This week the #puzzle is: Can You Reach the Edge of the Square? #average #geometry #trigonometry #integral

You start at the center of the unit square and then pick a random direction to move in, with all directions being equally likely. You move along this chosen direction until you reach a point on the perimeter of the unit square.

On average, how far can you expect to have traveled?

And for extra credit:

Let’s raise the stakes by a dimension. Now, you start at the center of a unit cube. Again, you pick a random direction to move in, with all directions being equally likely. You move along this direction until you reach a point on the surface of the unit cube.

On average, how far can you expect to have traveled?

Can You Reach the Edge of the Square?

Intermission:

I was the lucky person to be mentioned for my correct solution to the extra credit a week ago. Woohoo! And my request to not have my city/country translated into English was followed! Woohoo!

Highlight to reveal (possibly incorrect) solution:

Program Desmos

Let’s monte carlo this, baby!

For my program, WLOG, I assume I chose a direction in the upper right quadrant of the square, with an angle between 0 and 45 degrees.

Just to make sure, I also mess with a solution in Desmos, using integrals.

Both methods give me something like 0.561.

And for extra credit:

Desmos

Same program, new Desmos. Similar reasoning. Both methods give me something like 0.617.