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 (4n1) 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?
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 20050903 22:19:26 