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Subject: Re: Population of disjoint Attacksets

Author: Gerd Isenberg

Date: 15:10:02 06/01/04

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On June 01, 2004 at 17:45:50, Gerd Isenberg wrote:
>
>Btw. one needs only four odd/maj pairs to pack seven words into three binary
>digit words:
>
>one1,two1 := oddMaj(x1,x2,x3)
>one2,two2 := oddMaj(x4,x5,x6)
>ones,two3 := oddMaj(x7,one1,one2)
>twos,four := oddMaj(two1,two2,two3)
>
>But already 11 pairs to pack 15 to 4.
>
>one1,two1  := oddMaj(x1,x2,x3)
>one2,two2  := oddMaj(x4,x5,x6)
>one3,two3  := oddMaj(x7,x8,x9)
>one4,two4  := oddMaj(x10,x11,x12)
>one5,two5  := oddMaj(x13,x14,x15)
>
>one6,two6  := oddMaj(one1,one2,one3)
>ones,two7  := oddMaj(one4,one5,one6)
>
>two8,four1 := oddMaj(two1,two2,two3)
>two9,four2 := oddMaj(two4,two5,two6)
>twos,four3 := oddMaj(two7,two8,two9)
>four,eight := oddMaj(four1,four2,four3)
>
>Hmm, 1,3,11 pairs for 2**2-1, 2**3-1 and 2**4-1 words...
oups  1,4,11 of course

>Too lazy to look for a general formula nPairs(N) now ;-(

 N  nPairs N-nPairs
 3    1    2
 7    4    3
15   11    4
a first guess for 31
31   26    5

so may be for N = 2**n - 1
nPairs := N - n

But to prove that (by induction?) definitely exceeds my math skills.

Gerd



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