1.1: Dots and Boxes

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Page ID: 32425

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Here are some dots; in fact there are nine of them:

$9dots.png$

We’re going to a play an “exploding dots” game. Here’s the only rule for the game:

The 1←2 Rule

Whenever there are two dots in single box, they “explode,” disappear, and become one dot in the box to the left.

Example $\PageIndex{1}$: Nine dots 1←2 in the system

We start by placing nine dots in the rightmost box.

$Explode1a-300x87.png$

Two dots in that box explode and become one dot in the box to the left.

$Explode1b-300x83.png$

$Explode1c-300x84.png$

Once again, two dots in that box explode and become one dot in the box to the left.

$Explode1d-300x81.png$

$Explode1e-300x82.png$

We do it again!

$Explode1f-300x86.png$

$Explode1g-300x89.png$

Hey, now we have more than two dots in the second box, so those can explode and move!

$Explode1h-300x82.png$

$Explode1i-300x85.png$

And the rightmost box still has more than two dots.

$Explode1j-300x89.png$

$Explode1k-300x84.png$

Keep going, until no box has two dots.

$Explode1l-300x86.png$

$Explode1m-300x83.png$

$Explode1n-300x86.png$

$Explode1o-300x88.png$

After all this, reading from left to right we are left with one dot, followed by zero dots, zero dots, and one final dot.

Answer:: The 1←2 code for nine dots is: 1001.

On Your Own. Here’s a diagram showing what happens for seven dots in a 1←2 box. Trace through the diagram, and circle the pairs of dots that “exploded” at each step.

$Explode2a-300x85.png$

$Explode2b-300x81.png$

$Explode2c-300x83.png$

$Explode2d-300x84.png$

$Explode2e-300x87.png$

Answer:: The 1←2 code for seven dots is: 111.

Problem 1

Note: In solving this problem, you don’t need to draw on paper; that can get tedious! Maybe you could use buttons or pennies for dots and do this by hand.

Draw 10 dots in the right-most box and perform the explosions. What is the 1←2 code for ten dots?
Find the 1←2 code for eighteen dots.
What number of dots has 1←2 code 101?

Think / Pair / Share

After you worked on the problem, compare your answer with a partner. Did you both get the same code? Did you have the same process?

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