Ariadne’s String

This post is about a problem I learned about at Unsolved K-12, and was reminded of at Integer Sequences K-12. Both conferences were joint meetings of mathematicians and educators, organized by Gord Hamilton. Like several of my favorite problems from those conferences, this problem involved explorations on a lattice.

Here is the problem:
– You must get from the top left corner of an n by n lattice to the bottom left.
– Each step must take you in a straight line to a lattice point.
– Each line segment thus created must be longer than the previous one.
– Your path cannot cross itself.
For example, this is not a successful trip, as the player did not reach the destination:

Ariadne1
Taking the last step to the exit would violate the rules, as it would be shorter than the previous step.
 
On the other hand, this is a successful trip in six steps:
 
Ariadne2
The challenge is to get to the exit after taking the greatest possible number of steps. It turns out for example that for the 5 by 5 lattice, the maximum is nine steps. Can you find that solution?
 
Of course, you can pose the problem for a lattice of any size. The problem is a nice application of the Pythagorean theorem, and its puzzle-like quality should be engaging. Moreover, students can adjust the level of the challenge by tackling a smaller or larger square.

The figures above were created by Gord Hamilton. When introducing the problem to students, he tells a version of the story about Theseus, the Minotaur, and Ariadne — thus the title of this post. You should make up your own story.

–Henri

For more about work on lattices and geoboards, go to…
a blog post
Geometry Labs
and follow the links therein.

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