Logical

An interesting logical puzzle that wasted way too much of my time at work today:

Imagine that these six words have been written on a chalkboard: cat, dog, has, max, dim, tag. Three students—Albert, Bernard, and Cheryl—receive a piece of paper with a different letter written on it. Each letter is included in the secret word, which is one of the words on the board. The students need to determine, using logic, what the secret word is.

Their teacher asks Albert if he knows the secret word. After looking at his letter, Albert says yes. Bernard takes a moment to think, and then says yes, he also knows the secret word. Cheryl is the last to say yes. Which one is the secret word?

DON’T READ ANY FURTHER IF YOU WANT TO TRY TO WORK IT OUT FOR YOURSELF.















My logic goes as follows (with NO help from the internet – HONEST!):

Albert must have a letter that is unique across the 6 words.  Therefore the magic word CANNOT be tag – which has no unique letters.

The magic word also CANNOT be max.  If it was, then Albert must have had an x, in which case Bernard must have had an m or an a.  But if Bernard had an m, then he wouldn’t know if the magic word was max or dim.  And if Bernard had an a, then he wouldn’t know if the magic word was cat, has or max.  Therefore max cannot be the magic word.

The magic word also CANNOT be dim.  If it was, then Albert must have had an i, in which case Bernard must have had a d or an m.  But if Bernard had a d, then he wouldn’t know whether the magic word was dog or dim.  And if Bernard had an m, then he wouldn’t know if the magic word was max or dim.  Therefore dim cannot be the magic word.

This leaves the potential magic word as either cat, dog or has.

If the magic word was cat, then Albert must have had a c and Bernard must have had a t, in which case Cheryl must have had an a.  If the magic word was has, then Albert must have had an h or s and Bernard must have had an h or s, in which case Cheryl must have had an a.  But if Cheryl had an a, then Cheryl would not know whether the magic word was cat or has.  Therefore, by elimination, the magic word has to be dog.