## Listener 4256: *Boxes* by Radix (or *The Wrong Box!*)

Posted by Dave Hennings on 13 September 2013

I thought I’d got into the swing of the mathematical appearing on the penultimate Saturday of the three-monthly cycle, but yet again it was a surprise to see that my numbery brain needed to be brought to the fore. At least, it looked like a mathematical puzzle, except it was by Radix, and he only does standard crosswords, doesn’t he?

Well, it would appear not, and a quick look at the preamble indicated that this would be a mathematical that would be as difficult in its own way as Radix’s normal œuvres. For a start, we weren’t dealing with pure factorisations, but the sum of numbers, albeit in this case squares. Moreover, of the 72 entry cells, 26 were unchecked.

I guessed that the starting point would be the boxes with a side of single-digit length, albeit in base 24. That was the end of my guessing, although in hindsight I could have made a few deductions about the contents of some squares, such as box A side *a* which must begin AA or AB since the diagonal *d* was ABTE.

I then googled for help on determining a formula for the sum of squares but, as I was to find out, a number could be the sum of none or many pairs of squares. I also wondered if the fact that all four lengths *a*, *b*, *c* and *d* had no common factor should be the way forward, but that didn’t help me.

Finally, I turned to my trusty friend WolframAlpha. This helped me with the triangular numbers in Elap’s *Three-Square* back in November 2010…was that really three years ago? And so it was that it helped me here. Keying in a number told me everything I wanted to know about it, from the trivial (odd or even) through to whether it *couldn’t* be expressed as the sum of three squares and whether it *could* be expressed as the sum of two.

I’m afraid that I cannot detail the exact route that I took to crack this puzzle…except that it was a near vertical climb. I think the first clue that I solved was for box M, followed by K, P, I, L and J. And, of course, I mean ‘the first clue that WolframAlpha solved’. I dreaded coming to a dead end and having to go back to the beginning, and progress was very slow over the course of two days…two long days!

WolframAlpha was also a great help in converting a number in base 10 to its equivalent in base 24. For example “1024 in base 24” revealed “1:18:16_{24} (3 digits).

As I neared the end, two things were on my mind. The first was the message being spelt out by the last letters of the *c* lengths. It looked as though they were trying to say **Rotate eight**, so infinity was uppermost in my mind although I couldn’t really see how that was relevant. I was also becoming very worried about “any apparent ambiguity” that we were warned about in the preamble.

I eventually found it in the bottom right-hand corner with box T where side *b* could be BJ or QJ with the corresponding side *c* being QO or CO. There was still some work to be done on this, courtesy of the last letters of sides *c*, which had nothing to do with infinity, but spelt out **Total height in base two four**. No doubt adding up all the *c*s would give us the thematic word to be entered in the space beneath the grid.

Two quick sums using the two options for box T, and I had **CZWE** and, more likely, **CAME**. Hmmm…even CAME didn’t seem quite right. OK, I’d *arrived* at the end, but that was hardly thematic. I checked Chambers for an obscure meaning of the word and was rewarded with “a lead rod for framing a pane in a leaded or stained-glass window”, which was certainly obscure but even less thematic.

Perhaps Google would come to the rescue again. And it did: CAME – Computer Algebra in Mathematics Education, “an open, international organisation for those interested in the use of computer algebra software in mathematics education”. Perhaps Radix was telling me that he had expected me to write a computer program to help with the solution. I knew that was unlikely.

So, just as I dreaded, it was back to square one (almost) as I checked the notes I had made from my solving process. Half a day later, and I was nearing the bottom left corner and I found where I had overlooked the *two* options for box U. As well as MC and KF for sides *b* and *c*, there was AC and QF. By this time I had prepared a spreadsheet with the four dimensions of each box in base 24 and base 10, so it just needed me to change a couple of figures, and out popped 42509, which WolframAlpha told me was 3:1:19:5_{24} or **CARE**. Well, it was certainly something I had to take, but again it was hardly thematic. Of course, 90 minutes later I realised how much this puzzle was frying my brain as **CASE** was the correct translation.

I’m not really sure what to write next. As a feat of construction it was a masterpiece, but to say that it was a struggle would be an understatement. Perhaps I should just say that, as I wrote this blog, I realised that, with all the bits of paper and grids floating around, I copied the wrong version to my submission sheet (and the grid above) and all that work was in vain!!!

## Dave Russell said

Probably just a typo but I have the answer to U as J,AC,QF as does Shirley in her blog.

## gerrym said

Dave – check the last para of the blog!

## Dave Russell said

Sorry, should have read your blog first!

## Dave Russell said

… or rather, Dave’s blog – oh dear…