Exploding Dots

1.1 Base Machines

Lesson materials located below the video overview.

OPENING COMMENT: The entire text to this Exploding Dots course appears in the COMPANION GUIDE to the course. Plus ALL THE SOLUTIONS to the questions asked throughout this course, and some additional material, appear in it too. Check out the “Guides and Solutions” section at the home page.


Here is a story that isn’t true.

When I was a young child I invented a machine.  And this machine was nothing more than a row of boxes that extends as far to the left as I could ever desire. 


I called this machine a “ \(1 \leftarrow 2\) machine” and one would work this machine by placing dots into the rightmost box.

Placing one dot in into the machine doesn’t do much. It stays as one dot.


But in placing two dots into the machine, something exciting happens: Two dots in a box explode– kapow!–and produce one dot one place to the left.


The code for “two” in a \(1 \leftarrow 2\) machine is “no dots, no dots, … , no dots, one dot, no dots.” Let’s not bother writing the beginning lack of dots and just write this code for two as:


Placing a third dot into the machine (always the rightmost box) produces:


Putting in a fourth dot is exciting! One explosion is followed by a second explosion!


A fifth dot yields the code:


And hours of fun are to be had placing dots in a  \(1 \leftarrow 2\) machine and working out their codes.


Comment: One can place dots all at once in the rightmost box of the machine. For example, here’s what happens with six dots placed into the machine right away:


This shows that six has code: \(110\). (Is this what you expect given that \(5\) has code \(101\)?)


Question 1: a) Verify that the code for thirteen is \(1101\)  in a \(1 \leftarrow 2\) machine.  b) What is the code for fifty?


Recall that alll the solutions to these questions appear in the COMPANION GUIDE to this EXPLODING DOTS course.


Question 2: Could a number ever have \(100211\)  as its code in a \(1 \leftarrow 2\) machine?


Question 3:  (HARD!) Which number has code \(10101\)  in a \(1 \leftarrow 2\) machine?



But then one day, in this untrue story, I had an epiphany! Instead of playing with dots in a \(1 \leftarrow 2\) machine, I realized I could instead play with dots in a \(1 \leftarrow 3\) machine.


THE  \(1 \leftarrow 3\) MACHINE: Whenever there are three dots in any one box they “explode,” that is, disappear, and become one dot in the next box one place to their left. 


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


We have:

 THE  \(1 \leftarrow 3\) CODE FOR FIFTEEN IS: \(120\) .


Question 4: a) Show that the \(1 \leftarrow 3\) code for four is \(11\).  b) Show that the \(1 \leftarrow 3\) code for twenty is \(202\).


Question 5: What is the \(1 \leftarrow 3\) code for thirteen? For twenty-five?


Question 6: Is it possible for a number to have \(1 \leftarrow 3\) code \(2031\)? Explain.


Question 7: HARD CHALLENGE: What number has \(1 \leftarrow 3\) code \(1022\)?


Let’s keep going …


Question 8: What do you think the \(1 \leftarrow 4\) rule is?  What is the \(1 \leftarrow 4\) code for the number thirteen?


Question 9: What is the \(1 \leftarrow 5\)  code for the number thirteen?


Question 10: What is the \(1 \leftarrow 9\) code for the number thirteen?


Question 11:  What is the \(1 \leftarrow 5\)  code for the number twelve?


Question 12: What is the \(1 \leftarrow 9\) code for the number thirty?




Question 13: What is the \(1 \leftarrow 10\) code for the number thirteen? What is the \(1 \leftarrow 10\) code for the number thirty-seven? What is the \(1 \leftarrow 10\) code for the number \(273\)? What is the \(1 \leftarrow 10\) code for the number \(5846\) ?


BONUS: Here’s a cute little animation. Does it make sense to you?

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