Weekly Challenge 14: Mad Robot
The first part of this journey consists of a sequence of forward
motions followed by 90 degree turns. Although we could just halve
repeatedly to find the number of turns, we might consider the
possibility that the numbers in the problems might change, so look
for a general proceduce for finding the number of motions.
In doing this, we need to be careful about the number of forward
turns and note that the WHILE(DISTANCE > 0.001) condition is
evaluated after each halving.
The forward motion, if it occurs, following the $n$th turn will be
of distance $1000\times 2^{-n}$.This will occur for each $n$ for
which
$1000\times 2^{-n}> 0.001$
Taking logs gives
$\log(1000) -n\log(2)> log(0.001)$
Rearranging gives
$n <
\frac{\log(1000)-\log(0.001)}{\log(2)}=\frac{\log(1000000)}{\log{2}}\approx
19.93$
This first becomes invalid AFTER after 20 turns. Thus, the
path of the robot is determined by $20$ motions. The directions
that the robot travels in these 20 motions are alternately NE,
SE, SW, NW, .... Since there are 20 motions in total there are 4
motions in each of these directions.
Let $T$ be the total distance travelled NE. Then
$T=
1000\left(1+\frac{1}{2^5}+\frac{1}{2^9}+\frac{1}{2^{13}}\right)=1000\times
\frac{8465}{8192}$
Note that the distance SE will be half of this, since each motion
is half the length of the preceding NE motion. Similarly, total
distance SW will be one quarter of $T$; total distance NW will be
one eight of $T$. Each of these motions contributes a component in
the $x$-direction and component in the $y$-direction.
The final coordinates will therefore be
$$
(x, y) =
\left(\frac{T}{\sqrt{2}}\left(1+\frac{1}{2}-\frac{1}{4}-\frac{1}{8}\right),\frac{T}{\sqrt{2}}\left(1-\frac{1}{2}-\frac{1}{4}+\frac{1}{8}\right)\right)=\left(\frac{9T}{8\sqrt{2}},
\frac{3 T}{8\sqrt{2}}\right)
$$