Beam of Light

This month’s post is based on a problem that appears in Martin Gardner’s Sixth Book of Mathematical Diversions from Scientific American. Below (and here) is a Web Sketchpad model of an orderly forest. There is a tree at every point whose x– and y-coordinates are both integers. These are the green points. You are standing at the origin.

It is night and the forest is dark. You shine a strong flashlight towards trees in the first quadrant. The flashlight beam is represented by the yellow portion of the ray. The gray part of the ray represents the portion of the light’s path that is blocked by one or more trees.

You can drag point T to change the direction of the light. What do you notice? What do you wonder?

If you drag point T to (9, 6), you’ll see that the tree is blocked by the tree at (3, 2). There is another tree that obscures it as well, and you can see that tree by pressing Show other trees in path (if any).

Here are some questions to consider:

  • What can you say about the coordinates of trees that can be illuminated by the flashlight beam? Can you predict which trees can be lit without using the Web Sketchpad model?
  • Name five trees that you cannot see because the light is blocked by the tree at (5, 1).
  • If the flashlight beam cannot reach a particular tree, how can you determine the tree nearest to the origin that blocks the beam?
  • Name a tree you cannot see because 10 trees block the light to it.
  • If the y-coordinate of a tree is one more than its x-coordinate, can the flashlight beam reach the tree?
  • Think about trees whose x-coordinate is 31. Which of these trees are you not able to light?
  • If the tree at point (a, b) is visible, what, if anything, do you know about the tree at (b, a)?
  • In an infinitely large forest, is it possible to point the flashlight beam in a direction that does not illuminate a single tree? (Assume that point T is free to be dragged anywhere, not just to the lattice points.)

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