Dots n’ Boxes

Okay so I went to the lecture on Exploding Dots today, since Casey recommended it, AND IT ACTUALLY BLEW MY MIND.

First of all, the speaker had a really cool accent. To which my reaction was obviously:

But then I actually started listening to what he was saying, and even that was cool too! Basically he broke down every operation we know in math into boxes and dots. I’ll try and recount it as best as I can, but I think I’m still in shock at how easy it is.

We’ll make a box machine. Each box represents a base number, and any dot that goes in to the right most box, goes left. For example, say we make a 2 to 1 box machine. It’ll look like this!

[     ] [     ] [     ] [      ] [      ]

Isn’t it cute? Okay, so every time we put two dots into the right most box, they EXPLODE and show up as one dot in the next box to the left. Cool right? Okay, so if we put, say 12 dots in the first box, what’ll happen?

[     ] [     ] [     ] [      ] […………]

[     ] [     ] [     ] [……] [      ]

[     ] [     ] [  …] [      ] [      ]

[     ] [  .   ] [  .   ] [      ] [      ]

So, 12 then becomes 0,1,1,0,0. See how it works?

If we use base 10 for this concept, we can do some pretty awesome stuff with addition, subtraction, multiplication and division. SERIOUSLY. I’ll go over briefly addition, division, and division of polynomials (those were my favorites!)

Start with addition. Okay, tell me right now, what’s 245 + 372?! I can show you! If you do it the way the speaker taught us, just add each number that matches up without carrying anything over. So, 2+5 equals 7, 4+7 equals 11, and 2+3 equals 5. Now put these into your boxes with base ten!

[     ] [     ] [ 5   ] [  11  ] [   7  ]

Remember the exploding rule we had before with 2? With base ten, if you have ten dots they explode! So the 11 turns into 1 dot in the next box to the left, leaving one in the original box.

[     ] [     ] [  6  ] [   1  ] [   7  ]


Okay division! What’s 284 divided by 13? Make box machines for the two!

[     ] [     ] [     ] [   .  ] [  … ]

[     ] [     ] [  .. ] [ ……..] [  ….]

The trick to this is to find how many times the 13 box machine shows up in the 284 box machine. That is, how many times do you see one dot to the left of three dots? This is where the speaker used the wonderful colorful markers to circle pairs, but I’ll just save the trouble and figure it out for you (blogs can be lacking in these kinds of resources sometimes; it happens). It happens one time between the first and second box, and 2 times between the second and third. Treat those as base ten place markers. So, the one time between the first and second represents one, and the 2 times represents 20. So, 284/13 is 21 and a little extra, since all the boxes aren’t accounted for.

It gets cooler. You can do this with polynomials too. Just mark your boxes as 1, x, x², x³, and so on! Let’s try it.

(x4+2x³ + 4x² + 6x + 3) ÷ (x² + 3)

[   . ] [..   ] [ ….] [……] [ … ]

[     ] [     ] [  .  ] [     ] [  …]

So, just like before, you look for the smaller pattern in the bigger pattern, and based off of where the connections are, that’s how many of the number, x’s, x²’s, x³’s and so on. You end up with…..x² + 2x + 1! Go math!

One more thing though, I’mma relate this to Fibonacci, because why not?

So, first of all, 1/ (1-x-x²)? When you box it out, you get the Fibonacci sequence in polynomials. That is, 1 + x + 2x²+3x³+5x5+8x8+… and so on! It gets cooler.

The Hailstone Problem starts with any positive integer. If it’s odd, you use 3N + 1 to get another number, and if it’s even, you divide it by two (basically, (3/2)N + (1/2)). No matter what number you get, it always goes back to 4, 2, 1. WHAT WHY. Nobody knows!! If you take the 3/2 out of this problem and make a machine out of it (every 3 dots explodes and moves left as 2 dots), an interesting pattern happens. The first 2 digit representation (2 in the second box, 0 in the first) is 3, the first 3 digit representation is 6, the first 4, 9. The pattern is another Fibonacci sequence with every number in the sequence being the sum of the previous two digits!


After that, it’s up to people today to come up with why this matters and why the 4,2,1 cycle happens! If you figure it out let me know. But seriously though, reflecting on this. HOW COOL.


3 thoughts on “Dots n’ Boxes

    • No, thank you for coming to speak! I really enjoyed it. You made math fun (which is hard to do sometimes in a room full of math majors)! And thank you for including those links, I actually had some questions about it and now I have references!

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