominated glues’ to an ‘insulated gnome’, have no part to play in the scene whatsoever ;-)” However, we decided finally that the title must refer to the fact that we were going to be led to HA MP TO NC OU RT (Hampton Court) and the REAL TENNIS that is played there by the ‘boxed characters’.

That was a couple of days after I had checked whether Mountain Ledges qualified for his entry ticket to the exclusive Listener Setters Toping Club. He or she (there was a ‘hen party’ towards the end of the clues) left little doubt. We had ‘hectolitres’ in the very first clue! ‘Vast quantities: hectolitres – a thousand in this sense (6) giving us [H]L A K HS = LHAKHS. That was a fairly tough clue to start us off, already producing one of those rogue extra letters.

We found a less difficult boozy clue in ‘Underground beer down under (4)’ leading to the double definition TUBE then a rather appropriate clue for someone mixing his/her drink, ‘Brings up Aussie beer back in Scots’ yards (8)’ GROG thrown up in REES giving REGORGES. A pity that was an across and not a down clue and rather an evocative memory of what it used to be like outside the pubs on London Road in Glasgow on Saturday nights, but nevertheless, Mountain Ledges confirms his/her membership – See you in the bar next March, somewhere in the north, Mountain Ledges. Cheers!

The serious business of solving began and the grid was fairly speedily populated, top to bottom with a few head-scratching moments. TIPTOED or TYPTOED seemed to fit into 21ac. ‘Worked at Greek grammar  and washed over instruction to read on (7)’ That’s what I am doing at the moment as we are due at a Greek wedding and Christening in a month’s time but that didn’t help me to select TYED or TIED to go round PTO in that clue, until I looked in Chambers and found typto, to work at Greek grammar. I wonder how I will be able to sneak that new find into dinner conversation!

I suspect a few unwary solvers will carelessly opt for TIPTOED and maybe choose RIATA over REATA at 46ac ‘Run Tanzania with a lasso (5)’. However, most solvers would, by this stage of solving, have seen JEU DE PAUME helpfully appearing across the centre of the grid and telltale words, DEDANS, TAMBOUR and PENTHOUSE around the perimeter.

We were less sure of what those extra letters were adding up to as we had HA and RT in two corners then MP and TO produced by the very explicit clues ‘Income from church committee gathering money to repair a bell (11)’ C[M]OM + MEND A [TO]M and ‘Doctor buried with rupee in rock where Taj Mahal was built (5)’ AA with R round GP giving us AG[P]RA. It took Wikipedia (as usual) to prompt us that HAMPTON COURT was one of the surviving REAL TENNIS courts and a useful diagram there showed us how to join up those strangely asymmetrical dots (with the convenient instruction that we must not draw one of our straight lines along the perimeter). It was clear that REAL TENNIS would anglicise JEU DE PAUME, leaving all real words – how I appreciate that, as usual.

Another co-solver sent me lots of information about REAL TENNIS (of which I admit I know nothing). I am pasting it here:

Forgive me if you are already well versed in the mysteries of Jeu de Paume… but if not, I thought I’d give you some background material which might throw a bit more light on 4398… though I suspect you have long since completed your usual expert blog.

First, it is another example of Listener topicality… Australian Rob Fahey, the current World Champion, has held the title continuously since 1994. This week he is defending his title in Newport, Rhode Island, against American challenger, Camden Riviere. It is being played over three days. Riviere won the first leg on Tuesday by 3 sets to 1.   The second leg is tonight… and if Riviere wins all 4 sets, he will take the championship… it is overall the best of 13 sets.  If Fahey wins even one set tonight, then it will go to a final leg at the weekend.

The link below gives you free LIVE streaming, plus archived recording of what has been played already.

2016 Real Tennis World Championship

Before the event, I predicted a win for Riviere by 7 sets to 2!

I took the game up in 1954 at Oxford, captaining the University team in 1958.  I also joined The Royal Tennis Court club at Hampton Court in 1958 and was secretary there for several years in the 1970s.   This photo of mine, taken from the galley above the hazard side penthouse, gives a good idea of what the grid attempts to delineate.  As you will see, the projecting “tambour” wall on the near side of the court, does not extend all the way up to the net.

Huge speculation as to who the setter is!!! Do you know? He/she must be a player.

I thought this might interest you too … I wrote it in early 2008.

Nice one, Mountain Ledges (or Insulated Gnome, or whatever).  Many thanks.

## Listener No. 4398: Boxing Characters by Mountain Ledges

Posted by Dave Hennings on 3 June 2016

It seems that Listener setters’ pseudonyms are getting weirder and weirder. This week, we had Mountain Ledges, which was probably just an anagram of his name, Emanuel St Dingo perhaps… or perhaps not! His puzzle (or hers — Melanie Godstun?) seemed as though it should be straightforward enough with only nine clues requiring special attention. These would lead to one or more extra letters in the wordplay, resulting in some squares containing two letters. Replacing them with dots and joining them up would make everything clear. Oh, and a thematic phrase would need replacing.

I started on the across clues, and a dozen were slotted in within thirty minutes, and nicely spread round the grid. Thirty minutes later and an equal number of down entries were in. Unfortunately, none of the clues led to any extra wordplay letters, which was disappointing.

Now the reason why I was flummoxed by these special clues was that I got the letter counts all wrong. I thought that an entry count of (6) would lead to a 6-letter word + one superfluous wordplay letter, with one square containing two letters. It was only when I finally got 1ac Vast quantities: hectolitres — a thousand in this sense (6) giving HL (hectolitres) + A + K (thousand) + HS (in this sense) that I realised my stupidity: the two letters in some squares were formed by one from the across entry and one from the down!

With the grid nearly full, I could see that HAMP…NC…RT was trying to give me HAMPTON COURT, and I realised that I hadn’t identified the clues with more than one extra letter in the wordplay. Mountain Ledges was being extra devious with 12ac COMMENDAM and 39dn SEASE. These needed two letters to go into one unchecked square — C[M]OMMENDA[TO]M and SEA[OU]SE. Cunning stuff!

Luckily, the endgame was pretty easy. At first, I thought we might be threading our waay through the famous maze, but it didn’t take me long to see JEU DE PAUME across the centre of the grid. I hadn’t heard of this before, but Chambers gave me REAL TENNIS. It transpires that the game had originated in France back in the 16th century, or even before, but without rackets.

With PENTHOUSE, DEDANS and TAMBOUR around the grid, it was obvious how the five straight lines should be drawn. It all represented the plan of a Real Tennis court. There is a court in Holyport, the other side of Maidenhead from where I now live. It is fascinating to see how the layout developed and even more fascinating are the rules which seem to be outrageously complex. There’s also a reference to the club at Falkland Palace in the clue to 26dn Door at Falkland Palace stopped short international mountain-dweller (4) (YETI).

Thanks for some real entertainment this week, Mountain Ledges… or may I call you Emanuel or Melanie!