8 different rectangles with area in the teens forming an 11x11 square:
AABBCCCCCCC
AABBCCCCCCC
AABBDDDDEEE
AABBDDDDEEE
AABBDDDDEEE
AABBDDDDEEE
AAFFFFFFGGG
AAFFFFFFGGG
AAFFFFFFGGG
HHHHHHHHGGG
HHHHHHHHGGG
11112222333
11112222333
11112222333
11112222333
44556677333
44556677888
44556677888
44556677888
44556677888
44556677888
44556677888
11111111333
11111111333
22222222333
22222222333
44556677333
44556677888
44556677888
44556677888
44556677888
44556677888
44556677888
You have to use exactly eight rectangles, although I don't know for
sure how many different shapes can be used yet. This assumes you have
integer sides.
Some nice Mathematica code to simplify the search:
(*teen areas that will fit inside an 11x11 rectangle*)
teens = {14, 15, 16, 18};
(*The maximum number of any rectangle that will fit inside an 11x11 rectangle
max = 121/teens//Floor;
(*Brute force find all the combinations that add upto 121*)
(*This is fast: there are only 2688 possibilities*)
tab = Table[{i,j,k,l}.teens,
Evaluate[Sequence@@Transpose[{i,j,k,l}, max}]]];
Position[tab, 121]
yields {{3, 3, 1, 1}, {4, 1, 2, 1}} as the only combinations of areas
that work.
The two given above come from the second solution.
Brett Champion