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FG has posted a solution for the 12x1 rectangle, attaining 15/4 = 3.75.

More results:
11x1 rectangle is 101/27 = 3.740740...
7x3 rectangle is 56/15 = 3.733333...
7x2 rectangle is 67/18 = 3.722222...
8x1 rectangle is 26/7 = 3.714285...
6x1 rectangle is 85/23 = 3.695652...
7x1 rectangle is 11/3 = 3.666666...
5x1 rectangle is 25/7 = 3.571428...

I have removed specific solutions from this answer, as they have been superseded by FG's results.

Latest figures

FG has posted a solution for the 12x1 rectangle, attaining 181/48 = 3.3.7708333...

More results:
11x1 rectangle is 101/27 = 3.740740...
7x3 rectangle is 56/15 = 3.733333...
7x2 rectangle is 67/18 = 3.722222...
8x1 rectangle is 26/7 = 3.714285...
6x1 rectangle is 85/23 = 3.695652...
7x1 rectangle is 11/3 = 3.666666...
5x1 rectangle is 25/7 = 3.571428...

I have removed specific solutions from this answer, as they have been superseded by FG's results.

Latest figures

FG has posted a solution for the 11x1 rectangle, attaining 101/27 = 3.740740...

More results:
7x3 rectangle is 56/15 = 3.733333...
7x2 rectangle is 67/18 = 3.722222...
8x1 rectangle is 26/7 = 3.714285...
6x1 rectangle is 85/23 = 3.695652...
7x1 rectangle is 11/3 = 3.666666...
5x1 rectangle is 25/7 = 3.571428...

I have removed specific solutions from this answer, as they have been superseded by FG's results.

Latest figures

FG has posted a solution for the 12x1 rectangle, attaining 15/4 = 3.75.

More results:
11x1 rectangle is 101/27 = 3.740740...
7x3 rectangle is 56/15 = 3.733333...
7x2 rectangle is 67/18 = 3.722222...
8x1 rectangle is 26/7 = 3.714285...
6x1 rectangle is 85/23 = 3.695652...
7x1 rectangle is 11/3 = 3.666666...
5x1 rectangle is 25/7 = 3.571428...

I have removed specific solutions from this answer, as they have been superseded by FG's results.

Latest figures

FG has posted a solution for the 3x7 rectangle, attaining 56/15 = 3.733333...

More results:
7x2 rectangle is 67/18 = 3.722222...
8x1 rectangle is 26/7 = 3.714285...
6x1 rectangle is 85/23 = 3.695652...
7x1 rectangle is 11/3 = 3.666666...
5x1 rectangle is 25/7 = 3.571428...

I have removed specific solutions from this answer, as they have been superseded by FG's results.

Latest figures

FG has posted a solution for the 11x1 rectangle, attaining 101/27 = 3.740740...

More results:
7x3 rectangle is 56/15 = 3.733333...
7x2 rectangle is 67/18 = 3.722222...
8x1 rectangle is 26/7 = 3.714285...
6x1 rectangle is 85/23 = 3.695652...
7x1 rectangle is 11/3 = 3.666666...
5x1 rectangle is 25/7 = 3.571428...

I have removed specific solutions from this answer, as they have been superseded by FG's results.