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 2n+1 Statements (Posted on 2005-06-13)
1. At least 1 statement among these 2n+1 are true.
2. At least 3 statements among these 2n+1 are false.
3. At least 5 statements among these 2n+1 are true.
...
2n. At least (4n-1) statments among these 2n+1 are false.
2n+1. At least (4n+1) statements among these 2n+1 are true.

How many statements are true? Which?

 No Solution Yet Submitted by Dustin Rating: 4.0000 (7 votes)

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 Got it!!! Comment 15 of 15 |
OK, a little brute force and looking for patterns. Using the square brackets [] to mean round down, the solution is:

[n/8] + 1 + [(n+[(n+9)/8])/3]

The sequence, starting with n=0 is:
1,1,2,2,2,3,3,4,5,5,6,6,6,7,7,8,9,9,10,10,10,11,11,12...

I can't explain the logic, I just charted it out and noticed a pattern of 2 different sequences, one effected by yet a third sequence.

I made a couple of assumptions. One is that when a statement is undefined (as it is for n=4 and every 8 from then on; 12, 20, 28...) it is NOT true.

The second assumtion is that a statement is true unless it MUST be false.

For the second part of the question, the following statements are true:

1. Statement 1 is always true.

2. For a given n=x, if a statement is true, then it is true for all n>x.

3. Odd numbered statements become true when n is divisable by 8. ie: when n=8, statement 3 becomes true. When n=16, the next odd numbered statement, 5 becomes true.

4. Even numbered statements become true when n+z is divisable by 3. Initially z=1, then it increases by 1 for every time that (n+1) is divisable by 8.

 Posted by Basil on 2005-09-03 22:19:26

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