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.

Q.E.D.

Posted by Vee-Liem Veefessional on 2010-03-24 02:23:30