Place Value

# Other Rules

Let’s play the dots and boxes game, but change the rule.

**The 1←3 Rule**

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

### Example: Fifteen dots in the 1←3 system

Here’s what happens with fifteen dots:

**Solution: **The 1←3 code for fifteen dots is: 120.

**Problem 2**

- Show that the 1←3 code for twenty dots is 202.
- What is the 1←3 code for thirteen dots?
- What is the 1←3 code for twenty-five dots?
- What number of dots has 1←3 code 1022?
- Is it possible for a collection of dots to have 1←3 code 2031? Explain your answer.

**Problem 3**

- Describe how the 1←4 rule would work.
- What is the 1←4 code for thirteen dots?

**Problem 4**

- What is the 1←5 code for the thirteen dots?
- What is the 1←5 code for five dots?

**Problem 5**

- What is the 1←9 code for thirteen dots?
- What is the 1←9 code for thirty dots?

**Problem 6**

- What is the 1←10 code for thirteen dots?
- What is the 1←10 code for thirty-seven dots?
- What is the 1←10 code for two hundred thirty-eight dots?
- What is the 1←10 code for five thousand eight hundred and thirty-three dots?

**Think / Pair / Share**

After you have worked on the problems on your own, compare your ideas with a partner. Can you describe what’s going on in Problem 6 and why?