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10^n >= 7^(n+1)
ln(10^n) >= ln(7^(n+1))
...
n >= ln(7)/( ln(10) - ln(7) )
n >= 5.455696235812878344
n >= 6

$ perl -e'
    printf "chars: %2d   digits: %13.f   %s   dominoes: %11.f\n",
...
chars: 10   digits:   10000000000   >   dominoes:  1977326743
chars: 11   digits:  100000000000   >   dominoes: 13841287201
chars: 12   digits: 1000000000000   >   dominoes: 96889010407