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Puzzle - 2
Last
time I had sent the following puzzle that was sent to me by Tanmay, son of a
good friend of mine.
There are
250 statements, each of which is either definitely true or definitely false.
Statement number 'n' says that 'n' statements, out of these 250, are false.
For example, statement number 50 states that 50 statements are false. The
question is, how many of these 250 statements are true?
We disagree on its answer. Tanmay thinks that first 125 statements are true.
I think only one namely 249th is true. His reasons are as follows,
Let us take the 250th statement first. Obviously, it cannot be
true, as that would make all statements, including that one, false. Now for
the 249th statement. Again, if it is true, then all statements
preceding this statement are also true. Remember, the nth statement
says atleast n statements are wrong, so that atmost 250-n
statements are right. If n = 249, atmost 1 statements is right, but
our assumption that atleast 249 are wrong means that the statement that
atleast 1 statement is wrong is also true. Thus, 1st statement is
true; likewise, all the first 249 statements are true. As such, one can
proceed like this for each of the statements. Finally, the 125th
statement will not be a paradox. It will be noticed that this paradox will
exist even for the first 124 statements. Thus, 125 statements are correct.
I think only one namely 249th statement is true. My reasons are as follows
250th statement cannot be decided, as it becomes a paradox if you apply to
it. If 249th statement is true then this makes other statement to be false.
This is consistent with the other statements. Let’s take 1st statement that
says that only one statement is false. It is false as 249 statements are
false. Similarly 250th statement is false as it says 250 statements are
false but only 249 are false.
Difference lies in assuming whether nth statement says that 'atleast' or
'only' n statements are false. In fact puzzle does not use either of the
words. Tanmay thought that it gives impression of atleast whereas I thought
that it gives impression of only. I never thought that interpretation of
puzzle was so important. I will wait for your comments.
Here is another one
There is a virus. It multiplies to seven in an hour. There is one virus in a
bottle. It takes 50 hours to fill up the bottle. How much time was taken to
fill up the 1/7 (one-seventh) of the bottle? How much would it take to fill
up 1/7 of the bottle if to begin with there were 7 viruses. |