Hey this is Presh Talwalkar.
A teacher writes six words on the board: cat, dog,
has, max, dim, and tag. The teacher hands a
piece of paper to Albert another piece
of paper to Bernard and another piece of
paper to Cheryl. The teacher announces
that each paper contains a different
letter from one of the words on the board.
The teacher asks aloud, "Albert do you know the word?"
Albert replies, "Yes."
He does know the word.
The teacher then asked aloud, "Bernard, do you know the word?"
Bernard, after a moment of thinking replies, "Yes."
He also knows the word.
The teacher then asked if Cheryl
knows the word and Cheryl, after a moment
of thinking also replies, "Yes."
She also knows the word.
Albert, Bernard and Cheryl always aced their logic exams.
What is the secret word that their three letters spell?
Can you figure it out?
Give this problem a try and
when you're ready keep watching the
video for the solution.
We're going to analyze the letters in
each of the words in relation to the
responses that Albert, Bernard, and Cheryl give.
The teacher first asked if Albert
knows the word and he responds that he
does know the word.
This means that Albert's letter can
uniquely identify a word.
In other words,
the frequency of the letter that he got
is exactly one.
For example,
consider if albert had gotten the letter C.
The letter C only appears in the word cat
so Albert had gotten the letter C he would
be able to figure out that the word is cat.
In contrast suppose Albert had
gotten the letter A.
Well the letter A appears in multiple words,
so there would have been no way for Albert to figure out
which word was the word that they got.
Since Albert replied he was able to
identify the word that means he did not
get the letter A so we need to find out
which letters appear exactly once in
this list of words.
We can go through all of the words and we can figure out which
of these letters uniquely appear one
time in this list.
These are the letters: C, O, H, or S, X and letter I.
Notice in word tag the letters T, A and G
all appear multiple times in this list.
So since Albert replied that he could
identify the word, that means he didn't
have any of these letters. The word
cannot be TAG.
The teacher then asked Bernard if he knows the word.
Bernard, after a moment of thinking, also replies
yes so Bernard has heard Albert's answer
and wants to continue
the logic, so from this list of words, and
the letters that are remaining, Bernard
would be able to uniquely identify which
word they all had.
so we go through this list.
It could be the letter T.
It could be the letter G.
It could be the letter H,
if Albert had gotten the letter S also
it could have been the letter S if
Albert had gotten the letter H. These are
the possible choices the other letters
appear more than once in this list of
remaining letters that Bernard could
have gotten.
What this means is that the
word could not be 'max' or 'dim' so we can
eliminate those words.
Now Cheryl, who has
heard all of these, is then asked if she knows the word.
She looks at her letter
and thinks about what Albert & Bernard
said and also says, "Yes, I can figure out
what the word is." So, continuing the logic,
one of the remaining letters has to
uniquely identify the word. Well, there
are only two letters left there's a
letter A and there's a letter D.
The only unique letter left is a letter D because
the letter A appears in both the words
'cat' and 'has' so Cheryl must have gotten
the letter D and it could not have been
the word cat or has.
The secret word is 'dog.'
Albert would have gotten the letter
O, Bernard would have gotten the letter
G, and Cheryl would have gotten the
letter D. Did you figure it out?
Thanks for watching this video. Please subscribe
to my channel. I make videos on math and
game theory. You can catch me on my blog
"Mind Your Decisions" that you can follow
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catch on social media @PreshTalwalkar
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