Listen With Others

Are you sitting comfortably? Then we’ll begin

Listener No 4477, Word Squares: A Setter’s Blog by Elap

Posted by Listen With Others on 10 December 2017

The Theme

I started by creating a file containing all of the five-letter words from TEA 2.11. There were 12,125 words including proper nouns.

Out of curiosity I wrote a program to see how many ways ten of these words could be fitted into a 5×5 grid. It took three hours to find that there were 1,152,602,603 different filled grids (excluding reflections).

I then decided to see how many grids there were where the two diagonals were also words, and the answer was 575,560. This took 163 seconds.

I was unhappy with the number of unacceptable words in these, and so I produced a set of 10,508 words which did not contain any proper nouns. There would obviously be other unacceptable words, but it was easier to ignore grids which contained these than to go through the list deleting them (it was easy to remove the proper nouns because they began with a capital letter).

I reran my programs. There were now 229,858,145 grids (the run took 70 minutes) and 115,265 with the diagonals too (taking 71 seconds).

Since there were so many ways of fitting words into the grid, and since I wanted my final grid to be special in some way, it struck me that there was scope to see whether two of the grids could be anagrams of each other.

I decided that the grids should contain ten or fewer different letters so that they could be replaced by digits, thus forming the basis for a mathematical puzzle.

For each filled grid, I wrote the 25 letters (in alphabetical order), followed by the words in the grid, to a file. I then wrote a wee program which sorted the records into ascending order and which then scanned the file to see whether two adjacent records had the first 25 characters the same.

It turned out that there were six possible pairs of grids. Two of them contained similar words somewhere and three contained mildly rude words, but there was one pair which was acceptable. This was very satisfying.

Nature of the Clues

Since I was dealing with word squares, it seemed appropriate to use squares in the clues. The clues would consist of expressions in terms of letters whose values had to be deduced.

I wrote a program which determined which substitution of digits would produce the largest number of perfect squares in the grids. The idea was that some clue values could be squares or square roots – but how would the other clues work? After a lot of dithering, I decided to have some clue values as square roots of the entries, and some consisting of the required answer which was equal to the sum of the squares of two other values. The advantage of this is that the roots could be negative, which might fool some solvers (it fooled me too, because the first vetter spotted that I had two letters round the wrong way in one of the clues which led to a negative answer!).

In line with the theme of two squares, I decided that appropriate letter values could be numbers which could be expressed as the sum of two different squares.

The Hint

I needed a hint, though, which would appear when the letters were sorted by their values. Part of the hint would be the ten letters, in order, by which the digits 0 to 9 were to be replaced. Another part needed to indicate that the grids were anagrams of each other. But how? – there were not enough letters to convey the message. I then hit upon the idea of using lower- and upper-case letters in the clues so that a better hint could be constructed.

The ten letters, represented by 0 to 9, were ILAPCREMST. For the anagramming hint, I first thought of BYJUMBLING to indicate how the second grid was to be derived from the first, but I had also to indicate that the diagonals were words too. I was quickly using too many letters for the clues to be solvable without there being too many of them, and anyway I didn’t want an L in lower case because it would look like the digit 1.

I needed to say that there were TWENTY-FOUR words in the grids, but with ILAPCREMST that used three Ts already! Wait! What about TWO DOZEN?

The hint could then be ILAPCREMST……TWO DOZEN…… with some of the letters in lower case.

Some solvers, though, would assume that there were twenty words in the grids, and so a Y was needed to reinforce this. The word which contained the Y would have to indicate that the grids were anagrams of each other. Hmmm…

What have we got so far?

……ILAPCREMST……TWO DOZEN……

What word, containing Y, could indicate an anagram? If the word had more than, say, six letters we would again be heading for too many letter values for comfort. I needed a short word.

It was a few days later that VARY suddenly sprang to mind. It could be the positions of the letters in the first grid that would have to VARY to arrive at the second grid.

Our hint was now looking like one of these (some of the letters will be in lower case):

VARY ILAPCREMST TWODOZEN
VARY TWODOZEN ILAPCREMST
ILAPCREMST VARY TWODOZEN
ILAPCREMST TWODOZEN VARY
TWODOZEN VARY ILAPCREMST
TWODOZEN ILAPCREMST VARY

The chortle that my wife heard from the study was when I looked at the third option, ILAPCREMST VARY TWODOZEN, and a mean streak in me began to surface.

If there were at least four available letter values between the first T and the Y, wouldn’t some solvers assume that they were the first and last letters of TWENTY? Tee hee!

I changed the case of one of each of the duplicated letters to arrive at this hint:

I L A P C R E M S T V a r Y t W o D O Z e N

The trap was about to be set.

The Clues

I won’t go into details of the derivation of the clues (mainly because I have forgotten now), but the letter values were these:

I L A P C R E M S T V a r Y t W o D O Z e N
5 10 13 20 25 26 37 40 41 50 52 53 58 65 68 73 74 80 82 85 89 97

 
Intentionally, amongst the first values deducible were T = 50 and Y = 65, encouraging some solvers to jump to this conclusion:

  e
…… T W E N t Y ……
50 52 53 58 61 65

 
If this assumption is made, the values of e, L and O would be incorrect, as well as the values of W, E, N and t.

Solvers would most likely end up with these incorrect values:

Letter Correct
Value
Incorrect
Value
E 37 53
e 89 68
L 10 17
N 97 58
O 82 89
t 68 61
W 73 52

 
All the clues except for 7ac and 18ac (the most complex, and likely to be solved relatively late) would still work. Whether or not the correct or incorrect values are used, we have:

In 5ac, 13dn and 14dn O + t is 150
In 6ac and 10ac O – L is 72
In 6ac |N + R – C – e| is 9
In 15ac L + t is 78
In 19ac L + L + t – O is 6
 
In 2dn |A + t – N| is 16
In 3dn and 13dn e – W is 16
In 13dn |E – C – P| is 8
In 4dn |C + W – e| is 9

 
As a retired programmer, one of the lessons I learned early in my career was not to unnecessarily assume anything, and maybe this is a lesson to some of the solvers!
 

Advertisements

2 Responses to “Listener No 4477, Word Squares: A Setter’s Blog by Elap”

  1. Encota said

    A very interesting blog – many thanks!

  2. shirleycurran said

    Of course, I was one of the gullible solvers – I could hear that hearty chuckle from here!

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

 
%d bloggers like this: