For the list of 2n statements below, state how many are true in terms of n:

1: There is at least 1 true statement below this one.
2: There are at least 2 true statements below this one.
3: There are at least 3 true statements below this one.
...
n-1: There are at least n-1 true statements below this one.
n: There are at least n true statements below this one.

n+1: There are at least n false statements above this one.
n+2: There are at least n-1 false statements above this one.
n+3: There are at least n-2 false statements above this one.
...
2n-1: There are at least 2 false statements above this one.
2n: There is at least 1 false statement above this one.

It would seem to make sense for the middle n to be false, but the details seem elusive.

With n=1 the system is paradoxical, with the first statement claiming the second to be true while the latter claims the first to be false.

For n=2 it does seem the middle two are false while the first and last are true.

For n=3 it is still the middle 2 that are false while the top two and bottom two are true.

With n=4 it seems paradoxical again with the first and last two seeming to be true and the middle two seeming false, but the 3rd and 6th flip-flop endlessly, with no stable equilibrium.

For n=5 the middle 4 are false while the first 3 and last 3 are true.

For n=6, the first and last 4 are true and the middle 4 are false.

 

It would seem that in the limiting case, the top and bottom thirds (2n/3) are true while the middle third are false, but when n is not divisible by 3, the list is sometimes paradoxical and at other times something close to division into thirds. 

Posted by Charlie on 2012-12-07 16:59:56