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 11×11 monstrosity on the last slide 😉
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
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!
Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.
MP2 Think abstractly!
Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.
MP3 Work together!
Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.
MP4 Model reality!
Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.
MP5 Know the tools.
Students master the tools at their fingertips - whether it's a pencil or an online app.
MP6 Be precise!
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!
Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist 😉
MP8 Be lazy!?!
Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?