Determine which statements are True, and which statements are False

Question

Determine which statements are True, and which statements are False

1. There are 3 consecutive False statements
   
2. This is The First True Statement
   
3. There are equal numbers of True and False statements
   
4. There are not equal numbers of True and False statements
   
5. There are 4 consecutive False statements
   
6. The First and Last statements are False
   
7. There are 3 consecutive statements which are False, False, True.
   
8. There are 3 consecutive True statements
   
9. There are 3 consecutive statements which are True, False, True.
   
10. There are 4 consecutive True statements
    

This puzzle have more than 1 solution.

Answer 1

Final answer

There are five distinct possibilities: FTFTFFTTTT, TFFTFFTFFF, TFTFFFTTTF, FTTFFTTTFF, FTTFFTTFTF.

Proof

First, assume statement 5 is true.

This implies statement 1 is also true. So the four consecutive false statements cannot be 1,2,3,4; they must be either 6,7,8,9 or 7,8,9,10. So 7,8,9 are all false, which means there can be no sequences of statements of the form FFT or TTT or TFT.

So the sequence of statements 3,4,5 (ending in a true statement) must be FTT. But now statement 2 is either true or false, so the sequence 2,3,4 must be either TFT or FFT. Contradiction.

So statement 5 is false.

• Now assume statement 10 is true.

  Since statement 5 is false, the four consecutive true statements must be either 1,2,3,4 or 6,7,8,9 or 7,8,9,10. But if statement 2 is true, then statement 1 is false, so 1,2,3,4 cannot all be true. So 7,8,9 are all true. By our assumption, 10 is true and therefore 6 is false. We have FFTTTT for statements 5-10.

  Assume statement 1 is true, so that statement 2 is false. Note that exactly one of 3 and 4 is false, since they are opposites. So the three consecutive false statements must be 4,5,6, and the entire sequence is TFTFFFTTTT: six true statements and four false ones, making statement 3 false. Contradiction.

  So statement 1 is false, and there are no three consecutive false statements. Since 5 and 6 are false, this means 4 must be true, so 3 is false and 2 is true. Now the entire sequence is FTFTFFTTTT. There are no contradictions here, so this is a possible answer.

Let's see if there are any others: i.e. let's assume statement 10 is false.

• Assume statement 1 is true.

  This implies statements 2 and 6 are both false. Note also that exactly one of 3 and 4 is false.

  If 7 is false, then the sequence of false statements starting with 5,6,7 must continue until the end, so 8,9,10 are all false too, which means statement 5 is true, contradiction. So 7 is true, and we have TF??FFT??F.

  • If 9 is false, then 3 must be false (otherwise 1,2,3 would be TFT), so 4 is true, and 8 is false since there are no three consecutive true statements. This gives the entire sequence TFFTFFTFFF, which is another viable possibility.

  • If 9 is true, then the only way to get the three consecutive false statements required by 1 is if they are 4,5,6, so 4 is false and 3 is true. To make the counts of true and false statements agree, 8 must also be true, so we have TFTFFFTTTF, which is another viable possibility.

• Finally, assume statement 1 is false.

  Now 6 is true, so we have F???FT???F, and of course exactly one of 3 and 4 is true.

  • If 8 is true, then 6,7,8 must be the three consecutive true statements and 9 must be false (since we can't have four consecutive true statements). To avoid having a sequence TFT, 4 must be false, thus 3 is true, and then 2 must be true. So the entire sequence is FTTFFTTTFF, which is another viable possibility.

  • If 8 is false, then 9 must be true (otherwise 8,9,10 would be three consecutive false statements), so we have F???FT?FTF. If 7 were false, then the sequence 7,8,9 would be FFT; contradiction, so 7 is true. Now if 3 were false and 4 true, then the FFT sequence required by 7 must be 2,3,4, so 1,2,3 are three consecutive false statements, contradiction. So 3 is true and 4 false, and then 2 must be true to make the counts of true and false statements match. Thus the entire sequence is FTTFFTTFTF, another viable possibility.

Answer 2

This puzzle has surprisingly

five solutions: {{0, 1, 0, 1, 0, 0, 1, 1, 1, 1}, {0, 1, 1, 0, 0, 1, 1, 0, 1, 0}, {0, 1, 1, 0, 0, 1, 1, 1, 0, 0}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 0}, {1, 0, 1, 0, 0, 0, 1, 1, 1, 0}}.

There are no other solutions. Thanks @rand al'thor for pointing out a bug in how I selected 3-subsets.

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What happens is, that each of the 10 statements S1, S2, ..., S10, can be evaluated independently for any given 10-tuples. So, for example S1(1,1,1,1,1,1,1,1,1,1)=0, because in the 10-tuple {1,1,1,1,1,1,1,1,1,1} there are NO three consecutive 0s. The task is then to solve the equation [S1(x),S2(x),...S10(x)]=x which you can do by exhaustively going through all 1024 10-tuples. In essence, you are looking for a fixpoint of the vector-valued function F(x):=[S1(x),S2(x),...,S10(x)].

Answer 3

1. False

2. True

3. False

4. True

5. False

6. True

7. True

8. False

9. True

10. False

Or:

FTFTFFTFTF

Explanation:

7: There are 3 statements which are False, False, True. We could make it 5 as False, 6 as False and 7 as True. On 9, it's True False True. We could make 7 True (already), 8 false and 9 true. This fits in, completely. Now, let's start building up. At 2, it says that statement is true, so let's make that true. (don't remember the rest). I just realized this answer is wrong.

EDIT: I just realized this is wrong.