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HINGES - An Illustration of Gauss-Jordan Reduction - The HINGES Activity - Stage 1

Przemyslaw Bogacki

The "Puzzle Leaflet"

In a rather tongue-in-cheek fashion, I posted the following document on the course Blackboard site:

"HINGES" - a virtual 3D puzzle

An excerpt from a leaflet attached to a virtual puzzle called "HINGES" manufactured by DRB Enterprises.

At the beginning of the game, you will see a cube containing three planes: red, green, and blue, as in the left picture below.

(This cube is just a small portion of the R3 space, since the planes themselves are infinite).

Your objective is to perform a sequence of legal moves to get each of the planes parallel to a different side of the cube, as in the picture on the right above.

Each legal move is composed of these actions:

  • Choose one of the three planes to be the door plane, another to be the wall plane - the remaining plane is the ghost plane.
  • Using the hinge line (where the door plane intersects the wall plane), tilt the door plane to a new position (as far as you want), as if the ghost plane weren't there.
Here are three possible initial choices for the ghost plane -- shown semitransparent -- and the corresponding hinge lines -- shown in black.

I chose the name "HINGES" for this puzzle because selecting a hinge between two planes and rotating the "door" plane on the hinge are its key ingredients. Additionally, I wanted to avoid using any mathematical terminology in the name or in the leaflet that might suggest some ideas prematurely.

After reading the puzzle description, students were asked to try to think of a strategy to solve the puzzle in as few steps as possible.

I posed the following question, although I was not hoping for too many correct answers at this early stage:

"Which important linear algebra procedure is being illustrated by the HINGES puzzle?"

Przemyslaw Bogacki, "HINGES - An Illustration of Gauss-Jordan Reduction - The HINGES Activity - Stage 1," Convergence (June 2005)