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Listener No. 4386, Hailstorm: A Setter’s Blog by Elap

Posted by Listen With Others on 13 March 2016

The Theme

I have long been intrigued by the Collatz conjecture. Looking at Wikipedia, it has all sorts of names: Ulam conjecture, Kakutani’s problem, the Thwaites conjecture, Hasse’s algorithm and the Syracuse problem, none of which I had heard of. The Wikipedia entry continues, “The sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers, or as wondrous numbers.”

The conjecture states that if you take any positive whole number greater than one, and if it is even, halve it, or if it is odd, multiply by three and add one, and keep repeating the process on the resulting number, you will always end up with 1.

They are called hailstone numbers because they rise and fall rather like the movement of hailstones after they have been formed in a cloud.

It was while doing something totally unrelated to puzzles (designing a humane badger trap) that I suddenly wondered whether it was possible to fit a sequence of hailstone numbers into a symmetrical grid, or if not then a sequence with just one number missing (“Solvers are to write the missing number underneath the grid”).

The Grid

Once more resorting to my trusty 20-year-old Pascal compiler, I found that it was not too difficult to fit an all-but-one sequence into a 7 x 8 grid, and so I set myself the challenge of filling an 8 x 8 one.

I didn’t want any single-digit numbers or many with four digits, and after a little investigation I found that the number of lights in an 8 x 8 grid was far more likely to be 38 than any other number, given my criteria for the nature of the bars in the grid. I looked at various sequences, and the one which had 988 near the beginning had plenty of 2-digit numbers and no more than two 4-digit numbers. This sequence was the first one I tried when trying to create the grid.

I ran my antique Grid Generator specifying 38 lights and other strict criteria and provided a hailstone sequence of 41 numbers, all but three of which were to be fitted into each generated grid, filled grids being written to a text file for subsequent analysis.

Listener No 4386 Grid Analysis

The idea was that I would scan the text file (using another program) for filled grids for any where two of the three missed-out numbers were at the ends of the sequence, thus implying that there was just one number missing in the middle somewhere. If there was a filled grid which had no missing numbers (because all three omissions were at the ends of the sequence) then so much the better – I’d have a grid which contained a whole sequence. A complete sequence in the grid would be the ideal, but it was unlikely.

Many of the filled grids were based on the same base grid but with permutations of some of the entries. For example, 71 and 91 would be interchangeable if the first digits were unchecked. For my own satisfaction I wanted to find a grid that could be filled in only one way, but again this was unlikely.

Eureka! The 445,153rd grid contained a whole sequence and was fillable in only one way! I decided to let the run finish in case there was a more pleasing grid, and this took a few more days and quite a few kWh of electricity. This, and my wife’s wish to install an electric fence to keep out the badger, would clearly trigger an increase in my direct debit to Good Energy.

Out of over 25 million grids which fitted my criteria, 4,724 different ones were filled in 36,247 different ways. Out of these filled grids, 459 contained a hailstone sequence with one number missing, but 5 of the filled grids contained complete sequences. Two pairs of these were the same grid with two entries interchanged, but one grid contained a hailstone sequence which could be fitted in only one way. This was my chosen grid and was way beyond my expectations. I was so pleased that I decided to let the badger carry on as before, and anyway that electric fence wouldn’t be very nice for our neighbour’s numerous cats.

The Clues

What form would the clues take?

Since the theme was hailstone numbers, which many solvers could be unaware of, I decided to explicitly explain what they were in the preamble. The clues would then need to feature the hailstone sequence concept – but how? I thought of secretly making the letter values form a hailstone sequence, but that would have made the puzzle too easy for solvers who twigged, since given one hailstone number the rest of the letter values could be determined.

After a rest and some further thinking, during which time I thought further about the badger trap, I decided to use a hailstone relationship between upper- and lower-case letters. Since the preamble would remove the surprise of a hailstone sequence in the grid, I needed to have a further think. I created over six dozen variations of the same hint which said that the grid need to be filled with a hailstone sequence, and wrote a wee program which would show which, if any, of these could be used as a message in the completed grid, where A = 1, B = 2, etc. The number of digits in the hint had to be exactly 64, none of the lights could begin with a zero, and they all had to be different. Just one of my hints was successful, reading ‘PRODUCE 38 HAILSTONE NUMBERS FROM 988 AND FILL GRID’. It would have been nice to have the hint also request the erasure of the cells, but 64 digits was just not enough.

Having come up with this hint, I wondered whether it was acceptable – does ‘from’ include the 988? I looked up ‘from’ in Chambers, and one of the meanings is ‘beginning at’ and so the hint was equivalent to ‘PRODUCE 38 HAILSTONE NUMBERS BEGNNING AT 988 AND FILL GRID’ and I was happy that this implied that the sequence started at 988.

It is sometimes possible to fit the numbers into an empty grid without using any trial and error, but it is not possible to use the method in this case. I therefore decided to place the unchanged digits in shaded squares (they ended up as circles in the final puzzle).

The less-than-usual cross-checking of the digits could make cluing difficult, and so I arranged the clues in ascending order of their value.

I called the original puzzle ‘Hailstone’ but one of the vetters suggested that ‘Hailstorm’ would be better, and I agree!

Update

It is with much mournfulness that I must mention that our mighty mass of moving muscle met his maker after being mown down one misty morning by a moving motor vehicle.

However, my wife, who had recently read that badgers rarely live alone, has just found three more holes in the lawn. Such is life…
 

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