This week, Ilver’s back. His last Listener was his puzzle celebrating the 50th anniversary of *Doctor Who*. This week seemed to have another television theme with the American sitcom *Taxi*… or not! Whichever, we had down clues with wordplay leading to an extra letter. After that, there would be a quotation, a key, another quotation, a contradiction, a corection that would resolve the contradiction and finally an identity to be highlighted — in lower case, of course.

There was nothing about the across clues, which was vaguely worrying as there would normally be an “Across clues are normal” sentence. It was also a bit of a concern that there were only two cells where the left and right sides of the grid linked.

1ac *Beats time bars (6)* looked as though it could be TRACKS or TRAILS. I decided to find out which and switched to the downs. 1 was alternate letters of *tusk on a Thai ship* for TSOT[A]SI and 2 was R[V]ED ANT. 3dn was, of course, another surprise — while it was given as (11) in the clue, there were in fact only eight spaces! 4dn was KETA and that confirmed 1ac as TRACKS

A few more clues came, including 14ac *Roadway Model T car with old lady aboard (6)* with its nicely misleading surface reading for TARMAC. 13ac *To humiliate oneself brew tea with cold rag (7, two words)*, had me toying with CAGE RAT and RAGE ACT before finally settling for EAT CROW. That gave me CHAMPIONESS for 3dn, and the ONE could become 1 in the grid and then the PI could become Π. 10dn was PERIPLUS (8) for a five space entry, so PLUS became +. All this looked as though some sort of mathematical shenanigans was in front of us.

The northeast corner was a bit trickier, but the bottom right was fairly straightforward, with 19dn IMPISHLY enabling most of that quadrant to come together quickly. In the lower left quadrant I had a strange bit of luck. With 30dn *Baroque composer cycles to one old gypsy woman (4)*, I got CHAI ([B]ACH cycled + I) and then got interrupted. On returning, I absent-mindedly cycled CHAI to give AICH and looked that up to see **aich whow** “see **whow**“. Looking that up, it gave me the answer to 11ac EH WHOW, which I hadn’t solved yet and, bizarrely, didn’t have its own entry in Chambers!

By this time, I had most of the extra wordplay letters in the downs and could see that they would eventually spell out **A very interesting number**. I knew that this referred to 1729, the number of the taxi that Hardy had taken on visiting Srinivasa Ramanujan (although I’d forgotten the names of the two characters involved).

With the grid completed, it was time for the endgame. 1729 was the key used to find the first part of a quotation. It didn’t take long to write 172917291729… alongside the across clues, and taking the appropriate letters in turn gave **Beauty is the first test**. This turned out to be from the same mathematician, Godfrey Hardy, and continues “there is no permanent place in the world for ugly mathematics”.

Moving on, we had to find a contradiction to this latter part and its author using the same key. I tried the same treatment on the down clues as the acoss, and got TNOCBACRIFIMW…. That was gibberish. How about working backwards from the end of the across clues: SUASGYLCARIM…, more gibberish. I tried working backwards in reverse clue order for both acrosses and downs to no avail.

Now although we were told to use the same key, it may not be used in the same way. Perhaps 1729 would become 1/7/29 or 17/29 or 17/2/9. More letter counting, and I could see a brick wall and failure looming.

At the back of my mind, I had been toying with the “(24 cells in total)” given by the preamble. There were 20 across clues and 22 downs, so it really seemed unlikely that they would provide the next step — yet it didn’t stop me wasting a couple of hours trying. The grid, on the other hand, was 12×12 and was a more likely source. The 1st letter T then move on 7 squares in the first row for O, then 2 A then 9 N then EETRART in boustrophedon fashion. Nothing. The same for the downs — nothing.

And then the letters from square 1 T, square 7 H (row 1), square 2 E — THE — that looked promising. Continuing with square 9 W, square 1 O and so on through every row. I was presented with THEWORAHSTTMMAEICLDHARDY. HARDY at the end meant that I was home and dry and in between THE WOR and LD was an anagram of MATHEMATICS. Thus the apparent contradiction of the second part was that we were presented with an “ugly mathematics” (ie anagram) in “the world” and that needed to be corrected. I originally thought that the contradiction would be some sort of quotation and from a different author.

Of course, the π + 1 = 0 had been evident in the grid all along, and with the new letters in place, it could now be expanded to give e^{iπ} + 1 = 0. This is Euler’s Identity and is considered by many (most) mathematicians to be the most beautiful mathematical equation using, as it does, five “special” constants.

I suspect that the second part of this endgame may have caused a few problems, but luckily I’m not (I think) one of them. So thanks to Ilver for another excellent puzzle and a chance to investigate lots of other mathematical fun and games along the way (including 1+2+3+4+…=-^{1}/_{12}!).