A house of mirrors can be a dangerous place to live. Bump a wall and the whole house can fall around you – sharp shards everywhere. That’s why there are rules for anyone planning on building a house of mirrors.

Rule 1: EQUAL NUMBERS OF EACH COLOR. This house fails. There are only two dark blue squares.

Rule 2) All the colors must be DIFFERENT SHAPES. This house fails because the shapes are rotated, but the same.

Rule 3) All the colors must have MIRROR SYMMETRY. That’s true even if the color is not connected. The green color is in two parts and has mirror symmetry. Unfortunately, the brown shape does not have mirror symmetry, so this house of mirrors fails.

Rule 4) All the colors must have DIFFERENT LINES OF SYMMETRY. Can you see why this house of mirrors fails?

It fails because the red and blue shapes have the same line of mirror symmetry.

Summary of how to build a house of mirrors:

1) EQUAL NUMBERS OF EACH COLOR

2) DIFFERENT SHAPES

3) MIRROR SYMMETRY

4) DIFFERENT LiNES OF SYMMETRY

Now it is your turn. Try to build a 3×3 house of mirrors with 3 colors.

 

I think that is the only type of 3×3 house. Yes – you could color the rooms differently. Yes – you could rotate or flip the house, but it’s really the same blueprint.

That’s not true of 4×4 houses of mirrors. There are many. Try to find one before going to the next slide.

 

This is one 4×4 house of mirrors. I don’t know how many there are. 

The next 4×4 house of mirrors I’ll show you (think of me as your real estate agent) is one where all the lines of symmetry intersect at a single point.

I don’t know if you’d pay extra for that feature, but just thought I’d point it out. Wait… let me actually show you the lines of symmetry…

Here they are.

I can tell that you’re unimpressed. Maybe you want your home to have a bit more feng shui. The next 4×4 house I’ll show you has two colors that are totally connected. 

You see? The green and red rooms are totally connected. We’ll give this house a FENG SHUI score of two because there are two colors that are connected.

I wonder if any house with at least two rooms can get a perfect FENG SHUI score. That would mean each color is connected. If you find such a house contact me for a prize and recognition here. I don’t know if such a house exists.

Next look for a 5×5 house of mirrors with 5 colors.

Here is one of perhaps thousands. They’re difficult to find with a mediocre real estate agent like me. You should really find someone better.

Anyway, this house of mirrors has a feng shui score of 1 because of the red room.

Try to find a 6×6 house of mirrors. For the first time, you have a choice of how many colors your house should have. You could look for a 6-color house, but I’m going to show you the family planning alternative. A 4-color 6×6 house is sure to please!

Even though it has a feng shui score of zero, I think you should seriously consider buying this house of mirrors. The 6×6 neighborhood is definitely a nice place to live. Lots of space for the whole family.

7×7 would be better of course. 7 is prime, so I think you can convince yourself that only 7-color options are possible.

Not only is this 7×7 mansion big, it also boasts an impressive feng shui score of two (for the blue and yellow rooms.)

Next, I’ll show you the 11×11 palace of mirrors that I bought with the immense wealth accumulated from MathPickle.

Impressive on first glance. A feng shui score of 4 was all I saw. The sad fact is that I was duped. The reason I’m sitting in a pile of glass shards now – writing up this puzzle – is that the blue room doesn’t have mirror symmetry.

House of Mirrors

(Mikhail Veretennikov, 2019)

I was so happy when Mikhail contacted me with this extension of the spectral blaster puzzle into 2D. The solutions here are all his… except the impressive 11x11 monstrosity on the last slide 😉

Thank you Mikhail!

PS. Contact me: gord at mathpickle.com with any puzzling ideas. I’ll give you full credit and we can have fun letting the world discover your puzzle.

PPS. A prize for anyone who can find a house of mirrors with more than two colors and a perfect feng shui score. (see the presentation for an explanation 😉

PPPS. Below are three dodecahedra of mirrors. Which one will survive without shattering?

Joseph Howard found the first example of a house of mirrors with perfect feng shui. Spoiler alert: on the next slides you’ll see his beautiful solutions.

Just to recap: This house of mirrors has feng shui = 2 because the red and green are connected. The blue is not connected so it does not score a feng shui point. Neither is the orange. All shapes are the same size, but different shapes. They have different axes of mirror symmetry.

Joseph Howard’s first solutions were 10×10.

He sent two of these 10×10 solutions. Then he felt work pressure and went back to his job of being a real estate agent… but not before wondering if he could achieve perfect feng shui in a smaller house.

A few days later Bryce Herdt found this 8×8 solution.

Joseph Howard found two other 8×8 solutions and was pretty sure that no smaller house can have perfect feng shui.

Joseph’s second 8×8 solution.

However, Joseph’s most interesting thoughts came about in an error he made in his first email to me. Before his beautiful correct solution, there was a beautiful incorrect solution…

Your students might enjoy seeing this and finding out what went wrong…

Joseph initially thought that mirror symmetry OR rotational symmetry was acceptable in a house of mirrors.

Now THAT was an interesting mistake because…

It opens up a whole new real-estate market… Forget the old neighbourhood with their Houses of Mirrors. Welcome to Joseph’s new neighbourhood of Twisted Homes.

Twisted Homes must have rooms that are rotationally symmetric. The point of rotation for each room must be different, each must be a different shape. Again we ask if we can find a house with perfect feng shui. This home has feng shui of 2 because green and red are connected.

This is too good – we need a whole new puzzle to explore Joseph’s new neighbourhood…

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)