Can you solve the vampire hunter riddle? - Dan Finkel
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 diagonally – as the light
does here – does 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 stakes” test,
and you passed with flying colors.
