
If you want to see how Shalosh discovered (and proved!), (in 0.287 seconds!)
the solution to Math. Magazine problem #1868,
that asked for the number of ways of tiling with 4n+8 dominoes the region in the
discrete plane obtained by removing from the (n+4) by (n+4) square the central
nxn square, proposed by
CS Giant Don Knuth in the April 2011 issue, that, as it turned out,
was already proposed and solved, in 2004, by the human Roberto Tauraso, in Theorem 2.2. of
J. Integer of Sequence Article 04.2.3
the
input file
would yield the
output file

If you want to see the analogous theorems for
the number of ways of tiling with 4an+2a^{2} dominoes
the region in the discrete plane obtained by removing from the (n+2a) by (n+2a) square the central
n by n square, from a=1 (even you can do it!) to a=6 (not even Knuth can do it (by hand))
the
input file
would yield the
output file

If you want to see 150 deep theorems about the number of ways of
tiling with dominoes (also called dimers)
the region in the discrete plane obtained by removing from the
(c_{1}+n+c_{2}) by (d_{1}+n+d_{2}) rectangle the central
n by n square for 1 ≤ c1,c2,d1,d2 ≤ 4,
the
input file
would yield the
output file

If you want to see 38 deep theorems about the number of ways of
tiling with MONOMERS and DIMERS
the region in the discrete plane obtained by removing from the
(c_{1}+n+c_{2}) by (d_{1}+n+d_{2}) rectangle the central
n by n square for many cases of 1 ≤ c1,c2,d1,d2 ≤ 3,
the
input file
would yield the
output file

If you want to see explicit expressions for the number of ways of
DOMINOtilings crosses whose central square is a 2 by 2 square and
a 4 by 4 square, and that have equal arms of length n, for all n,
the
input file
would yield the
output file

If you want to see explicit expressions for the number of ways of
DOMINOtilings of crosses whose central rectangle has sidelengths up to 4,
and that have equal arms of length n, for all n,
the
input file
would yield the
output file

If you want to see explicit expressions for the number of ways of
MONOMERDIMER tiling crosses for 1 by 1, 1 by 2, 1 by 3, and 2 by 2 rectangles,
and that have equal arms of length n, for all n,
the
input file
would yield the
output file

If you want to see generating functions for the sequences enumerating
the number of DIMER (i.e. domino) tilings of a by n (or a by 2n, if a is odd) for
a from 1 to 9, then
the
input file
would yield the
output file

If you want to see generating functions for the sequences enumerating
the number of MONOMERDIMER (i.e. domino) tilings of a by n for
a from 1 to 7, then
the
input file
would yield the
output file

If you want to see 81 deep theorems (rigorously proved!) giving the BIVARIATE generating functions
for the number of DIMER (domino) tilings the
region in the plane obtained by removing from the
c[1] + m + c[2], by , d[1] + n + d[2] rectangle
the central m by n rectangle, for 1 ≤ c[1],c[2],d[1],d[2] ≤ 3
the
input file
would yield the
output file

If you want to see 81 deep theorems (rigorously proved!) giving the BIVARIATE generating functions
for the number of MONOMERDIMER tilings the
region in the plane obtained by removing from the
c[1] + m + c[2], by , d[1] + n + d[2] rectangle
the central m by n rectangle, for 1 ≤ c[1],c[2],d[1],d[2] ≤ 3
the
input file
would yield the
output file