# 2.2: Other Rules

- Page ID
- 9829

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

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 \(\PageIndex{1}\): *Fifteen dots in the 1←3 system*

Here’s what happens with fifteen dots:

**Answer:**-
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?