[seqfan] Re: The Savannah Math

David Seal david.j.seal at gwynmop.com
Fri Dec 27 11:24:34 CET 2019


> On 26 December 2019 at 16:49 Ali Sada via SeqFan <seqfan at list.seqfan.eu> wrote:
> ...
> I excluded the empty sets because including them wouldn't give me a unique
> sequence. I didn't know how to word the definition in a way that makes this
> clear.

Sorry, I don't see why you think including the empty sets wouldn't give you a unique sequence.

And it seems to me that the empty sets are inherently there in what you're doing - after week 1's animal arrival, you have three possibilities, namely F or H or Z, and then after week 2's update, F becomes H, H becomes the empty set and Z remains Z. Three possibilities existed before the update, and they resulted in three different animal populations after the update, so there are still three possibilities.

> As for the hungry lions, H+Z=F. A hungry lion eats one Zebra to become a fed lion.

I take it that you mean that HZ becomes F during a week's update - the third option of the three I listed. For the sake of using clear notation, I would express that as HZ --> F or H+Z --> F rather than H+Z=F - it's a change to the state and it doesn't work in both directions (a fed lion doesn't do the reverse transition into a hungry lion and a zebra again!), rather than an equality which does.

What does F+Z become during the update? You seem to have answered that when you wrote:

> > 3.    In the third week, the combinations of the second week change:
> > FH becomes H; FZ becomes F; 2H disappears; F becomes H; and 2Z remains 2Z.

in which the "FZ becomes F" looks fairly clear: F+Z --> F. But that creates a problem about what F+H+Z becomes during an update: if the fed lion eats the zebra, the hungry lion dies:

  F+H+Z = (F+Z)+H --> F+- = F

but if the hungry lion eats the zebra, the fed lion merely becomes hungry:

  F+H+Z = (H+Z)+F --> F+H

I.e. if you really do mean that "FZ becomes F" and that HZ becomes F, you've got an ambiguity about what happens to FHZ. You can resolve that ambiguity by saying whether the fed lions or the hungry lions get first pick of the zebras - or indeed there's a third option: lions hunt and feed co-operatively and there's enough meat on a zebra to feed two lions, so F+H+Z --> 2F. (That third option is probably the most 'realistic' one, but I do realise that that doesn't matter to you, so I'm just noting it as an option, not arguing in its favour.)

The other possibility that strikes me is that "FZ becomes F" might have been a mistake: an alternative is that the fed lion doesn't bother hunting and merely sleeps off its meal, only becoming a hungry lion while the week goes by: F+Z --> H+Z.

Basically, what is needed is not a few specific cases of how a week passing affects the animal population, but a general rule: what does a population f fed lions + h hungry lions + z zebras, or (f,h,z) for short, become after a week? Some options are (hopefully with all the formulae correct by now...):

* Each lion eats a zebra if it can, fed lions get first pick of the zebras if there aren't enough for all the lions, which gives the following update rule:

  if f>=z:           (f,h,z) --> (z,f-z,0)
  if f<z and f+h>=z: (f,h,z) --> (z,0,0)
  if f<z and f+h<z:  (f,h,z) --> (f+h,0,z-f-h)

* Each lion eats a zebra if it can, hungry lions get first pick of the zebras if there aren't enough for all the lions, which gives the following update rule:

  if h>=z:           (f,h,z) --> (z,f,0)
  if h<z and f+h>=z: (f,h,z) --> (z,f+h-z,0)
  if h<z and f+h<z:  (f,h,z) --> (f+h,0,z-f-h)

* The lions hunt as a pack and eat a zebra if there is one, and there is enough meat on a zebra to feed them all, which gives the following update rule:

  if f+h=0:          (0,0,z) --> (0,0,z)
  if f+h>0 and z=0:  (f,h,0) --> (0,f,0)
  if f+h>0 and z>0:  (f,h,z) --> (f+h,0,z-1)

* Each hungry lion eats a zebra if it can, fed lions don't bother this week, which gives the following update rule:

  if h>=z:           (f,h,z) --> (z,f,0)
  if h<z:            (f,h,z) --> (h,f,z-h)

All of those rules say that (0,1,1) --> (1,0,0), i.e. H+Z --> F, and the first three say that (1,0,1) --> (1,0,0), i.e. F+Z --> F. So based on what you've said, I can't tell which of those rules it is (and there are doubtless other possibilities besides those - they're merely the ones that have occurred to me...) - but they do produce different results for (1,1,1).

The fourth one of those rules is different - it says that (1,0,1) --> (0,1,1), i.e. F+Z --> H+Z, so that's only compatible with what you've said if "FZ becomes F" was a mistake. I only mention it because it is a way of resolving the question of whish lion gets the zebra if there are both fed and hungry lions present.

To sum this up, I can't tell from what you've written what happens to the general population (i.e. f fed lions, h hungry lions and z zebras, with each of f, h and z a non-negative integer but no other restrictions) over a week, before the next animal arrives. Without that knowledge, I can't tell what happens to your sequence - it simply isn't well enough defined - and simply knowing what happens to some specific populations like H+Z or F+Z isn't enough to tell me what happens to the general population. 

> For example, let's take the combination F3Z. We have a fed lion with three
> zebras. In the next week, the fed lion becomes a hungry lion, eats one of
> the zebras, and becomes a fed lion. The resulting combination is F2Z. The
> next week it becomes FZ, then it becomes F, then H, then it dies.

So the update for a week is:

F: if there is a Z, F becomes H and then eats a Z to become F again; otherwise, F just becomes H.

H: if there is a Z, H eats a Z to become F; otherwise, H dies.

Z: number of Zs is reduced by the number that are eaten by Fs and Hs.

And in terms of the (f,h,z) positions, that example is (1,0,3) --> (1,0,2) --> (1,0,1) --> (1,0,0) --> (0,1,0) --> (0,0,0). That's still compatible with any of the first three rules above, so I'm afraid it doesn't help me to decide which of them you mean.

The simplest example that distinguishes between them is FHZ, or (1,1,1) in (f,h,z) notation:

* Under the first rule, the F becomes H and eats the Z to become F again, while the H dies, and then the F becomes H and dies over the following two weeks: (1,1,1) --> (1,0,0) --> (0,1,0) --> (0,0,0). Basically, the F update above happens *before* the H update, with different zebras being eaten during each of them.

* Under the second rule, the H eats the Z to become F, while the F becomes H, and then the F takes two weeks to become hungry while the H dies in the first of those weeks: (1,1,1) --> (1,1,0) --> (0,1,0) --> (0,0,0). Basically, the F update above happens *after* the H update, with different zebras being eaten during each of them.

* Under the third rule, the F and the H eat the Z between them to become 2F, and the n the 2F become 2H and die over the following two weeks: (1,1,1) --> (2,0,0) --> (0,2,0) --> (0,0,0). Basically, the F update above happens *simultaneously* with the H update, with the same zebra being eaten during both of them.

So it would be helpful if you could say what happens to the FHZ example - it would hopefully resolve which of those rules it is (or possibly indicate that it's some other rule that I haven't thought of...).

David



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