The Logical Lie Detection - Puzzle

Three Paley brothers and three Thomson brothers operate a company that manufactures lie detectors. Three of these six men always tell the truth, and three always tell lies; neither set of brothers consists exclusively of liars. 

Some recent statements from the six men are recorded below. 

Can you find the six men's full names, and tell which men tell the truth and which tell lies?

1. Alan: "Both my brothers tell lies."

2. Boris: "Both my brothers tell the truth."

3: Chuck: "Alan and Boris are both liars."

4. Dalman: "Chuck and I are brothers."

5. Edwin: "Boris and I are brothers."

6. Finney: "Edwin tells the truth."

7. Finney: "Boris is one of the Paleys."

Click here is the SOLUTION of the puzzle! 

The Logical Lie Detection - Solution

What was the puzzle?

The statement given by all the six persons are - 

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1. Alan: "Both my brothers tell lies."

2. Boris: "Both my brothers tell the truth."

3: Chuck: "Alan and Boris are both liars."

4. Dalman: "Chuck and I are brothers."

5. Edwin: "Boris and I are brothers."

6. Finney: "Edwin tells the truth."

7. Finney: "Boris is one of the Paleys."

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1] Since as per given information, neither set of brothers consists of entirely all liars (hence truth tellers), 3 liars (or 3 truth tellers) must be distributed as (2, 1 or 1, 2) among 2 groups of 3 brothers.

2] Boris says his both brothers are truth tellers. If his statement is true then there will be 3 brothers telling the truth which is impossible. Hence, Boris is a liar.

3] If Alan's statement (1) is truth then Chuck must be lying in his statement (3). And if Alan is lying then Chuck must be telling the truth. 

 That is one of the Alan or Chuck is a truth teller and other is a liar.

4] So far we got 2 liars (Boris and Alan/Chuck) and 1 truth teller (Chuck/Alan). Since, there are total of 3 liars & 3 truth tellers in total, there must be 2 truth tellers and 1 liar among Dalman, Edwin and Finny.

5] If Finney is lying then his statement (6) suggests that Edward is also liar. But we can't have 2 liars among Dalman, Edwin and Finney as deduced above. Hence, Finny must be telling the truth and hence Edwin.

6] The true statement (5) of Edwin suggests that Edwin and Boris are brothers and as per truth teller Finney, surname of Boris & hence of Edwin is Paley.

7] As Finney is telling the truth (and hence Edwin), the Dalman must be lying. This way, we have 2 truth tellers and 1 liar among Finney, Chuck and Dalman as deduced in step 4.

8] The lying statement (4) of Dalman suggests that Chuck and Dalman are not the brothers. Hence, one of them is Paley and other is Thomson.

9] So third brother of Boris and Edwin must be either Chuck or Dalman. So, Alan and Finney must be brothers having surname Thomson.

10] Since, Finny is telling the truth the statement (1) of Alan (suggesting both of his brothers are liars) must be a lie. 

11] And if Alan is lying then Chuck must be telling the truth (STEP 3).

12] Now, if Chuck is third brother of Boris and Edwin Paley, then statement of Boris (2) would be true and all Paley brothers would be truth tellers which is impossible.

13] Hence, Dalman who is liar (STEP 7) must be third brother of Boris and Edwin Paley.

14] Obviously, since Chuck isn't brother of Dalman, he must have surname Thomson like Alan and Finney.

CONCLUSION : 

Full Name : Alan Thompson,    Behavior : Liar
Full Name : Boris Paley,          Behavior : Liar
Full Name : Chuck Thomson,   Behavior : Truth teller
Full Name : Dalman Paley,      Behavior : Liar
Full Name : Edwin Paley,         Behavior : Truth teller
Full Name : Finney Thomson,   Behavior : Truth teller
 

The Race Between 2 Brothers

Zachary challenges his brother Alexander to a 100-meter race. Alexander crosses the finish line when Zachary has covered only 97 meters.

The two agree to a second race, and this time Alexander starts 3 meters behind the starting line.

 
If both brothers run at the same speed as in the first race, who will win?

He will win the second race for sure! 

Source 

Winner of The Race Between 2 Brothers

What was the race of?

Let's assume that, Alexander completes the first race in time 't'. That means, he reaches at the finish line after running 100m after time 't' since start of the race. In the same time interval, Zachary could reach only 97m.

Now, in second race too, Alexander covers 100m once again in time interval 't' & Zachary runs 97m distance. Since, Alexander started 3m ahead of start line, at this point of time both are at the same point with 3m left to complete the race. 

 
Since, Alexander had won first race with faster speed & speed of both are unchanged in second race, it's clear that Alexander will take less time to cover leftover 3m distance. Hence, Alexander will be winner of the second race. 

MATHEMATICAL APPROACH:

Let's suppose Alexander took 10s to complete the first race. Then, his speed is 100/10 = 10m/s.

In 10s, the Zachary could run only 97m. So, his speed is 97/10 = 9.7m/s.

In the second race, their respective speeds are unchanged but Alexander has to run 103m to reach at the finish line compared to 100m of Zachary.

Hence, time taken by Alexander to reach at the finish line = 103/10 = 10.3s and that taken by Zachary = 100/9.7 = 10.30929s.

It's clear that Zachary needs more time to finish this race too. Hence, Alexander will be the winner in this race as well. He beats Zachary by 100 - (10.3x9.7) = 0.09m.