# Posts Tagged ‘Elap’

## Listener No 4477: Word Squares by Elap

Posted by Dave Hennings on 8 Dec 2017

Last year’s Elap problem was about the Collatz conjecture and hailstone numbers. (A Youtube channel that I have recently come across is Numberphile which deals with a whole host of fascinating mathematics — see here for one of its Collatz videos.)

This week, what looked like a fairly interesting set of just 22 clues. Where there was a single algebraic expression, the answer was its square; where there were three expressions, the first was the sum of the squares of the other two and was the answer. Each letter was less than a hundred and the sum of two different non-zero squares.

As I’ve said before, there is normally just one starting point for a mathematical puzzle. Here, it was unlikely to be a clue like 13dn AD + oo, A + C + e – W, O + t – M – T (4) but more likely to be 12ac (which had two clues) D/P (2) and A (3) or 16ac I (2).

Anyway, before going any further, I constructed a little table of the sums of squares less than 100:

1 2 3 4 5 6 7 8 9
1 X 5 10 17 26 37 50 65 82
2 5 X 13 20 29 39 52 67 85
3 10 13 X 25 34 45 58 73 90
4 17 20 25 X 41 52 65 80 97
5 26 29 34 41 X 61 74 89
6 37 40 45 52 61 X 85
7 50 53 58 65 74 85 X
8 65 68 73 80 89 X
9 82 85 90 97 X

It would have probably been easier just to list out the distinct 28 values, but the table did just fine.

Starting with 12, where the first two digits and the whole number were squares, that had to be 169, 256 or 361, with A = 13, 16 or 19. However, only 13 was the sum of two squares (or SOTS as I put in my notes), so one down and 21 to go.

It was fairly near the beginning of the whole process that I remembered a numerical (Arden’s Square-bashing back in 2010, I think) where the correct solution depended on realising that the square root of a number can be positive or negative. I wondered if this would happen here.

I then seem to forget all about that until I solved 2dn A + t – N (3) which was 13 + 68 – 97 giving -16. A short while later, I got to the end of the puzzle anyway so no real negative square issues.

Except, I had both E and a equal to 53!

Drat!

Luckily, I didn’t have to go right back to the beginning, and found 13 Y, R – C, E – C – P (2) to be the culprit. Changing E from 53 to 37 fixed the problem and I breathed a sigh of relief.

Mind you, the bottom half of each grid looked a bit sparse and presumably the letters in numerical order would help resolve it: they spelt out ILAPCREMSTVarytWoDOZeN. At first I wondered if I’d got the first bit wrong and it should be ELAP….

This could be split more clearly into the three parts required by the preamble: ILAPCREMST for the letters 0–9; Vary — what needed doing to the letters in the first grid to give the second; and Two Dozen giving the total number of 5-letter words in the final grids, which I assumed would be across, down and diagonal, excluding upwards and backwards.

A short while later, after a bit of letter-matching and shuffling, I ended with the required number of words. Some were a bit weird and needed checking in Chambers, especially CEILI, ARERE and TRASS.

About par on the stopwatch for a mathematical for me, and rewarding to get to the end without too much back-tracking! Thanks, Elap.

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## Word Squares by Elap

Posted by shirleycurran on 8 Dec 2017

I’ve been dreading the Friday download all week. Yes, this OCD of Listener solving reaches the point where that three-monthly numerical can disrupt our existence for days. When we saw the length of the preamble (several times the number of letters in the clues) and tried to get our heads round what we were being instructed to do, we were truly discombobulated. This was not going to be easy.

Could Elap hope to have his Listener Tippler’s Club membership renewed with such an offering? I searched through his clues and found DeW – that’s one of the world’s top selling whiskies isn’t it? (It was also one of the last clues we solved when we had slogged for about five hours and found that D = 80, e = 89 and W = 73 which gave us 9216 to enter). I wonder whether that PASTS in Grid 1 was a careless spelling of PASTIS. Benefit of the doubt to Elap so “Cheers”. See you with the Pastis in Paris?

The other Numpty soon worked out that of the 34 available digits between 2 and 98 that could be the sum of two squares, 2,8,18,32,72 and 98 were not available as they are all the sums of identical squares, so we were left with 28 potential integers that had to be the equivalents of the letters I,A,P,C,R,M,S,T,E,Y,O,D,Z,L,N,V,W and t,o,e,a,r (there would be six left-overs). That didn’t seem quite so daunting and we set to work with him filling the usual mountain of paper and complaining at my slowness with the calculator.

Initially the grid fill went well but we hit our first brick wall when we found that we had E = 53 and o = 53.  We had O at 89 and t at 61 at this stage and things had been looking good, but it was not to be. I don’t think my O Level maths teacher ever told us that a negative number squares to a positive, but the other Numpty knows that sort of thing and with lots of cursing, we extricated ourselves from our mess which meant rethinking O, t and N among others.

Enough – you wouldn’t be reading this if you hadn’t completed the puzzle. So on to the endgame.

Once our grids were complete (well, all the clues were in) it took five minutes to order the letters by increasing numeric values and we found the ten letters that must replace the digits 0 to 9 – ILAPCREMST, the hint that we had to VARY the positions of the letters in the first grid to arrive at the second grid and the information that we were looking for TWO DOZEN five-letter words.

At this point, I should admit that during our flailings, the word TWENtY had obligingly appeared and totally misled us about how many words were going to appear in the word squares, but TWO DOZEN! That is an achievement in itself and, of course, required two words going diagonally in each grid or some going in two directions (actually I found 26 by counting STIME and EMITS, TRAMS and SMART as well as the four diagonals, with SAIRS being a valid Scottish word but I suppose that is just nit-picking).

Converting the number grids to word grids was almost fun. That’s how I like my crosswords – WORDS! – but we were faced with gaps and were told that we had to VARY the positions of the letters in the first grid to produce the second. Crossword compiler told me that there was only one way to complete grid one and that gave SEERS on the last line and the spare letters T,M,S,S,I,S to use to complete the second grid

Ah, the HARE. He was there in 2d, running round like a headless chicken or burying his head in the sand at the thought of a numerical crossword, and there was a MARA doubling back on himself at the top of the second grid but it was another solver who actually told me how to fill the second grid. I had to anagram or jumble those 25 letters of the first grid and that gave SILLIEST TRAP SECRETES A MARA. Why didn’t I spot that for myself? Simples!

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## Hailstorm by Elap

Posted by shirleycurran on 11 Mar 2016

Collatz conjecture? I, the Listenernumericalcrosswordiphobe had never heard of it but the other Numpty had. I find it difficult to conceive that a professional mathematician would devote his mental energy to developing and exploring such a conjecture. However, Elap’s puzzle clearly confirmed it for the case in point, so it can be added to all that tremendously useful intellectual baggage that the Listener crossword provides and that I can slip into the next lagging dinner conversation.

Through habit I scan the clues to confirm Elap’s continued membership of the Listener drinky club but no, he almost manages RUM in his clues and there is a very small gt, but that is the sum total of his alcohol, so, in view of the daunting preamble, we have to compensate and pour ourselves a couple of large ones and get going on the solve. Cheers, Elap, anyway.

The 1dn NNNN was an easy way in, followed by 8ac NP+PP. This allowed one to have a look at the possible ‘Collatz accomplices’ for N=2 (just 1 and 4) and P=7 (just 2 and 22), making things less intimidating. It was possible to nibble away at the clues without a lot of options to try and track, and (which seems rare nowadays, no need for a spreadsheet) but it was easy to go astray- with G as 160, I took a long time and much muttering to notice that 53 was an option for g, not just the hastily assumed 80!

There were a couple of hitches like this and muttered retreats before the more numerical Numpty finally had a grid-fill and we had to work out what to do next. Fortunately, my skills at decoding are fairly limited and the alphanumeric option is the one that springs to mind. There was even a useful hint there too, since we were told that this decoding had to be done in a ‘thematic direction’. I’ve rarely seen snow or hail fall in any dramatically different direction from down, so we worked downwards and speedily got PRODUCE. My only useful contribution was to spot that the 2-digit 38 appeared next and we were told that there were to be 38 entries, so we were able to complete the message: HAILSTORM NUMBERS FROM 988 AND FILL GRID.

We carefully listed the 38 relevant hailstorm numbers then enjoyed the jigsaw puzzle of fitting them in with just enough hints given by those circled numbers that were already there. (Yes, I have to admit, albeit a numerical Listener puzzle, this was fun!) The quarterly Listener number puzzle without spreadsheets, heaps of discarded worksheets, chewed pencils and frayed temper? And one with two grids to be completed with two different methods. This must be a numerical triumph. Many thanks, Elap.

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## Listener No. 4386: Hailstorm by Elap

Posted by Dave Hennings on 11 Mar 2016

SCENE: The Editor’s office, December 2015.

There is a desk in the centre of the room. There is an in-tray on the desk containing a letter. There is a clock on the wall. It reads 11:00.

There are two chairs: a big one is behind the desk and a small one in front of it. Editor is sitting on the big chair and Sub-editor on the small one. Next to the small chair is a stool. Elap is sitting on it; he is hugging his briefcase.

Elap: Nice in-tray.

Editor: Thanks.

Editor takes the letter. As he does so, the in-tray slowly disappears. Elap either doesn’t notice or is used to that sort of thing happening.

Editor: This is the letter I mentioned on the phone. (Reading from letter) “The amount of Listener real estate that is wasted by the mathematicals having small grids and a puny set of clues is unacceptable. Do something about it… or else!” It arrived a week ago after the last mathematical.

Sub-editor (to Editor): That was yours, wasn’t it?

Editor (ignoring Sub-editor): Out of all the mathematical setters, you were the only one who said they had something in the pipeline that might help.

Elap (removing a thick sheaf of about 50 A4 papers from his briefcase): Yes, you’re lucky. I’ve just finished my latest effort. Given the title of the puzzle, I managed to con the Meteorological Office into giving me use of their old supercomputer for six months this year. It produced this.

Editor: We need to fill eighteen column inches, not three newspapers.

Elap: Oh, this is just the computer output. Here’s my preamble. (He takes another wadge of about 40 sheets from his briefcase and hands it to Editor.) I’m sure you can whittle it down a bit.

Editor (reading page 1): “The Collatz conjecture is a problem posed by the German mathematician, Lothar Collatz, in 1937. It is also called the 3x+1 mapping, 3n+1 problem, Hasse’s algorithm (after Helmut Hasse), Kakutani’s problem (after Shizuo Kakutani), Syracuse algorithm, Syracuse problem, Thwaites conjecture (after Sir Bryan Thwaites), and Ulam’s problem (after Stanisław Ulam).

“Collatz was born on 6th July 1910 in Arnsberg, Westphalia. In 1937 he posed the famous conjecture, which remains unsolved. The conjecture can be summarized as follows. Take any positive integer n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called “Half Or Triple Plus One”, or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness. The sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.

“For example, the sequence for the number 27 is as follows: 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, …”

DISSOLVE to clock which now reads 12:25.

Editor (still reading): “…the second part explains how solvers must apply the results of the first part, after erasing the contents of all but the circled cells. In each version of the grid, the 38 entries are different and none of them starts with a zero.”

The last page contains a grid and some clues. Editor puts it down on the desk. His eyes have glazed over.

Editor: Well, it’s not a particularly large grid…

Sub-editor (glancing at last page): …but bigger than yours was…

Editor (ignoring Sub-editor): …and there could be more clues…

Sub-editor: …more than yours had…

Editor (glaring at Sub-editor): …but with a somewhat cut-down preamble it should do the trick. Thanks, Elap.

###### Any similarity to actual events is entirely unlikely.

An Elap mathematical again this time. Last year’s (no. 4347 Pairs) used squares or numbers which were concatenated squares. With Elap, I always think back to his Three-square puzzle in 2010 which needed us to realise that all rows and columns consisted of triangular numbers. I nearly failed on that one, and I knew that an Elap endgame could be the cause of potential grief.

This week’s puzzle was based on the Collatz conjecture where x/2 or 3x+1 repeatedly would lead us to 1. 1dn looked as good a start as any, NNNN being 2 digits had to be 16, not 81 which would have 8ac as 1• but had to be greater than 1dn. So N was 2, which meant that 8ac was 2P + P² and only P=7 gave a number 6•.

From there, progress was fairly quick, with 2dn, 4dn, 10ac, 26dn, 10ac and 28dn leading the way. There were a couple of long pauses as I progressed, and care had to be taken to ensure that all the options for the hailstone numbers were accounted for. I made one mistake with H=28, where I initially overlooked h=9 as a possible option; luckily h was 56.

The grid was completed in about two hours, and all we had to do was “decode it (in a thematic direction)”. For about twenty minutes I tried to use the values of all the numbers in the clues — N=2, R=3, m=4, t=6, etc. I realised that the thematic direction would be vertical, either all down, or down, up, down.

Luckily, it didn’t take too long to realise that it was simple alphabeticala positions that led to Produce 38 hailstone numbers from 988 and fill grid. I listed out the required numbers, a bit puzzled by there being 50, including 988 and 1.

1114 was the first one in the grid, using two of the numbers left from satge 1. 1672 therefore occupied the second 4-digit space, and from there the grid-fill was easy… until near the end. I had options for some of the 2-digit entries — 5dn could be 10, 13, 16 or 19. Most of the options for these digits would fill gaps in the main body of hailstone numbers, and I realised that “from 988” meant just that.

In reading about hailstone sequences, I was particularly struck by 27, a seemingly innocuous number, but requiring 111 steps to reach 1 and climbing to 9232 on its way.

Thanks to Elap for another excellent puzzle, and thanks to Collatz (or Hasse or Kakutani or Thwaites or Ulam) for his fascinating mathematical conjecture. I await its proof with interest.

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## Pairs by Elap

Posted by shirleycurran on 12 Jun 2015

The penultimate week of February, May, August and November – the dreaded numericals. I download it and dash away, handing over to the other Numpty. Not much chance of even checking for Elap’s membership of the Listener Setter’s toper’s crowd. The clues read ‘FLY!’ (which I do) and Grr! which, I admit, is my usual reaction to the hated things – though, of course ‘bottoms up, Elap!’ and you get PALE, or, with the top removed, just ALE, so I suppose he has automatic membership.

Numpty No 1 applies his usual logic and Zipf’s law of frequencies which suggests that the likely ranking is going to be I, Y, E, X, L, G, S, F, T. Very soon there is a restriction on I and Y. Y can’t be greater than 9 (19ac) and I has to be less than 10 (13dn) so I and Y are 1,  4, or 9. Clue 13 also suggests that I is not 1, so I and Y are either 4 or 9. He proceeds in this way and soon can guess the first hint: BY ITSELF (?YI?????).

My messy calculations

With a few hiccups, he struggles to a full grid and completes what seems to be the hints: BY ITSELF, GAP RDX J NUM Z. What on earth can that mean? To a computer man, it sounds like code. We puzzle over it for too long then decide to simply ignore that strange string of letters and simply put all the numbers from the grid in ascending order and try multiplying them by themselves and inserting a GAP.

This task has fallen to me and I idly multiply the first number by itself (13 X 13 = 169) and put a gap into it, giving 16 9. Wasn’t this puzzle all about squares? It is perfectly obvious that if I take the square root of those two numbers, I get 4 3, remove the gap, and that gives another number that appears in my list – 43!

I try it with the next number in my list. 19 X 19 = 361. The square roots of 36 1 are 6 1. Remove the gap and yet another number from my list appears. Eureka! With a shout of glee, I tempt the other Numpty back (he has gone to bed by this time!) and we work our way through the list until only 1771 is left. That squares to 3136 441, which gives square roots of 56 21. That goes under the completed grid.

What a compilation! I hope Elap will give us a setter’s blog to tell us how long it took him to find numbers that would work and dovetail into a symmetrical grid, and why he didn’t tease us with some unlikely phrase made from the un-hinting letters,  RDXJNUMZ (which are used in clues) and CHKOVW (which are not). Perhaps in Welsh or Serbo Croat?

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