Can a 4x11 rectangular grid of square blocks be covered (having no overlaps), with multiple copies of the tetromino (including rotations, but not reflections) as shown below?
If so, give an example. If not, provide a proof that this cannot be achieved.
Note: Each of the 44 square blocks of the 4x11 grid has the same shape and size as each of the 4 square blocks of the tetromino.
Lengthwise, you need to sum to 11. Now, 11=2+2+2+3=2+3+3+3. So there are two posibilities, each involving at least a three unit side.
But having a three unit side lengthwise is impossible. Here is why:
Suppose you lay out this way,
_ _ _
Then, no configuration would fill up the squares in the 4 unit side. This would leave either a 2x2 gap, or a single square with a 3x1 gap, or a shape with is non-tetromino.
Note that any parts each of these 4x3 block must be filled in independently, otherwise it would affect other parts and results in gaps straight away.