Logo turtle bouncing in a red and blue circle with 135-degree turns

[Yankel]

Hello all,

I have a question, would like to have your opinion on the matter.

In the Logo programming language, I draw a red blue circle and put a turtle in it. I have specified that the turtle will move forward constantly, but, when approaching a white background (the page's background), it will turn right by 135 degrees. As a result, every time the turtle comes to exit the circle, it turns around and moved forward another way, staying in the circle. In this way the turtle never escapes the circle.

The question is: Is there a starting position (for the turtle itself and/or his head) that will allow him to escape?

Thank you !

View attachment 7237

 

[I like Serena]

Hello all,

I have a question, would like to have your opinion on the matter.

In the Logo programming language, I draw a red blue circle and put a turtle in it. I have specified that the turtle will move forward constantly, but, when approaching a white background (the page's background), it will turn right by 135 degrees. As a result, every time the turtle comes to exit the circle, it turns around and moved forward another way, staying in the circle. In this way the turtle never escapes the circle.

The question is: Is there a starting position (for the turtle itself and/or his head) that will allow him to escape?

Thank you !

Without loss of generality we can assume that the turtle escapes at the top.

\begin{tikzpicture}[>=stealth,shorten >=2pt,,shorten <=2pt]
\fill[cyan!50] circle (2);
\draw[help lines] (-3,-3) grid (3,3);
\draw circle (2);
\fill (0,2) circle (0.07);
\draw (2,0) -- (-1,3);
\draw[->, thick] (2,0) -- (0,2);
\draw[->, thick] (0,2) -- (2,2);
\draw[->] (0,2) +(62.5:0.7) node {$135^\circ$} +(135:1) arc (135:0:1);
\fill (0,0) circle (0.07);
\draw[<->] (0,0) -- node[below, xshift=4] {\small$\frac 12\sqrt 2$} (1,1);
\end{tikzpicture}

If the turtle starts somewhere on the first vector, or in the area that's above it, and moves to the top, it will escape.
And it's not possible to escape in 2 hits or more.

More generally, if the turtle is within distance $\frac 12 \sqrt 2$ of the center, it cannot escape.
And otherwise, it can escape, but only if it is directed at a point where the given angle is $135^\circ$ or less.