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Logic Puzzle Test

Once again you are on the Island of Knights, Knaves and Normals. In case you've forgotten, here Knights always tell the truth, Knaves always lie, and Normals sometimes lie and sometimes tell the truth. A new piece of information you have just discovered is that the inhabitants live in a caste system. Knights form the highest caste, Normals rank just below them, and Knaves rank last.

Trying to make your way to the airport to get out of here for good, you meet three residents: Andy, Betty, and Chris. You know that one is a Knight, one a Knave, and one a Normal. However, until they speak you are uncertain about who is which.

Andy says, "Betty is of higher rank than Chris."

Betty says, "Chris is of higher rank than Andy."

To settle the matter once and for all, you ask Chris, "Who has higher rank, Andy or Betty?"

Figure out the only possible answer Chris can give.

Use our three step logical problem solving procedure to prove Chris' answer. Be certain to proofread your work for clarity, spelling, punctuation, capitalization, complete sentences, legibility, neatness, and conciseness.

The Solution

Possible solutions (the correct one is highlighted)

Chris says that they are of equal rank.
Chris says that Andy has the higher rank.
Chris says that Betty has the higher rank.

Fact list

Knights always tell the truth.
Knaves always lie.
Normals sometimes lie and sometimes tell the truth.
One individual is a Knight, one a Normal, and one a Knave.
Knights are the highest caste.
Normals rank just below Knights.
Knaves are the lowest ranking caste.
Andy says, "Betty is of higher rank than Chris."
Betty says, "Chris is of higher rank than Andy."

Evaluation

1. Since we know that the castes are hierarchical, and that each is represented in the group, it is impossible for Andy and Betty to be of equal rank.
2. If Andy is telling the truth, he has to be the Knight or the Normal. This would mean that Betty must be the Normal or the Knight. Regardless, Chris would be the Knave. If Andy is lying, he has to be the Knave or the Normal. This would mean Betty must be either the Normal or the Knave. Here, Chris would have to be the Knight. So whether Andy is lying or telling the truth,

Chris cannot be the Normal.
1. If Betty is telling the truth, she has to be the Knight or the Normal. This would mean that Chris must be the Normal or the Knight. Regardless, Andy would be the Knave. If Betty is lying, she has to be the Knave or the Normal. This would mean Chris must be either the Normal or the Knave. Andy would have to be the Knight. So whether Betty is lying or telling the truth,

Andy cannot be the Normal.
1. Since neither Andy nor Chris can be the Normal, Betty has to be. Therefore if Chris is a liar, he must be the Knave. Andy, then, has to be the Knight. In this case, Chris would lie and say that Betty has the higher rank. If Chris is telling the truth, he would have to be the Knight. Andy, then, has to be the Knave. Here Chris would tell the truth and say that Betty has the higher rank. Thus, whether Chris is lying or telling the truth,

He has to answer that Betty has the higher rank.

This problem is an adaptation of one presented by mathematician Raymond Smullyan in his book, What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles (Prentice-Hall, 1978).