1978 AHSME Problems/Problem 22
The following four statements, and only these are found on a card:
(Assume each statement is either true or false.) Among them the number of false statements is exactly
Solution
There can be at most one true statement on the card, eliminating and
. If there are
true on the card, statement
("On this card exactly four statements are false") will be correct, causing a contradiction. Therefore, the answer is
, since
are false and only the third statement ("On this card exactly three statements are false") is correct.
Solution 2
If all of them are false, that would mean that the th one is false too. Therefore,
is not the correct answer. If exactly
of them are false, that would mean that only
statement is true. This is correct since if only
statement is true, the card that is true is the one that has
of these statements are false. Therefore, our answer is
.
~Arcticturn