Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
A 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
Investigate how you can work out what day of the week your birthday
will be on next year, and the year after...
You sent in a large number of solutions to
this problem but many of them only considered the spider moving
horizontally or vertically but not diagonally across the sides of
the room. I have included below a net of the room with the path of
the spider as it takes the shortest route over the ceiling. I hope
it helps you to see what was going on. However the spider could
walk around the walls to get to the fly or along the floor -
risking a human foot!
Andrei of Tudor Vianu National College
calculated the distance the spider must travel in the original
problem and has worked out when it is best to go via the floor
instead of the ceiling.
The shortest distance is $7.40355$ m, i.e. the first situation.
This is because the dimension "wide" is larger than the dimension
"high"and the spider starts from the middle of the wall.
Now I analyze the situation when the fly goes down the wall. In
the first case, with the fly fixed, it was situated in the upper
middle of the face, so it was better for the spider to go on the
top of the box. When the fly arrives at the middle (height) of the
box, it is the same for the spider to go over the top or over the
bottom of the box. When the fly goes still further to the bottom,
the spider should go on the bottom of the box.