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:
On the other hand, this is a successful trip in six steps:
Of course, you can pose the problem for a lattice of any size. According to Charles R Greathouse IV and Giovanni Resta, the optimal solutions for square lattices with n = 1, 2, ..., 9 are: 0, 2, 4, 7, 9, 12, 15, 17, 20. (OEIS A226595) [Correction: Greathouse & Resta were solving a closely related problem: the path could start and end anywhere in the lattice.]
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.
For more about work on lattices and geoboards, go to...
- a blog post
- Geometry Labs
...and follow the links therein.