Twisted Homes are haunted. Ghosts and ghouls drift in slow-moving vortices and walking around the drafty rooms can be disorienting. 

Vampires have four rules to build a twisted home.

Rule 1: Each room must be the same size. This home fails because the green room is too big and the purple one is too small.

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 😉

Twisted Homes

(Joseph Howard, 2019)

Joseph Howard is a real estate agent in real life, but he plays with mathematics in his free time. Twisted Homes happened whenever Joseph played around with another puzzle (House of Mirrors) and came up with a solution that included shapes that had rotational symmetry. That was wrong, but like all great mistakes it inspired us to create something new.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

 
MP1 Toughen up!

This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

 
MP6 Be precise!

This is where our students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.

(http://www.corestandards.org/Math/Practice/)

Please use MathPickle in your classrooms. If you have improvements to make, please contact us. We'll give you credit 😉

Gordon Hamilton

(MMath, PhD)

 

Lora Saarnio

(CEO)