This week the #puzzle is: When Will You Cross Your Path? #geometry #angle

Anita the ant is going for a walk in the sand, leaving a trail as she goes. First, she walks 1 inch in a straight line. Then she rotates counterclockwise by an angle 𝝋, after which she walks another 2 inches. She rotates counterclockwise an angle 𝝋 again, after which she walks 3 inches. She keeps doing this over and over again, rotating counterclockwise an angle 𝝋 and then walking 1 inch farther than she did in the previous segment.

At some point during her journey, she crosses over her initial 1-inch segment. By “cross over,” I am including the two end points of that first segment.

Anita realizes that 𝝋 was the smallest possible angle such that she crossed over her 1-inch segment. (Among the ants, she’s known for her mathematical prowess.)

How long was the segment along which she first crossed over the 1-inch segment? Your answer should be a whole number of inches.

And for extra credit:

It’s time for you to check Anita’s work. What was the measure of angle 𝝋?

Remember, this was the smallest possible angle for each turn such that she crossed over her 1-inch segment at some later point.

When Will You Cross Your Path?

Highlight to reveal (possibly incorrect) solution:

Desmos. Image 1. Image 2.

My method is to recreate Anita’s path in Desmos and play around with it. Along the way I captured a couple of interesting images. When trying to find the angle, I zoomed in and added digits, until I thought I had enough.

Solutions: The segment is 4 inches long. The angle is 138.59°.