Alex, Bert, and Carl know a specific number 1 to 9. From the statements below can you determine the number knowing one of them made three true statements and one of them made three false statements?
Alex: The number is less than or equal to 5.
Bert: The number is 2, 4, 7, or 9.
Carl: The number is even.
Alex: Bert's first statement is false.
Carl: The number is at least 5.
Bert: The number is odd.
Carl: Alex's first statement is true.
Alex: The number is not 1, 2, 8, nor 9.
Bert: Carl made exactly one true statement.
Problem: Finding a number N(1-9) that Alex, Bert, and Carl know based on the statements they made and the following conditions: C1 One of them made 3 true statements. C2 One of them made 3 false statements.
Statements made
1) N is less or equal to 5. 2) N is either 2, 4, 7, or 9. 3) N is even. 4) Statement #2 is false. 5) N is greater or equal to 5. 6) N is odd. 7) Statement #1 is true. 8) N is neither 1, 2, 8 or 9. 9) Carl only made one true statement.
About the statements: Six of them(1, 2,3, 5,6, and 8) are about the number N and the others(4, 7, and 9) about the statements made. Alex made statements #1,4,8; Bert made #2,6,9 and Carl made #3,5,7
Statements #1 and #7 are mutually conditional. If one is true the other is also true and vice versa. Statements #2 and #4 are contradictory. The truth of one implies the falsehood of the other and vice versa. So are statements #3 and #6.
Solution: Based on the data given above build a truth table(3X9) with rows 1, 2, and 3 representing Alex(abbre.A), Bert(B), and Carl(C) respectively and the columns representing the statements from 1 to 9.
Making the assumption that statement #1 is true we can distribute the remaining values of the truth table(omitted here) of the rest of them as follow:
Statement# Truth value 1. T 2. F 3. T 4. T 5. T 6. F 7. T 8. F 9. F
We see from the truth value of Statement#1 is true that of #7 is also true. Thus, we have a T in row 1 and a T in row3. To satisfy condition C2 we must put 3F's in row2. But that make us to put T's in row 3 col. 3;in row 1 col. 4, and in row 3 col. 5 Then we put an F in row 1 col. 8. The first 2 because statements #2 and #4 are contradictory, and the same is true for statements #3 and #6. The next one (a T in row 3 col.5) because #9 is false. And in row 1 col. 8 because of condition C1. Now because statement #8 is false, N must be in the set {1,2,8,9}. Because statement#3 is true N=1 and N=9 are eliminated, leaving us with 2 and8. N=2 cannot be because statement #2 is false. Choosing N=8 will make statement #1 false, contrary to assumption. Therefore, our choice of a T for statement#1 is wrong.
Reverse the values in the truth table by assuming that statement #1 is false. Below is the new distribution of values:
Statements and truth values
1. (F) 2.(T) 3.(F) 4.(F) 5.(T) 6.(T) 7.(F) 8.(F) 9.(T)
Since statement #8 is false, N is in the set{1,2,8,9}. Because statement #1 is false, N=1 and N=2 are eliminated, leaving us with 8 and 9 as choices. But since statement #3 is also false, 8 is also eliminated. So the only choiceleft is 9.
ANSWER 9
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