Percolation is Odd Stephan Mertens, Otto-von-Guericke University Cristopher Moore, Santa Fe Institute
<latexit sha1_base64="ra7591rNlC9TFfYc3oiajUfMOE=">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</latexit> The Total Number of Spanning Configurations is Always Odd Height 1 2 3 4 5 6 7 1 1 1 1 1 1 1 1 2 3 7 17 41 99 239 577 Width 3 7 37 197 1041 5503 29089 153769 4 15 175 1985 22193 247759 2764991 30856705 5 31 781 18621 433809 10056959 232824241 5388274121 6 63 3367 167337 8057905 384479935 18287614751 868972410929 7 127 14197 1461797 144769425 14142942975 1374273318721 133267613878665
: # configurations with k occupied sites A n , m ( k ) 10 19 10 15 10 11 10 7 1000.0 k 10 20 30 40 50 60
nm ∑ z k A n , m ( k ) R n , m ( z ) = k =0
nm nm ∑ ∑ z k A n , m ( k ) p k (1 − p ) nm − k A n , m ( k ) R n , m ( z ) = P cross ( p ) = k =0 k =0
nm nm ∑ ∑ z k A n , m ( k ) p k (1 − p ) nm − k A n , m ( k ) R n , m ( z ) = P cross ( p ) = k =0 k =0 1.0 0.8 0.6 0.4 0.2 5 × 5, 11 × 11, 22 × 22 0.2 0.4 0.6 0.8 1.0
nm ∑ z k A n , m ( k ) R n , m ( z ) = k =0
nm R n , m ( − 1) = ∑ A n , m ( k ) − ∑ ∑ z k A n , m ( k ) R n , m ( z ) = A n , m ( k ) k =0 k even k odd
nm R n , m ( − 1) = ∑ A n , m ( k ) − ∑ ∑ z k A n , m ( k ) R n , m ( z ) = A n , m ( k ) k =0 k even k odd n 1 2 3 4 5 6 7 8 m � 1 � 1 � 1 � 1 � 1 � 1 � 1 � 1 1 1 � 1 1 � 1 1 � 1 1 � 1 2 � 1 1 � 1 1 � 1 1 � 1 1 3 1 1 1 1 1 1 1 1 4 � 1 � 1 � 1 � 1 � 1 � 1 � 1 � 1 5 1 � 1 1 � 1 1 � 1 1 � 1 6 � 1 1 � 1 1 � 1 1 � 1 1 7 1 1 1 1 1 1 1 1 8
k odd k even partial matching
k odd k even partial matching
The Odd One Out
The Odd One Out 2 ⌋ n + ⌈ ⌊ 2 ⌉ m m
R n , m ( − 1) = ∑ A n , m ( k ) − ∑ A n , m ( k ) = ( − 1) ⌊ 2 ⌋ n + ⌈ 2 ⌉ m m k even k odd n 1 2 3 4 5 6 7 8 m � 1 � 1 � 1 � 1 � 1 � 1 � 1 � 1 1 1 � 1 1 � 1 1 � 1 1 � 1 2 � 1 1 � 1 1 � 1 1 � 1 1 3 1 1 1 1 1 1 1 1 4 � 1 � 1 � 1 � 1 � 1 � 1 � 1 � 1 5 1 � 1 1 � 1 1 � 1 1 � 1 6 � 1 1 � 1 1 � 1 1 � 1 1 7 1 1 1 1 1 1 1 1 8
Other Matching Proofs A square integer has an odd number of divisors 2 n The number of binary trees with leaves is odd p = x 2 + y 2 A prime has an odd number of representations p = 4 n + 1
Shameless plug To put it bluntly: this book rocks! It somehow manages to combine the fun of a popular book with the intellectual heft of a textbook. Scott Aaronson, UT Austin This is, simply put, the best-written book on the theory of computation I have ever read; one of the best-written mathematical books I have ever read, period. www.nature-of-computation.org Cosma Shalizi, Carnegie Mellon
Shameless plug N E W ! I M P R O V E D ! To put it bluntly: this book rocks! It somehow manages to combine the fun of a popular book with the intellectual heft of a textbook. Scott Aaronson, UT Austin This is, simply put, the best-written book on the theory of computation I have ever read; one of the best-written mathematical books I have ever read, period. www.nature-of-computation.org Cosma Shalizi, Carnegie Mellon
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