Rule 2: Each room must have rotational symmetry. This home fails because the red room has mirror symmetry but not rotational symmetry and the purple room has no symmetry at all.

Rule 3: Each room must be different.

Now that’s not quite enough… we need to say a bit more…

We also need to say that these two rooms are really the same. It is true that the “S” and “Z” don’t look identical, but if gravity fails and one room gets flipped over, you can see that they might be considered the same. This is certainly how vampires think about it. So we have:

Rule 3: Each room must be different. (gravitationally flipped rooms are considered identical)

This twisted home satisfies the first three rules, but it fails on a fourth. Can you guess what it is?

Rule 4: ???

Rule 4: All rooms must be connected.

This house fails because the blue room and orange rooms are not connected.

Now I’ll be honest – I have not found a solution to Joseph Howard’s puzzle yet. Vampires are generally pretty intelligent, so it is not surprising that if a twisted home exists, they are keeping its existence secret.

The attempt on the left was my first attempt. You can see that I failed to get all the shapes different (the purple and orange shapes are the same) and I failed to get every room connected (both blue rooms are disconnected.)

This second attempt worked a bit better. I only violated the fourth rule that all shapes must be connected, but other than that it is good.

Because I found it so difficult, I decided to get organized… trying to solve the 4×4. See if you can do it before you go to the next slide.

There are only five tetrominoes. Unfortunately, only three of these have rotational symmetry so 4×4 cannot be filled by rotationally symmetric rooms. A 4×4 twisted home is impossible – even for vampires.

Try 5×5 before going forward.

I realized that instead of trying to build rooms inside the 5×5 or 6×6 frame of a home, I should maybe try to build the rooms that have rotational symmetry first – and then try to squish them in…

If this is going to work in the 6×6 frame, you must choose six of the hexominoes. You very quickly discover that it is impossible.

What about 8×8? Try it before proceeding…

These are all the octominoes with rotational symmetry. Do any eight of them fit into the 8×8 square. I have failed to find a solution, but I’ve also failed to find a proof that it doesn’t work.

For those of you wanting to build an imperfect home, why don’t you allow yourself to break one and only one rule and then try to satisfy the broken rule with as many rooms as possible.

I let myself forgot about rule 4 (all rooms need to be connected) and came up with a solution that has three rooms connected. That’s a failure, but it’s not a catastrophe 😉

What a beautiful result from Trevor Hoffman (aka The Little Viridian Fighter)… 

Trevor’s 9*30 rectangle can be scaled to become a square.