A Spectral Blaster Ring is what you see if you get hit with a spectral blast. What doesn’t kill you makes you stronger ūüėČ If you have not seen the slide show on Spectral Blasters you should see that before you continue.

The same four rules apply to¬†Spectral Blaster Rings as you already know. Let’s illustrate these rules through examples that violate them. This ring violates RULE 1: Equality. There are not an equal number of all the colors that are present.

Does this work? It satisfies RULE 1: equality. However, it violates RULE 2: different patterns. The red and purple patterns are the same… just rotated.

Does this work? It satisfies RULE 1: equality and RULE 2: different patterns. However, it violates RULE 3: mirror symmetry. Neither the red nor purple has mirror symmetry.

Does this work? It satisfies RULE 1: equality¬†and RULE 2: different patterns¬†and RULE 3: mirror symmetry.¬†However, look at the axes of symmetry…

RULE 4: different axes

The purple and red mirror symmetry axes are the same. RULE 4 is violated.

Those are the four rules.

What kinds of rings work? What ones fail? Try to blast¬†this 18 ring as a class. Use three colors. I don’t think it can be blasted¬†with six colors.

Did I successfully blast this 15 ring?

All colors are symmetric (RULE 3) and the axes are different (RULE 4) and an equal number of each color is used (RULE 1).

But look at the yellow pattern…

Compared to the green pattern. They are the same ;-(

 

RULE 2 is violated.

Can a 15 ring be blasted with 5 colors and then with 3 colors?

Did this 8 ring get blasted successfully?

Everything looks good… except that we don’t have enough reds ;-(

Try to blast an 8 ring.

Did I at least manage to blast this 16 ring successfully?

Alas no.

The axes of symmetry of black and purple are the same ;-(

RULE 4 is violated.

Try to blast a 16 ring. Enjoy blasting different sized rings.

Are any rings blastable with exactly two colors? Which rings of size 2n blastable with n colors? Why are rings that are the product of an odd number, D>1, and an even number E>2 difficult or impossible to blast with E colors?

Spectral Blaster Hit

(MathPickle, 2017)

Please explore Spectral Blaster before looking at this puzzle. Find puzzle sheets and the above slide show here.

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 ūüėČ

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?

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

Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos ūüėČ

Gordon Hamilton

(MMath, PhD)