See how May and Ray, friends of Goldfish & Robin explain the workings of 1 <–3 machine: Kids Explain Math For Kids.

Instead of playing with a \(1 \leftarrow 2\) machine, I realized I could play with a \(1 \leftarrow 3\) machine (again written and read backwards, a “three-one “machine). Now whenever there are three dots in a box, they explode away to be replaced with one dot, one box to the left.

Here’s what happens to fifteen dots in a \(1 \leftarrow 3\) machine.

First there are five explosions in the first box, with each explosion making a dot in the second box to the right. Then three of those dots explode away. This leaves behind two dots and makes one new dot one place to the left. We thus see the code \(120\) for fifteen in a \(1 \leftarrow 3\) machine.

Here are some more questions you might or might not choose to ponder. 

4. a) Show that the code for four in a \(1 \leftarrow 3\) machine is \(11\).

     b) Show that the code for thirteen in a\(1 \leftarrow 3\) machine is \(111\).

     c) Show that the code for twenty in a \(1 \leftarrow 3\) machine is \(202\).

5. Could a number have code \(2041\) in a \(1 \leftarrow 3\) machine? If so, would the code be “stable”?

6. Which number has code \(1022\) in a \(1 \leftarrow 3\) machine?

We can keep going!

7. What do you think rule is for a \(1 \leftarrow 4\) machine? What is the \(1 \leftarrow 4\) code for the number thirteen?

8. What is the \(1 \leftarrow 5\) code for the number thirteen?

9. What is the \(1 \leftarrow 9\) code for the number thirteen?

10. What is the \(1 \leftarrow 5\) code for the number twelve?

11. What is the \(1 \leftarrow 9\) code for the number twenty?

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