In this Mini Mathematical Universe your students will try to figure out the four rules that are needed to calibrate their spectral blaster. They collectively make any guess of a sequence of colours and then you will tell them if they violated one of the four rules and which one. If they violated more than one undiscovered rule (which will probably happen a lot) you will give them the lowest numbered rule that has been broken.

Let’s see it in action.

Eleven students each suggest one color. The twelfth student says “try firing the Spectral Blaster.”

Does this pattern work? No. Why? Because it violates… RULE 1: All colors present must appear equally often. (equality rule)

Note: This pattern violates ALL the rules, but you only say one rule at a time.

Twelve students each suggest one color. The thirteenth student says “try firing the Spectral Blaster.”

Does this pattern work? It satisfies RULE 1 (equality rule), but No – it still fails. Why? Because it violates… RULE 2: All color patterns must be different (different pattern rule.)

One child wishes to try something similar and then tries firing the Spectral Blaster, but it still does not work. The blue pattern is different from the other two, but they are the same so it still fails because of RULE 2: All color patterns must be different (different pattern rule.)

The next group suggests the following pattern which does not violate Rule 1 (equality rule) or RULE 2 (different pattern rule,) but it does violate RULE 3 (all patterns must have mirror symmetry.)

Blue and purple do have mirror symmetry, but not red.

The next student group suggests the following pattern which satisfies RULE 2 (different pattern rule) and RULE 3 (all patterns must have mirror symmetry) as you can see on the next slides…

Red has mirror symmetry.

Blue has mirror symmetry.

Blue has mirror symmetry.

Unfortunately it fails RULE 1 (equality rule.)

The next group suggests this little pattern. Does this work?

RULE 1 is certainly satisfied (equality rule.)

Red has mirror symmetry.

Blue has mirror symmetry.

Purple has mirror symmetry.

Unfortunately the patterns for all colors are all the same… two squares separated by two squares.

This violates RULE 2 (different pattern rule.)

The next group of students suggest another little pattern. Does it satisfy RULES 1, 2 and 3?

Red is symmetric.

Blue is symmetric – and the two blue squares are separated by only two spaces. Red was separated by three. This means the red and blue patterns are different.

Purple is symmetric – and the two purple squares are separated by only one space.

RULE 1, 2 and 3 are satisfied. Unfortunately RULE 4 is not.

What do you think RULE 4 is? Many wrong, but interesting answers should flow. Cap the discussion after a minute or two.

RULE 4: The lines of mirror symmetry must all be different (different axes.)

Unfortunately in spectral blast above – the red and purple lines of mirror symmetry are the same. The good thing is – that’s all the rules! Can you find a pattern that satisfies everything.

Does this work? Are all four rules satisfied?

PS. We could make another rule: there need to be at least two colors, but I’ll leave it up to your students if they ever discover the trivial solution of a single colour pattern.



Red has mirror symmetry.


Blue has mirror symmetry.


Purple has mirror symmetry.

This is our first successful shot from our spectral blaster.

Find a solution using twenty spaces and five colors.

Here is one solution.


Being hit by a spectral blast doesn’t kill, but it is disorienting. Here is a sample of what a temporarily blinded individual sees. For another, related puzzle on Spectral Blaster Rings look here.

Spectral Blaster

(MathPickle, 2017)

Get your students deducing the mysterious rules that need to be satisfied in order to fire their spectral blaster. Until the four rules are discovered students should remain together as a class. Then they should go off to try to solve and explore. Here are the four rules for reference. Do not let your students see this list!

Rule 1: Equality

Rule 2: Different Pattern

Rule 3: Mirror Symmetry

Rule 4: Different Axes

The copyright for the Ray_Gun_Mark_II_model_BOII is from Call of Duty. MathPickle’s use of this image is legitimate as it does not interfere with the selling of the game and is for noncommercial purposes and is a part of my general commentary and critique of computer based games as opposed to board games. Parents need to get their children involved in slow, methodical problem solving. Doing this problem solving in a group rather than looking at a computer screen has social advantages. If there is one thing that parents should keep from Call of Duty and other violent shoot-em-ups – it is the violence. That can be exciting for boys… but try to move it into a slow paced strategic game like Memoir ’44.

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 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, etc.

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?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)