SCENE: *The Editor’s office, December 2015.*

There is a desk in the centre of the room. There is an in-tray on the desk containing a letter. There is a clock on the wall. It reads 11:00.

There are two chairs: a big one is behind the desk and a small one in front of it. **Editor** is sitting on the big chair and **Sub-editor** on the small one. Next to the small chair is a stool. **Elap** is sitting on it; he is hugging his briefcase.

**Elap**: Nice in-tray.

**Editor**: Thanks.

**Editor** takes the letter. As he does so, the in-tray slowly disappears. **Elap** either doesn’t notice or is used to that sort of thing happening.

**Editor**: This is the letter I mentioned on the phone. (*Reading from letter*) “The amount of Listener real estate that is wasted by the mathematicals having small grids and a puny set of clues is unacceptable. Do something about it… or else!” It arrived a week ago after the last mathematical.

**Sub-editor** (*to ***Editor**): That was yours, wasn’t it?

**Editor** (*ignoring ***Sub-editor**): Out of all the mathematical setters, you were the only one who said they had something in the pipeline that might help.

**Elap** (*removing a thick sheaf of about 50 A4 papers from his briefcase*): Yes, you’re lucky. I’ve just finished my latest effort. Given the title of the puzzle, I managed to con the Meteorological Office into giving me use of their old supercomputer for six months this year. It produced this.

**Editor**: We need to fill eighteen column inches, not three newspapers.

**Elap**: Oh, this is just the computer output. Here’s my preamble. (*He takes another wadge of about 40 sheets from his briefcase and hands it to ***Editor**.) I’m sure you can whittle it down a bit.

**Editor** (*reading page 1*): “The Collatz conjecture is a problem posed by the German mathematician, Lothar Collatz, in 1937. It is also called the 3x+1 mapping, 3n+1 problem, Hasse’s algorithm (after Helmut Hasse), Kakutani’s problem (after Shizuo Kakutani), Syracuse algorithm, Syracuse problem, Thwaites conjecture (after Sir Bryan Thwaites), and Ulam’s problem (after Stanisław Ulam).

“Collatz was born on 6th July 1910 in Arnsberg, Westphalia. In 1937 he posed the famous conjecture, which remains unsolved. The conjecture can be summarized as follows. Take any positive integer n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called “Half Or Triple Plus One”, or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness. The sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.

“For example, the sequence for the number 27 is as follows: 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, …”

*DISSOLVE to clock which now reads 12:25.*

**Editor** (*still reading*): “…the second part explains how solvers must apply the results of the first part, after erasing the contents of all but the circled cells. In each version of the grid, the 38 entries are different and none of them starts with a zero.”

*The last page contains a grid and some clues. ***Editor** puts it down on the desk. His eyes have glazed over.

**Editor**: Well, it’s not a particularly large grid…

**Sub-editor** (*glancing at last page*): …but bigger than yours was…

**Editor** (*ignoring ***Sub-editor**): …and there could be more clues…

**Sub-editor**: …more than yours had…

**Editor** (*glaring at ***Sub-editor**): …but with a somewhat cut-down preamble it should do the trick. Thanks, Elap.

###### Any similarity to actual events is entirely unlikely.

An Elap mathematical again this time. Last year’s (no. 4347 *Pairs*) used squares or numbers which were concatenated squares. With Elap, I always think back to his *Three-square* puzzle in 2010 which needed us to realise that all rows and columns consisted of triangular numbers. I nearly failed on that one, and I knew that an Elap endgame could be the cause of potential grief.

This week’s puzzle was based on the Collatz conjecture where *x*/2 or 3*x*+1 repeatedly would lead us to 1. 1dn looked as good a start as any, NNNN being 2 digits had to be 16, not 81 which would have 8ac as 1• but had to be greater than 1dn. So N was 2, which meant that 8ac was 2P + P² and only P=7 gave a number 6•.

From there, progress was fairly quick, with 2dn, 4dn, 10ac, 26dn, 10ac and 28dn leading the way. There were a couple of long pauses as I progressed, and care had to be taken to ensure that all the options for the hailstone numbers were accounted for. I made one mistake with H=28, where I initially overlooked h=9 as a possible option; luckily h was 56.

The grid was completed in about two hours, and all we had to do was “decode it (in a thematic direction)”. For about twenty minutes I tried to use the values of all the numbers in the clues — N=2, R=3, m=4, t=6, etc. I realised that the thematic direction would be vertical, either all down, or down, up, down.

Luckily, it didn’t take too long to realise that it was simple alphabeticala positions that led to **Produce 38 hailstone numbers from 988 and fill grid**. I listed out the required numbers, a bit puzzled by there being 50, including 988 and 1.

1114 was the first one in the grid, using two of the numbers left from satge 1. 1672 therefore occupied the second 4-digit space, and from there the grid-fill was easy… until near the end. I had options for some of the 2-digit entries — 5dn could be 10, 13, 16 or 19. Most of the options for these digits would fill gaps in the main body of hailstone numbers, and I realised that “from 988” meant just that.

In reading about hailstone sequences, I was particularly struck by 27, a seemingly innocuous number, but requiring 111 steps to reach 1 and climbing to 9232 on its way.

Thanks to Elap for another excellent puzzle, and thanks to Collatz (or Hasse or Kakutani or Thwaites or Ulam) for his fascinating mathematical conjecture. I await its proof with interest.