A bug is placed at one corner of a wire frame in the shape of a cube. This cube currently does not have any sugar for the hungry bug.
The bug crawls along the 12 wires of the frame searching for something to eat. At each of the 8 corners the bug randomly chooses one of the 3 wires to follow next (including the one it just traveled).
The moment the bug crosses the first wire to the next corner, a piece of sugar is placed on the original starting corner. At some point, the bug will return to the starting corner and reach the sugar.
What is the probability that the bug will have visited all 8 corners by the time it returns to the starting corner?
totCt=0; succ=0;
for trial=1:1000000
bug='b';
while bug(end)~='a'
switch bug(end)
case 'a'
r=randi(3);
switch(r)
case(1)
bug(end+1)='b';
case(2)
bug(end+1)='c';
case(3)
bug(end+1)='d';
end
case 'b'
r=randi(3);
switch(r)
case(1)
bug(end+1)='a';
case(2)
bug(end+1)='e';
case(3)
bug(end+1)='f';
end
case 'c'
r=randi(3);
switch(r)
case(1)
bug(end+1)='a';
case(2)
bug(end+1)='f';
case(3)
bug(end+1)='g';
end
case 'd'
r=randi(3);
switch(r)
case(1)
bug(end+1)='a';
case(2)
bug(end+1)='g';
case(3)
bug(end+1)='e';
end
case 'e'
r=randi(3);
switch(r)
case(1)
bug(end+1)='h';
case(2)
bug(end+1)='b';
case(3)
bug(end+1)='d';
end
case 'f'
r=randi(3);
switch(r)
case(1)
bug(end+1)='h';
case(2)
bug(end+1)='b';
case(3)
bug(end+1)='c';
end
case 'g'
r=randi(3);
switch(r)
case(1)
bug(end+1)='h';
case(2)
bug(end+1)='d';
case(3)
bug(end+1)='c';
end
case 'h'
r=randi(3);
switch(r)
case(1)
bug(end+1)='e';
case(2)
bug(end+1)='g';
case(3)
bug(end+1)='f';
end
end
end
if length(unique(bug))==8
succ=succ+1;
end
end
produces
succ
succ/trial
succ =
128945
ans =
0.128945
This is slightly more than 1/8, and the latter is within the range of possibilities though the next run was also high:
succ =
129077
ans =
0.129077
and the next:
succ =
129077
ans =
0.129077
(That was weird; it actually happened twice in a row)
then:
succ =
128772
ans =
0.128772
then:
succ =
128644
ans =
0.128644
So it looks like it's definitely higher than 1/8
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Posted by Charlie
on 2023-12-05 10:41:28 |
computer solution via simulation
