
# 1.2: 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 $$\PageIndex{1}$$: Fifteen dots in the 1←3 system

Here’s what happens with fifteen dots:

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

## Problem 2

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

## Problem 3

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

## Problem 4

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

## Problem 5

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

## Problem 6

1. What is the 1←10 code for thirteen dots?
2. What is the 1←10 code for thirty-seven dots?
3. What is the 1←10 code for two hundred thirty-eight dots?
4. 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?

1.2: Other Rules is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michelle Manes via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.