THE STAMP QUESTION...




 
Here are four ways to get you started! 
a
a
( å S(Z(xc1)(yd1) ) + ( å S(Z(xc1)(yd1) +
(yc1)(xd1) )
c = d ^{} c ≠ d^{
}
S
S
Let n be a positive real integer. Let a be the number of polyominoes of n. Take polyomino p(s). Place it in rectangle q(s), q(s) being the smallest rectangular grid possible p(s) can fit in. q(s) has dimentions c by d. The maximum number of different ways p(s) can be placed in q(s) (when c = d) is 8, the minimum 1, and when c does not = d, the maximum is 4 times, and minimum 1. Call the number of times p(s) fits in q(s) Z.
Take rectangle R. The number of times q(s) fits in it (when c = d) is (xc1)(yd1), and when c does not = d, (xc1)(yd1) + (yc1)(xd1). Multiply this equation by Z, and that is how many different places p(s) can be placed in R. Sum all the other polyominoes with it (using the sigma function) and there is your answer (see above). So for 4 connected stamps to fit in a 4 by 3 grid would be:
(8(32)(41) + (42)(31)) + (41)(31) + (x0)(y3)(y0)(x3)
which can be simplfied as 28y + x(19y28) + 33 which comes out as 65.
Incidentally, the number of different ways a block of 6 connected stamps can fit in a sheet 8 by 9 is 7755.