Go Bugs Go!

In our previous post, we learned how to construct a regular polygon with 15 sides using compass and straightedge. The interesting part of the construction is that it is based on the integer solutions of the Diophantine equation $$3x + 5y = 1.$$

We have also learned about the measuring liquid puzzle - "how to measure out exactly 1 liter of water using a 3-liter jug and a 5-liter jug." Again, we showed that the answers to the puzzle correspond to integer solutions of the equation $$3x + 5y = 1.$$

Today, let us consider a third problem that is related to the equation $$3x + 5y = 1.$$

We will state the problem and pose some questions for the reader to think about.

A game of running bug. The rule of the game is, each time we can move the bug either 3 steps or 5 steps to the left or to the right.

Questions are:

  • How can we move the bug from a point $A$ to a point next to it? Is there a relationship between the moves and the equation $3x+5y=1$?

  • Prove that, from a point $A$, after a number of times, we can move the bug to an arbitrary point $B$ on the line.

  • Instead of letting the bug running on a line, we change the game so that the bug is moving on a circle. Is there a relationship between this game and the construction problem?

Happy thinking and see you again in our next post!

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