## Listener 4177: Number Plates by Xanthippe

Posted by erwinch on 9 March 2012

A while back Xanthippe advertised on the Crossword Centre’s Message Board for test solvers to check his new numerical puzzle and presumably this is it. However, it turned out to be a bit of a hybrid and the preamble even cited Chambers as the primary reference, which I believe to be a first for a numerical Listener.

Our initial grid entry could have been the first digit of

**k (= 1?)**but mine was**a**from**2iac:****30a (three)**(I have spelled out the answer lengths here to avoid confusion). With**a**prime and crossing**30a**, only**a = 13**fitted so**2iac = 390**With 1iac a palindrome (four), the final digit was also 1,**1iac = 1??1****xdn: 2au (four)**

**rdn: x / u (two)**

So

**r = 2a = 26****8iac: Prime (two) = 6? = 61 or 67; t & q = 1? or 7? (qdn)**

**x = 26u**so u must be in the range 39 to 99

**(7iac) SQRT(2(x – u)) = SQRT(50u)**(this must be an integer)

**udn: SD is odd (two)**so

**u = 50, 72 or 98; SQRT(50u) = 50, 60 or 70**

But

**t > u (tdn)**and**7iiac: 10q (three)**so**t & q = 70; 7iiac = 700;****u = 50; x = 1300; SQRT(2(x – u)) = 50**The final three digits of each row were to be replaced by letters. For row 7 (

**700**) this could only be**ILL**, which fits the given definitions cross or sick.**6iiac: (p – 7)^2 + 7 (four) = ???7**

The final digit of p equals the first of q so p = ?7 and p – 7 = ?0

p is not prime and can only equal 57,

**p = 57, 6iiac = 2507**The final three digits of row 6 (

**507**) could be**RAI**with definition music (African).**4iiiac: SD = k (two)**

The maximum value of 4iiiac is 99 (SD = 18) but k is prime so

**k = 11, 13 or 17****sdn: (n + 1)u (three) = (n + 1)50 = ?00**since n is prime;

**n < 19 (= 11, 13 or 17); s = 600, 700 or 900**

**7iac: 127n – SQRT(2(x – u)) (four) = 2???; 127n – 50 = 1347, 1601 or 2109**

The only fit is

**7iac = 2109; s = 900; n = 17; 2nu = 1700 (12iac); C (prime) = 31, 61 or 71****9iac: SD = 10 (five) = ?0100;**

**w (prime) = 9?; 9iac = 90100; w = 97**

**vdn: n – 5 (two);**

**v = 12**

**3iac: (v + 1)^2 (three) = 169**

**5iac: anv / 2 + 10 (four) = 1336**

**8iiac: Palindrome (five) = 10501**

**jdn: Even (three) = 162**

**cdn: j + 10 (three) = 172**

**2iiac: 2o (four) = 7?02**

**3iiac: b – 2a – p (four) = b – 83 = 20?3 or 20?9 (g is prime)**

**b = ?09? = 2092;**

**b = 2092; 1iac = 1221; 3iiac = 2009**

**y is prime = 2? = 23 or 29**

**10iiac: 10(z + 1) = 230 or 890 so z (a palindrome) = 22 or 88**

**11iiac: s – 4z – 10 = 890 – 4z = 802 or 538**

The first digit of 11iiac is the end of B but B is even so

**y = 23; z = 22; B = 28; 10iiac = 230; 11iiac = 802; 12iiac: k + 10z (three) = 23?**The grid now looked like this:

Shortly after this I found that the 38 two-letter entries and 12 two-digit numbers in the grid would represent the 50 US states.

Finishing off the numbers in the grid was fairly straightforward and this was my favourite part of the puzzle. Whether it was by design or luck, just as you seemed to be at an impasse another opening would appear. My total working did not fill a single side of A4 and a calculator was the only aid required. However, I was unaware at this point of the mistake in row 7:

My least favourite part of the puzzle was checking each of the US states – this bordered on the tedious. [With my brother, we had a go at trainspotting in the Sixties and each had an Ian Allan book of numbers – we barely stuck at it for two weeks.] I had all the states listed from AK to WY and checking those that formed the down entries in columns 5, 6 and 7 was simple enough. But when it came to the numbers in the first two columns I made a mistake and had to backtrack. Fortunately I did not have to go back too far but had it been any further I might have given up thinking that this was not worth wasting time on. I could also see a problem looming – three 21’s in rows 1, 4 and 7. I kept on rechecking the working for these figures until twigging that this was the ambiguity that was to be resolved.

**7iac = 127n – SQRT(2(x-u)) = 2159 – 50**but I had naturally taken the principal square root, the positive rather than the other negative value. Using the negative value results in**7iac = 2159 + 50 = 2209**. The 22 gave us NY and the two central 21’s were NC and NM. Finally, the central pairs of 09’s (AK and WV) and 10’s (CA and MA) completed the set:[With thanks to demonplates.com]

So, the final grid displayed twelve past and future car number plates under the current UK system, except that the letter I is never used since it would be seen as identical to the number one. This would have confused some plates with the previous prefix system (

**H110 GAM**may have been issued under this system in 1990/91). But what thematic link was there between number plates and states? I could only think of one, that Xanthippe had invented the rhyming slang:**number plate, state**. So states would be referred to as numbers, as indeed they all were in the interim grid. Perhaps a tenuous link as these things go but I think acceptable.To sum up, I have now forgotten the tedium of checking the states and will long remember this as an inspired idea expertly adapted as a Listener. I certainly hope that Xanthippe can find the time to blog the development of the puzzle – that should be most interesting.

As a footnote, I see that the twelfth plate with the 00 date is not due to be issued until September 2050. Looking at the DVLA site, the following plates and

**CL00 GAL**or**CL00 BOT**, if available, are likely to cost in excess of £1000 at today’s prices. Will there be any takers from the crossword world I wonder?I was once driving with work colleagues visiting from India and they asked if there was any significance in cars having yellow or white number plates. It is so we know that we are not on the wrong side of the road of course – always follow the yellow!

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## Paul Taylor (@aPaulTaylor) said

I really enjoyed this puzzle, up until the square-root ‘ambiguity’, which I think I have a bit of a problem with. True enough, positive real numbers have two real square roots, one positive and one negative. But the square root

function— √x or SQRT(x) — always denotes the principal square root. This isn’t just a convention: if it can take either of two values, it’s not a function. I don’t think saying ‘SQRT means “square root of”’ in the preamble is enough to get you out of this.Also, the ‘solvers must resolve an ambiguity’ instruction in the preamble came after the statement ‘after the grid is filled with numbers…’, implying that the ambiguity is not in the numerical part of the puzzle (in fact ‘implying’ seems a little weak here — that is what it says).

Lastly, the ambiguity didn’t really seem necessary. I can sort of see the reasoning behind it, insofar as it ensures you know exactly what’s happening with the state codes, but it was surely there was a way of doing it not so utterly ill-matched with the rest of the puzzle.

Sorry. Rant over. Like I say, I thought it was a really nice puzzle otherwise. Maybe I’m just annoyed because it took so long for the penny to drop.

## erwinch said

I must admit that I am somewhat surprised at the reaction to this square root business. I have long considered such tricks to be an integral part of the thematic puzzle and would sorely miss them if they were to be removed no matter how artificial or unnecessary they may appear. SQRT here had to be seen as a Listener function where no formal rules apply.

There again, I did feel aggrieved by Elgin’s mirror image trick in

Asylag(Listener 3758) since I had spent quite some time looking at the completed grid in a mirror but had failed to spot anything significant. I suppose that no one enjoys being tricked.However, these puzzles are not industrial processes where the ideal is to minimise the time spent by solvers so that they can use the saved time more productively. I did not particularly enjoy checking the fifty states but do not consider it time wasted although I did waste about 20 minutes looking for my trusty printout of all primes below 10,000. This had been generated using BBC basic decades ago and I kept on forgetting that I had put it in an old Chambers Thesaurus so had tried elsewhere. It eventually turned up in my Brewer’s Modern Phrase & Fable where it was returned after checking that 3701 (o) was prime.

## Phil Caine said

I see you had 2 possibilities ANY and ADD at row 9 in your early working. I looked in Bradford’s and came up with ADD and AND as synonyms for JOIN and I assumed that this was the ambiguity. The down number 23 gave either DElaware or NEbraska.

The other 23 was in the NW quadrant and both states checked with the across possibilities IDaho and INdiana.

I decided that Chambers and not Bradford’s was the bible and chose ADD giving a correct entry.

## Paul Taylor (@aPaulTaylor) said

I certainly don’t have a problem with evil tricks, but they have to be fair evil tricks, and for me this one just edged over the line. I think this could have been resolved to my satisfaction with just some different wording in the preamble (something like “Solvers will likely have to amend one number to accord with a less usual interpretation of one of the clues…” or whatever).

## erwinch said

Yes, you are quite right Paul; a trick cannot be considered completely fair unless everyone agrees that it is so the preamble here should probably have been more exact. This would also have helped solvers such as Phil above and others who thought that ADD or AND defined by join was the ambiguity.

They had found the synonyms under

Join(er), Joinedin Bradford’s, which includes many lists under these broad headings. Bradford’s has been known for its mistakes and inaccuracies and we once even had a puzzle that exploited many of them. One example from my 5th edition is Haiti being listed as an island whereas it is no more an island than say Wales is.Anddid not immediately spring to my mind as a synonym for join but I have given it some thought and have come up with the following:Russia and/join China in veto.

Perhaps it works better if the first partner is more definitely pluralistic:

Morecambe & Wise and/join Des O’Connor in Panto.

Even here it does not look like a seamless substitution since I can see subtle differences in meaning. And appears to be a bald statement of union while join implies something deeper such as

the more unaccountable and corrupt regimes there are in the World then the better for us.## Phil Caine said

Thanks Erwin. That’s more or less the conclusion I came to although I was considering AND = JOIN in the logic/maths context of (say) a Venn diagram which is in Chambers.

Regarding the actual anomaly in this puzzle I thought that as it was the only way to include NY, there wasn’t really a dilemma and I’m surprised at all the fuss. However Xanthippe has ‘form’ when it comes to tricks. I was marked incorrect on his/her last puzzle when I submitted the cut out bits, lightly cellotaped them to the grid:-(