This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
A sheet of paper has statements numbered from 1 to 100. Statement n says
"exactly n of the statements on this sheet are false." Which statements are
true and which are false? What if we replace "exactly" by "at least"?
From _Litton's Problematical Recreations_
It is tempting to argue as follows:
At most one statement can be true (they are mutually contradictory).
If they are all false, statement 100 would be true, which is no good.
So 99 are false, and statement 99 is true.
If replaced by "at least", and the "real" number of false statements is
x, then statements x+1 to 100 will be false (since they falsely claim
that there are more false statements than there actually are). So, 100-x are
false, ie. x=100-x, so x=50. The first 50 statements are true, and statements
51 to 100 are false.
However, there is a hidden and incorrect assumption in this argument.
To see this, suppose that there is one statement on the sheet and it
says "One statement is false" or "At least one statement is false,"
either way it implies "this statement is false," which is a familiar
paradoxical statement. We have learned that this paradox arises because
of the false assumption that all statements are either true or false.
This is the hidden assumption in the above reasoning.
If it is acknowledged that some of the statements on the page may be
neither true nor false (i.e., meaningless), then nothing whatsoever can
be concluded about which statements are true or false.
This problem has been carefully contrived to appear to be solvable (like
the vacuous statement "this statement is true"). By changing the
numbers in some statements and changing "true" to "false," various
circular forms of the liar's paradox can be constructed. A much
more complicated version of the same problem is:
1) At least one of the last two statements in this list is true.
2) This is either the first true or the first false statement in the
3) There exist three consecutive false statements.
4) The difference beween the numbers of the last true statement and
the first true statement is a factor of the unknown number.
5) The sum of the numbers of the true statements is the unknown
6) This is not the last true statement.
7) Each true statement's number is a factor of the unknown number.
8) The unknown number equals the percentage of these statements which
9) The number of different factors which the unknown number has
(excluding 1 and itself) is more than the sum of the numbers of the
10) There are no three cosecutive true statements.
What is the number?
The incorrect but plausible solution is:
By 2, either way 1 must be false, and then so must both 9 and 10.
6, if false, says "This is the last true statement", which gives
a paradox, thus 6 must be true, and so must 7 and/or 8.
7 and 8 cannot both be true, as the number had to be a multiple
of 6,7,8 , that is a multiple of 168 (by 7), and less than 100 (by 8)
If 8 is false, then 3 is true (8,9,10 is false), if 8 is true, then
3 is false (3 cannot be true when both 6 and 8 are true, then there
are no three consecutive statements left).
So we have either
A) F X F X X T F T F F
B) F X T X X T T F F F
In A), 4 and 5 must be true (by false 10), and 2 may be true or false.
So by 5 the number shall be either 27 (2+3+4+5+6+7) or 25 (3+4+5+6+7).
None of these can fullfill 8, though, so A) is out leaving us with B)
Now, (by 7) 3,6 and 7 are factors, the number must be a multiple of
42, as 2+3+4+5+6+7=27, 5 must be false.
By false 10, 2 and 4 must be true, that is 5 shall be a multiple
of the number. Now the number must be a multiple of 2,3,4,5,6 and 7,
that is a multiple of 3*4*5*7=420. 420 has 22 different factors
and the sum 2+3+4+6+7 = 22, so the only multiple of 420 that
fulfills false 9 is 420.