Well, all right, this is also for you illogicians out there.

I hand you a sheet of paper with 100 numbered statements on it. They read like this:

- 1. Exactly 1 of the statements on this sheet is false.

2. Exactly 2 of the statements on this sheet are false.

3. Exactly 3 of the statements on this sheet are false.

...

99. Exactly 99 of the statements on this sheet are false.

100. Exactly 100 of the statements on this sheet are false.

*n*on the sheet reads: "Exactly

*n*of the statements on this sheet are false."

**Question**: Which statements are true and which are false?

There is, of course, a bonus. Replace "exactly" with "at least" all 100 times. Try the same question. Does your answer change? Why and how and wheretofore?

There is, of course, a second bonus. Still with the 100 "at leasts", add a 101st statement. ("101. At least 101 of the statements on this sheet are false."). Try the same question. Does your answer change? Why and how and wheretofore hereunder?

## 4 comments:

1. Statement 99 is true. The rest are false.

2. First 50 statements are true, the rest are false.

3. No true statements and no false statements.

#3 is wrong in the previous comment. Correct answer: first 50 true, the last 50 false. 51st is neither true nor false.

The previous comment corrects a mistake with another mistake. The 51st statement is simply false. The answer simply does not change.

A formal solution here.

Case 1: The word 'exactly' is used.

All the 100 statements are disjoint statements (i.e., none of them implies any of the others, in fact, any one of them implies none of the others!), hence only one of them can be true. Which means 99 of them must be false, which means

statement 99 is true, while the rest are false.Case 2: The word 'atleast' is used.

Two observations:

1) If statement

nis true (2 <=n<= 100), statementn-1 is also true.2) If statement

nis false (1 <=n<= 99), statementn+1 is also false.Therefore, there is a number

nsuch that statements 1 throughnare all true, while statementsn+1 through 100 are all false. The strongest of the true statements is statementn, which means atleastnstatements are false. Also, since statementn+1 is false, atmostnstatements are false. Together, they imply that exactlynstatements are false. Now, we also know that 100-(n+1)+1 = 100-nstatements are false. Hence, we have the equation 100-n=n, orn= 50. Thus,statements 1 through 50 are all true, while statements 51 through 100 are all false.Case 3: The word 'atleast' is used, with 101 statements. Then the above equation becomes 101-

n=n, which yieldsn= 50.5, which is absurd, hence in this case, the set of statements is not self-consistent, andno solution can be found.Post a Comment