Listen With Others

Well, this quarter I was actually geared up for this quarter’s mathematical puzzle — so often they catch me by surprise. However, I wasn’t geared up for the setter. Having expected Nod or Zag to make a reappearance, I was faced with Smudge, and he didn’t ring any bells regarding a previous puzzle. I was surprised, therefore, to find that he had set one of the tough puzzles of 2016 over two years previously — No 4388 Cycle 20% More, all about Gilbert and Sullivan’s The Grand Duke, not a mathematical at all.

This week, we had a series of properties in a list, such as square, cube, triangular number, Fermat prime. Each grid entry had to be associated with exactly one of these properties, as did the clue number at which it was entered. We were also provided with information about numbers divisible by their reverse, perfect numbers and integers which are the sum of two squares, all of which “may be helpful”.

I loved the clueing technique here, similar in many ways to Piccadilly’s The Properties of Numbers — II last year. I made a list of the clue numbers and then went through annotating each to show which of the given properties applied to them. I gradually teased out some grid entries but, two hours later, I reached a dead end — my first. This was, I think, because I had put 56 as a definite for 10dn, rather than just a possible.

I decided to be a bit more organised on my second attempt, and created a grid with clues down the left and properties/occurrences across the top:

This made it much easier to tick off the entries as I resolved them or to mark properties that didn’t apply to a clue.

I also used my favourite mathematical tool, WolframAlpha, to identify the following:

• tetrahedral numbers, n(n + 1)(n + 2)/6: 1, 4, 10, 20, 35, 56, etc
• Mersenne primes: 3, 7, 31, 127, 8191, etc
• Fermat primes: 3, 5, 17, 257, 65537, etc
• Perfect numbers: 6, 28, 496, 8128
• Numbers whose reverse is divisible by the number: 1089, 2178 and palindromes

Unfortunately, none of this prevented me from diving headlong towards dead-end number two, which was overlooking 901 as a possible entry for 34ac (reverse of a prime but not a prime).

Third time lucky, and my grid looked like this:

And I managed to successfully complete the puzzle like so:

Thanks for an excellent mathematical, Smudge.