[sparknlaunch12]

Really great image however I was confused by one comment on the page suggesting there are more than 892 variations.

"I was confused at first because so many partitions are not on the poster. I see that this poster has culled horizontal and vertical symmetries, but the entry is titled “the 892 unique ways to partition a 3×4 grid.” Given this, I think a poster with all 3,164 partitions would have been (counter intuitively) more elegant. Or if the poster had just been titled “892 unique ways to partition a 3×4 grid.”

[joejohnson]

There are 892 unique arrangements of unit rectangles in a 3x4 grid when you consider two arrangement which are rotations or mirror images of one another to be the same arrangement. When you count each arrangement and any rotations/mirror-images of that arrangement, there are 3,164 partitions.

Example: This partition:

  +-+---+-+
  | |   | |
  | |   +-+
  | |     |
  | |     |
  +-+-----+

...is just a rotation of this partition:

  +-+---+-+
  |     | |
  |     | |
  +-+   | |
  | |   | |
  +-+---+-+

There are 3,164 partitions if we say the above figures are distinct; if we call the above figures the same partition (but shown in two different orientations) then there are only 892 partitions.

[sparknlaunch12]

That makes sense. Thanks for the explanation.