May 02, 2014

Horse 1666 - DIVE

This is Horse 1666. Great fires of London!
Sir W. Pen and I to Tower- streete, and there met the fire burning three or four doors beyond Mr. Howell’s, whose goods, poor man, his trayes, and dishes, shovells, &c., were flung all along Tower-street in the kennels, and people working therewith from one end to the other; the fire coming on in that narrow streete, on both sides, with infinite fury. Sir W. Batten not knowing how to remove his wine, did dig a pit in the garden, and laid it in there; and I took the opportunity of laying all the papers of my office that I could not otherwise dispose of. And in the evening Sir W. Pen and I did dig another, and put our wine in it; and I my Parmazan cheese, as well as my wine and some other things. 
- Diary of Samuel Pepys, 4th Sep 1666

Samuel Pepys who would become an MP, once worked under the then Exchequer, George Downing; for whom Downing Street is named. Pepys revolutionised the Royal Navy, is mentioned in Newton's "Philosophiæ Naturalis Principia Mathematica" but is most famous for the diary he kept for nine years... and burying cheese...
Pepys was somewhat obsessive and it is that vein, that I continue Horse 1666.

A game with which I was quite obsessed with for a while was 2048.
2048 is a game in which you mash together two identical tiles to form an new one. These tiles are in ascending powers of 2 until you get to the 2¹¹ tile which is 2048; hence the name of the game.
There is a variant of this on an 8x8 grid in which the same rules apply except that you need to get to the 2¹⁴ tile which is 16384.
I eventually worked out that if you play this game for long enough, the tiles through a process of spontaneous arrangement will eventually arrive at something approaching a normal distribution with some 2ˣ at the very centre. Once when I was listening to a Liverpool match over the internet and played for just over 2 hours, I got to the 2²° tile which was 1048576.

I've subsequently found this brilliant list of 2048 variants:
But the best that I've found there is DIVE:

DIVE feeds all sorts of nerdiness quotients and requirements because rather than 2048's condition of mashing together identical tiles, provided that one is a factor of another, the two tiles will mash. That is to say that 11 will mash with 55 to produce 66 but 22 will not mash with 55. Both 11 and 22 will mash with 66 as both are factors and the resulting tiles are 77 and 88 respectively.

Perhaps most satisfyingly of all, DIVE is more about management of factors that 2048 could ever hope to be. Because new tiles which are generated are all the "prime" factors of the tiles already on the board, careful management can result in the elimination of unwanted primes from the available list. Curiously, if a some power of a prime like a square or a cube etc. is a factor and no other multiple of said prime currently exists on the board, then that is used instead. Numbers like 25, 8 and 169 will on rare occasions crop up.

It is the accidental appearance of unwanted primes where the frustration in this game truly lives. Numbers like 13, 17, 19 & 37 which appear, will not form nice multiples whereas 4, 8, 6, 9, 10, 15, 25, 30, 60 etc. are things which are readily useful.

Typically I'll get a score between about 700-1000 but I'm very much aware that the way to push forward to scores of 2000+ is to manipulate the tiles to start generating massive primes. To date, the biggest prime that the game has ever generated for me is 727 but the biggest prime which I've ever matched with another is 349.

Like 2048, I suspect that had the game had a far larger board to play on (say 8x8) that it too would nominally fall into a normal distribution pattern. Also like 2048, I suspect that the highest possible score is something approaching 23³ or 12167 simply because I don't think that it would be technically possible to go much beyond that without generating 2s. As a result of the way that the game operates, the fewest number of tiles that you can have on the board are the initial pair of 2s; beyond that there are at least three tiles always. 2s though are a very small prime (the smallest) and 2 is a factor of every even number; so that must have some limiting process on the game.

DIVE like 2048 is a game with a limiting process as time goes on. Clearly insanity has no limits, even as 2ˣ approaches ∞.


Anonymous said...

I might have a problem.

Unknown said...

Please join our ranks. The world record holder, biggiemac42, with a score of 232533, and I, Catzee (HS 68815) would love to have another expert around.