Time for MATH MINUTE! (provide your favorite theme music
here).
Get out
your paper & pencil, because I have a new puzzle for you!!
Hints/Responses: PUZZLE #87 Indiana Smith and the Museum Mirrors
Mystery
A. The
Museum of Modern Sculpture was having problems.
Several
times in recent months something or someone was removing pieces of priceless
sculpture from the main gallery at night.
The
gallery was a long narrow room with mirrors down one side and windows down the
other side.
What
to do?
Call
in Indiana Smith to check out the situation of course.
One
night he quietly slipped into the gallery with his trusty torch (flashlight).
He knew from his studies of mathematics and physics that he could shine the light
on the mirror and see around and behind the sculptures.
Also,
and more importantly, he knew exactly where any objects (including those which
did not belong there) were in the gallery by the Principle of Reflection*.
This
gave him not only the exact location of something, but also the shortest route
to the something around the sculptures by going directly to the mirror at a
special point and then to the object.
How
did Indiana Smith find the shortest path to the mystery something?
Hint:
The following sketch was found on a crumpled piece of paper on the floor the
next morning. What does it mean?
- -- - - - - - - - -
- - - - - - - - - - - - - - - - --- - -
- -
- - - - - - - -- -- -- - - -- - - - - - - - - - -- - - - --- - - --
A review of the
sketch found above shows 3 angles which are all the same, plus some sides of 2 triangles
which are equal.
This
follows by using reflection in the mirror as follows.
First,
if you reflect the position of IT (what is IT by the way?), also marked as
point B, in the mirror you get point B/. This is what Indi sees when he looks
in the mirror. The position of IT is reflected to appear to be at point B/, and
the shortest route from Indi to B/ is the straight line through A.
If
you can connect the position of Indi and B/ with a straight line which crosses
the mirror at point A, this makes the angle between the line from Indi to A and
the mirror the same as the angle between the line AB/ and the mirror.
But
the reflection of the triangle made by A, B/, and the point C where the line BB/
crosses the mirror determines another triangle made by A, B, and C, and the 2
triangles are equivalent ! (why?)
So
the angle between Indi and A to the mirror is the same as the angle between IT
and A to the mirror.
This
is what the Principle of Reflection tells us.
This
means the fastest route for Indi to get to IT is to go to point A at the mirror
and turn so that the angle is the same with the mirror, and then go directly to
IT.
By
following this plan Indi was able to catch IT, who btw happened to be a
disgruntled employee dressed as a piece of sculpture.
Case
solved!
B. Indiana
Smith finds himself in the Mohave Desert without any water, and he must get across
140 miles of desert as quickly as possible in order to rescue the precious
crystal blob from the villain NAZTY.
In
order to do this he has to get some water and fast.
He
recalls that a steam runs along side the area where he finds himself. He estimates
that he is 100 miles from the steam, the crystal blob is 40 miles from the
steam, and the distance between him and the blob is 140 miles if they were at
the stream.
What
is the shortest route to the stream and to the blob?
Hint:
Make a sketch of the geometry of the puzzle like Indiana Smith did in Part A
above.
Here is a sketch of
the geometry of the above sketch, with the reflection principle added. As you
can see the shortest route from Indi to the reflected blob, blob/, is the
straight line through point A, so the shortest route from Indi to the blob is
the to the point A (to get some water) and then to the position of the blob.
Simple.
BONUS PUZZLE: Using
the ideas from above, see if you can solve this puzzle.
A
cowgirl has her horse out in the area near the barn as below, and she wants to
take it to the barn. But first she wants to take it to the pasture to feed it
some grass, then take it to the stream to get some water, then take it to the
barn.
What
is the shortest route to accomplish this sequence of tasks?
Hint: Check out the
following geometric analysis of this setting with some reflections added, as
used above.
*Principle
of Reflection: When a ray of light bounces off a flat mirror the angle of the
incoming ray with the perpendicular to the mirror is equal to the angle of the
outgoing ray with the perpendicular to the mirror. This means that the angles
of the ray with the mirror are also equal.
Have
fun!
Send
your comments, ideas and solutions before Monday to the email below, and
in the subject line be sure to put
MM in the subject line
Visit
us here online at:
http://www2.potsdam.edu/parksjm/MM1.1.htm
to see the results every Friday.
See you next time on MATH MINUTE! (theme music fades out here).