# 207 geometry/topology/fixed.point.p

A man hikes up a mountain, and starts hiking at 2:00 in the afternoon
on a Friday. He does not hike at the same speed (a constant rate), and
stops every once in a while to look at the view. He reaches the top in
4 hours. After spending the night at the top, he leaves the next day
on the same trail at 2:00 in the afternoon. Coming down, he doesn't
hike at a constant rate, and stops every once in a while to look at the
view. It takes him 3 hours to get down the mountain.

Q: What is the probability that there exists a point along the trail
that the hiker was at on the same time Friday as Saturday?

You can assume that the hiker never backtracked.

geometry/topology/fixed.point.s

```         /\  /\               -----
/  \/  \              | | |
/   /\   \             | | |
/   /  \   \            | | |
\  /----\  /         ---|.|.|---
\/|    |\/          |  | | |  |
-----------          -----------
|         |             |   |
-----------             -----
```

--
stein.kulseth@tf.tele.no [X.400] stein.kulseth@nta.no [internet]

Place block A onto the x-y plane so that four of its corners are at
(0,0), (0,1), (4,0), (4,1) (I give x and y coordinates only because
the z coordinate will always be obvious). Place block B so four of
its corners are at (2,1), (2,2), (6,1), (6,2). Now place block C with
one 4x1 face on the x-y plane with one corner at (0,1) (tangent to
block A) and tangent to block B at (2,1). Note that the angle between
block A and block C is arctan(1/2), and a corner of block C will be at
a point with approximate coordinates (3.5777, 2.7888). Call this
point P.

Now place an identical configuration of blocks on top of the first
three as follows: block D with corners at (3.4,0.4), (4.4,0.4),
(3.4,4.4), (4.4,4.4), block E with corners at (2.4,2.4), (3.4,2.4),
(2.4,-1.6), (3.4,-1.6), and block F with one corner tangent to D at
(3.4,4.4) and one side tangent to E at (2.4,2.4).

If you have been plotting this on graph paper, then the following
will be clear:

Every block touches every other in its own layer. And A and B each
touch D and E, and block C touches F. Point P falls under block D, so
blocks C and D touch, and by symmetry so do blocks F and A. And the
edge of block C intersects the edge of block E at (2.4,2.2) so blocks
C and E touch, and by symmetry so do blocks F and B. Done!

-- David Karr (karr@cs.cornell.edu)

All the blocks are placed with their 2x4 face UP, although any face up
would have worked, as it turns out. The blocks are called AAAA BBBB CCCC,
etc.

```      AAAA
AAAA /_______
BBCC \
BBCC
BBCC
BBCC
/\
||
```

The two arrows point to the intersection of AC and BC.

Now take block "D" and place the top edge along the diagonal (between the
two arrows) so that the block extends SOUTH EAST of the line. Likely now
the block does not touch either A or B. So slide the block towards the
NORTH WEST so that it just touches A and B. You can now easily place
blocks E and F perpendicular to block "D" so that they both touch all of
ABC.....QED
--
Guy Cousineau

Continue to: