#MathArtChallenge 95: Magic Squares

The Challenge: Create a magic square. Bonus points if you make it a physical thing.

Magic Squares go WAY back. I’d heard of them, but hadn’t really dug in until I started reading George Gheverghese Joseph’s The Crest of the Peacock. Here’s the gist:

  • Using a square grid, place the counting numbers (equal to the number of spaces in the grid) in a manner such that every row, column and diagonal sum to the same value.
    • So for a 3×3 grid, you want to place the numbers 1 through 9. The sum of each row, column and diagonal will be 15.
    • For a 4×4, place the numbers 1 through 16, the sums will be… ? (It’s more fun if you work it out.)

Part of my attraction to this is that Joseph shares Yang Hui’s description of how to form a 3×3 magic square, and it felt like a story that begged to be made with legos. So I did that:

Yang Hui’s work on magic squares was published in 1275, and even he said he was “passing on the works of earlier scholars and would make no claim to originality,” (Peacock, 209).

Spoiler alert…. there’s only one 3×3 square. I would love to lego up a 4×4, but I don’t have enough legos currently, but you’re all far more creative than I and I’m sure you can sort out a way to explore a bit with the 4×4 and the 5×5.

Magic squares have been around for a….while. There are examples of them in China back to 190 BCE and in India as early as 587 CE. The Chinese examples were connected to an emperor Yu (Joseph calls him “semimythical”) and are just beautiful.

Yu magic squares

Hui went so far as to create magic circles. Joseph notes that some of his explanations for how he derived the higher-number magic squares and circles were “either absent or cryptic to the point of obscurity”. I would love to see some of you illustrate his magic circle… but I’ll also be thrilled just to see some magic squares.

IMG_2032

Things to ponder:

  • How many 4×4 squares are possible? What about 5×5? Can you create an algorithm for deriving them?
  • What impact does it have on the creation if there is a center square (like in a 3×3 or a 5×5 grid) and when there is not (like a 4×4 grid)?
  • How could you use my video (and, please, definitely legos or blocks that the students can play with themselves) to help students think about means/averages?

#MathArtChallenge 90: Sand Piles

The Challenge: Explore a toppling sand pile.

Here is my explanation (note, I’m pretty sure I have a mistake in toppling 20)

Here is a video that Numberphile put together describing them as well:

Reflection questions:

  • Is it possible for a pile to topple differently? Does it matter if different stages appear different or is it necessary for the toppling to be consistent?
  • What would an isometric (triangular) grid toppling looks like? How about a hexagonal one?
  • How might you meld this with computer programming?
  • Is there a less or more illuminative coloring for the sand piles? What kinds of coloring may make the sand pile structures more obvious or less?

#MathArtChallenge 76: Decagon & Pride Flag

Here is my first attempt at a more thoughtful #mathartchallenge. Pushback, thoughts, additions you’d like to see all welcome. You can comment below, or here, or talk to/DM me on twitter.

June is Pride month. I try to always acknowledge and honor that in my classroom. Last year, I put up this display in my classroom window:

This year, it made sense to create a rainbow #mathartchallenge to post for my kids to see. Having seen pride flags with Black and Brown stripes, noting it as a nod to a more inclusive pride, I picked a decagon design that I wasn’t totally sure how to create from Arts and Crafts of the Islamic Lands , and voila.

Except, of course, that I messed up. I didn’t do my research. ANY research. The original rainbow flag is 6 colors: red, orange, yellow, green, blue and violet. There are new designs that include Black and Brown stripes, and as of 2018, designed by Daniel Quasar, there is this flag:

Daniel Quasar flag

This flag includes not just the Black and Brown stripes, but also nods to the transgender pride flag. You can read about it here. 

I also want to speak the names of Tony McDade, a Black trans man killed by police, and Nina Pop, a Black trans woman killed in her apartment, both within the last few weeks. 

When I posted my ill-informed rainbow, I hadn’t necessarily planned it to be a #mathartchallenge. However, seeing the collaboration of Tina Cardone and Xi Yu, I think it makes for a good exercise. If you follow the tweet, you can see their solution – the tutorial Xi refers to is also here.

You may also notice they actually tagged their creation with “Queer #BlackLivesMatter”, which I failed to do.

Here is a better version.

MAC 75 Decagon pride flag

Better, I think because it honors (as best I could match) the actual colors of the pride flag, including those represented by the transgender flag. Not great yet, because as any decagon has 10 parts, I just ran out of space for all the colors (an 11 sided shape is rather challenging to construct). I would like to note that in the trans flag, the white color represents gender neutral or non-defined gender. Given the white paper, it seemed the easiest shade to not include. By no means do I mean to exclude gender neutral or non-defined gender people in this representation.

If you choose to participate in this #mathartchallenge, here are some questions I’d like you to reflect on:

  • The larger shape here is a decagon (10 sided shape). Where else do you see the theme of 10 appear here?
  • How many? What did you count and how did you count it?
  • What angles are represented here?
  • What shapes do you see that are not decagons?
  • How do the shapes interact with each other? What can you say about the relationships between them?
  • Can you find an 11 sided design that would better represent Daniel Quasar’s 11 color-flag?
  • How do symbols, like flags, interact with identity at large? With your personal identity? What thoughts about representation are present or missing here?

#MathArtChallenge Day 28: Paper Tube Designs

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

Today, courtesy of the extraordinary Mark Kaercher:

 

#MathArtChallenge Day 27: Golden Icosahedron

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Create your very own Golden Rectangle Icosahedron!

(Yes, I absolutely meant “rectangle”, but twitter doesn’t give you an edit button.)

MATERIALS NEEDED: Cardboard, markers, scissors/box cutter, string/ribbon/thread

 

 

#MathArtChallenge Day 26: Golden Spiral

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Draw a golden spiral,

MATERIALS: Grid paper? Writing utensils.

Use grid paper. (Or not? Might be a neat wonky one if you don’t?)

Start with a 1×1 square and another next to it.

Using a 2 unit long side, create a 2×2 square.

Then a 3×3, etc.

When it’s as large as you like, draw connected quarter circles to form your spiral!

 

#MathArtChallenge Day 25: Tessellations

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Great your very own tessellation from a small piece of cardboard.

  1. Start with a square (Cereal or pop box cardboard works great) Mine was 2”x2”
  2. Cut a piece from one side and tape it to an adjacent side.
  3. Repeat for remaining 2 edges.
  4. Trace!

MATERIALS NEEDED: Pencil, paper, and cardboard from a cereal box or a pop box or something similarly thin but stiff enough to trace.

#MathArtChallenge Day 24: Polygon Midpoints

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE:

  1. Draw a polygon.
  2. Mark the midpoint of each side.
  3. Connect the midpoints of each side to make a new polygon.
  4. Repeat.

Don’t sleep on the quadrilaterals here. They do something rather surprising and beautiful!

MATERIALS NEEDED: Whatever you prefer!

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#MathArtChallenge Day 23: Brunnian Links

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Build a set of 3, 4 or maybe even 5 Brunnian links.

MATERIALS NEEDED: Your call. Joey already did the 3 link one with straws. I used crocheted yarn. Shoelaces?

And then because I realize it’s not nice to not pull apart the 4 for you:

DAY23 MAC STILLS

#MathArtChallenge Day 22: Extending Day 14…

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Dave Richeson came up with a brilliant extension for Day 14

MATERIALS NEEDED: paper (grid?) and pencil!

2020-04-05 20.39.04