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Words count (Posted on 2014-03-25) Difficulty: 4 of 5
(D3)How many subsets of (a,b,c, x,y,z) contain no letters neighboring each other in the English ABC (26 letters)?

(D4)How many "words" (i.e, ordered subsets from (a,b,c, x,y,z)) contain no neighboring letters that neighbor each other in the English ABC (26 letters)?

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts re: solution to part 1; thoughts on part 2 | Comment 7 of 8 |
(In reply to solution to part 1; thoughts on part 2 by Charlie)

Not sure what significance it has, if any, but the only other "pattern" I see in the higher columns of your table is that n(a,s) is evenly divisible by 2(a-s+1).  Unfortunately I don't see a pattern in the rest of the divisors that would lead to a closed-form solution for n(a,s).  For example, for n(a,5):

n(5,5) = 14 = 2*7

n(6,5) = 124 = 4*31

n(7,5) = 582 = 6*97

n(8,5) = 1928 = 8*241

n(9,5) = 5110 = 10*511 = 10*7*73

n(10,5) = 11604 = 12*967

n(11,5) = 23534 = 14*1681 = 14*41*41

n(12,5) = 43792 = 16*2737 = 16*7*17*23

Edited on March 26, 2014, 9:04 am
  Posted by tomarken on 2014-03-26 09:01:02

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