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   Author  Topic: Infinite Checkerboard  (Read 8403 times)
Eric Yeh
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Re: Infinite Checkerboard  
« Reply #25 on: Aug 8th, 2003, 10:09am »
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o ok, i guess i got them mixed up; its been years.  strange tho, i thought i remembered thinking the "normal" checkers version was cooler bc of the limit, but guess i may have mixed up my mnemonic...
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Re: Infinite Checkerboard  
« Reply #26 on: Aug 8th, 2003, 12:32pm »
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Actually, the limit it converges to is 5, but you can never reach it (even a perfect sequence of moves would be infinite). So it's kind of pretty, but ultimately useless Sad
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Re: Infinite Checkerboard  
« Reply #27 on: Aug 8th, 2003, 12:36pm »
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well, not quite "useless" -- you need to see that every finite set is under 5 or else the answer would (potentially) change.  so in my opinion it is pretty AND useful (and hence truly pretty by my defn of it  Wink ).
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Re: Infinite Checkerboard  
« Reply #28 on: Aug 8th, 2003, 1:33pm »
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I guess I was misleading in what I said. What I should have said was: the series converges to 1 if you pick a position on the fifth row empty row as the starting point.
 
But can anybody come up with a way of generating the infinite sequence of moves that would allow you to move to the fifth row? I suspect you'd end up with a finite number of moves, then a well-defined infinite sequence of moves, and then another finite sequence of moves. That would be quite a feat!
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Eric Yeh
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Re: Infinite Checkerboard  
« Reply #29 on: Aug 8th, 2003, 1:53pm »
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yeah i thought about that at the time -- its actually impossible bc, since you need EVERY piece to get to the fifth row (thats where the beauty of the exact convergence comes in), and furthermore every piece has to go forward only, youre going to have infinitely many pieces that are too far away horizontally to "help the effort" "in time".  anyway, that is a tiny bit hand wavy but very easy to formalize, so im sure you geniuses can handle it  Wink
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Re: Infinite Checkerboard  
« Reply #30 on: Aug 11th, 2003, 10:55am »
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Okay, point taken. You couldn't actually solve it. But there are some constructions that can be made that are infinite in extent, but doable in practice (given infinite time or when taking the limit). For instance:
 
000000 ... 00000000
001111 ... 11111110
000000 ... 00000100
000000 ... 00000100
000000 ... 00000000

 
can be transformed to:
 
000000 ... 00000000
100000 ... 00000000
000000 ... 00000000
000000 ... 00000000
000000 ... 00000000

 
Now this is not particularly helpful for the task, since we can't push that interesting bit at the end off to infinity. But perhaps there is something else we can do.
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Re: Infinite Checkerboard  
« Reply #31 on: Aug 12th, 2003, 1:03pm »
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james,
 
ye sorry i was actually thinking about the diagonal case again.  for that it is easy to show that for any finite set S of checkers with value = sum(board), S cannot be achieved.
 
ok, now on to your intended question, for the horizontal case, which i think is interesting.  first, one has to come up w a good interpretation, which i think amounts to a set of allowable transfinite transformations.
 
first interpretation:  allow any infinite string of 1's:
 
0 0 1 1 1... -> 1 0 0 0...
 
this is per your msg, and i think is the most "basic" one that needs to be allowed, and hence must be included in our set of allowable transformations.
 
i played with this forumalation for a bit and am pretty convinced that it is not solveable, but i also couldnt come up w a proof.  think i might if given a bit more time but dont quite have it.  instead i cheese out and add another allowable transformation on an infinite diagonal sequence of 1's:
 
0 0 0 0...      0 1 0 0                      0 0 0 0
0 1 0 0         0 0 0 0                      1 0 0 0
0 0 1 0     -> 0 0 0 0  or equivlently  0 0 0 0
0 0 0 1         0 0 0 0                      0 0 0 0
...       ...
 
and the puzzle is then easily solveable.
 
comments?
« Last Edit: Aug 12th, 2003, 1:42pm by Eric Yeh » IP Logged

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Re: Infinite Checkerboard  
« Reply #32 on: Aug 12th, 2003, 1:04pm »
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agh whats up w yabb squeezing spaces.  o well hopefully you see what i mean there.
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Re: Infinite Checkerboard  
« Reply #33 on: Aug 12th, 2003, 1:39pm »
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Eric,
 
The regular posting script squeezes out spaces, but the modification posting script doesn't. So if you want to put the spaces back, try modifying your post. The preview should show the spaces back in, and once you save, they should stay.
 
I'll see if I can figure the problem out with the diagonal and orthogonal transfinite moves. I was actually thinking that the orthogonal one I suggest is not very fundamental. A better fundamental move would be this one:
 
1 0 0 0 0 0 0 => 0 1 0 1 0 1 0 1
 
But this can't replace the other move I suggested because it's not very useful. Although I like the fact that my original suggestion is just this superposition:
 
   0 1 0 0 0 0 0 => 0 0 1 0 1 0 1
+ 0 0 1 0 0 0 0 => 0 0 0 1 0 1 0
= 0 1 1 0 0 0 0 => 0 0 1 1 1 1 1
 
But of course this isn't valid in checkers ...
« Last Edit: Aug 12th, 2003, 1:45pm by James Fingas » IP Logged

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Re: Infinite Checkerboard  
« Reply #34 on: Aug 12th, 2003, 1:43pm »
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cool thanks, now its fixed.  wow you are really expert here now  Wink
 
i am sure you will get it very easily -- its almost trivial in that formulation  Tongue  i am disappointed in myself for my cheat, haha!
« Last Edit: Aug 12th, 2003, 1:47pm by Eric Yeh » IP Logged

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Re: Infinite Checkerboard  
« Reply #35 on: Aug 12th, 2003, 1:46pm »
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ps  FWIW, try using the diagonal transformation only finitely many times.  maybe even minimize it (i think my soln is optimal).
 
pps  if you define the "infinite case" to mean that for arbitrarily small epsilon you can systematically place less than epsilon worth of extra checkers on the board and still reach your goal, then you cover both of these transfinite transformations (and a hell of a lot more im sure).  FWIW  Tongue
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Re: Infinite Checkerboard  
« Reply #36 on: Jan 26th, 2008, 2:21am »
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on Aug 8th, 2003, 1:33pm, James Fingas wrote:
But can anybody come up with a way of generating the infinite sequence of moves that would allow you to move to the fifth row? I suspect you'd end up with a finite number of moves, then a well-defined infinite sequence of moves, and then another finite sequence of moves. That would be quite a feat!

 
Sorry for coming late, but this is exactly what you were looking for. I hope you'll like it.
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