An empirical observation about this kids' problem going around, which math folk can try to beat me to explaining: https://www.nytimes.com/2017/05/15/us/math-counts-national-competition.html …
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I kept thinking about this involution, but it's not obvious to me why it works.
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For each chick, 1 of nhbrs switches, so single-pecked chicks and non-or-double-pecked chicks swapped. Same # of non-pecked, double-pecked.
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Ah, beautiful. All props to you.
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Other observations I made, for what it's worth: With even chickens, swapping every alternate chickens' directions keeps peck # unchanged.
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And the total number of unpecked is equal to the total number of contiguous blocks of chickens facing same direction, of size > 1 and < all.
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So close to seeing it, and yet fell just short. Oh well. Math.
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With n total chickens, the probability of x unpecked chickens is ((n choose 2x) + g(n, x) * (n/2 choose x)) * 2^(1 - n), where g(n, x) is…
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0 if n is odd -1 if n is even and x is odd 1 if n is even and x is even (x assumed integer; whole probability should be taken 0 otherwise.)
End of conversation
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