It seems totally preposterous now but I once asked Radix in an e-mail if he might be related to Ix – a son perhaps. I had harboured the similar idea that Leo might be a scion of the mighty Leon but must accept far more likely explanations such as Leo having been born in August or simply given that name. Anyway, this was Leo’s fifth Listener since 1996 and the title suggested a mathematical theme – my favourite.

Knowing how long setters spend compiling their puzzles, I hate to describe them as easy and prefer to think that I must have been mentally in tune with the clueing style. I certainly was this week and was able to rattle through the clues in a single sitting. Extra letters spelt out a seven-word message:

**Letter count gives a second magic square.**

Identifying the extra material that was to help with filling the nine empty cells that appeared was a bit more problematical. For example, in 39ac, I originally boxed as surplus the entire second half of the clue – underlined here:

What preceded IBM in 2001 – majority had late date unfortunately missing

I had known that HAL precedes IBM by one letter in the alphabet (see the interesting Wikipedia article on

HAL 9000) so thought that the first half sufficed as a very neat clue. I started by counting the letters in the surplus material and we had 39 here (the same as the clue number!) However, the across and down numbers never matched and added together, or otherwise manipulated, did not form a 3×3 magic square anyway, as far as I could see. There was also the matter that the letter count was to give a

*second* magic square when we had yet to see a first.

Finding the first magic square was the most enjoyable part of the puzzle for me. The extra material in 40dn (

**adult entertainment**) was to match that in 39ac, which of course was

**majority = 18**.

**Two little ducks** at 41ac took longer than it should have to fathom – the delightful

**22** to match

**catch**. I spent the 1972 summer vacation working at Kingston’s Top Rank bingo hall but the callers did not use the likes of

*two little ducks*. That would have been: all the twos, twenty-two then two and three, twenty-three, etc – I won a cash prize for selling the most ice cream anywhere in the Top Rank organisation in August 72. So, here is our first magic square:

In the meantime, I had been looking at the initial grid and found that there were four possibilities that could result in real words throughout the final grid: **twenty-one**, **twenty-two**, **thirty-one** or **thirty-two**. Finishing the puzzle followed swiftly: **letter count** (two = 3, five = 4, etc)** gives a second magic square**:

To adjust the initial grid, we had to look at the totals:

**1st magic square = 45**

**2nd magic square = 21**

**Title = 24**

**24 + 21 = 45**, which I found to be *totally satisfying*.

So, **twenty-one** was entered to give the final grid:

This puzzle prompted me to have a closer look at 3×3 magic squares. I wondered for example how many such squares might be formed from the numbers 1 to 30, ignoring rotations and reflections and with no number repeated. A friend was kind enough to write a program that determined that there were no less than 346 squares that fitted this bill. I would therefore conclude that they are not particularly magical at all. Any straight sequence of nine numbers can be arranged into a magic square as can an alternate sequence, every third number and indeed every nth. The middle number of the sequence always appears in the centre of the square and the totals are three times this number.

Our second magic square was a straight sequence (3 to 11) with 7 in the middle and the total 21. The first magic square was a little more interesting but still had a regular distribution about the middle number, 15:

**2** 3 4 **5** 6 7** 8** 9 10 11 **12** 13 14 **15** 16 17 **18** 19 20 21 **22** 23 24 **25** 26 27 **28**

A search on the Net found that this first square is known as an

Alphamagic Square. Note also that the general formula for a 3×3 magic square was first devised by Edouard Lucas who featured in

*City Tour* by Mango (Listener No.3977) in August last year.

I do not know if the theme was found on the Net or elsewhere but this was a first-class adaptation into the crossword format and tremendous fun – thank you Leo. I had thought the title clumsy but it can’t be a coincidence that it requires precisely **24** letters to spell **five hundred and seventy-six** – nice one!