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  • The greatest challenge a vampire hunter can take on

  • is to bring sunlight into a vampire's lair.

  • You've stealthily descended into the darkness of a vampire cave,

  • setting a sequence of mirrors as you go.

  • When the sun reaches the right angle in the sky,

  • a focused beam of light will ricochet along the mirrors,

  • strike your diffuser,

  • and illuminate the great chamber where the vampires sleep.

  • You set the final mirror

  • and sneak through an opening in the corner of the great chamber.

  • The diffuser must be wall-mounted,

  • but the walls are crowded with coffins,

  • which you don't dare disturb.

  • The only open spots are in the other three corners of the room.

  • The light will enter through the southwest corner at a 45 degree angle

  • and bounce off the perfectly smooth metallic walls

  • until it hits one of the other three corners.

  • But which corner will it hit?

  • You know the room is a rectangle 49 meters wide and 78 meters long.

  • You could probably find the answer

  • by drawing a scale model of the room and tracing the path of the light,

  • but the sun will be in its place in just minutes,

  • and you've got no time to spare.

  • Fortunately, there's a different way to solve this puzzle

  • that's both simple and elegant.

  • So in which corner should you place the diffuser

  • to flood the vampire lair with sunlight?

  • Pause the video if you want to figure it out for yourself.

  • Answer in 2

  • Answer in 1

  • You could tackle this problem by examining smaller rooms,

  • and you'd find a lot of interesting patterns.

  • But there's one insight that can unravel this riddle in almost no time at all.

  • Let's draw the chamber on a coordinate grid,

  • with the Southwest corner at the point (0,0).

  • The light passes through grid points

  • with coordinates that are either both even or both odd.

  • This is true even after it bounces off one or more walls.

  • Another way of thinking about it is this:

  • since the light travels at a 45 degree angle,

  • it always crosses the diagonal of a unit square.

  • Traveling 1 meter horizontally changes the x coordinate

  • from even to odd or vice versa.

  • Traveling 1 meter vertically changes the y coordinate

  • from even to odd or vice versa.

  • Traveling diagonallyas the light does heredoes both at once,

  • so the x and y coordinates of any points the light passes through

  • must be both even, or both odd.

  • This observation is more powerful than it seems.

  • In particular, it means that we have a way to identify the kinds of points

  • the light won't ever go through

  • If one of the coordinates is even and the other is odd,

  • the light will miss them.

  • That means it'll miss the top two corners of the room,

  • since those points have one even and one odd coordinate.

  • The Southeast corner is the only option for the diffuser.

  • And indeed, when that precious beam of sunlight enters the hall,

  • it bounces between the walls

  • and strikes the Southeast corner, spot on.

  • The vampires, sensing the intrusion,

  • burst from their coffins and turn to dust in the light.

  • It was a “high stakestest,

  • and you passed with flying colors.

The greatest challenge a vampire hunter can take on

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