SAT1000 - P934

Logic Level 1

Four girls $\textup{Alice,Betty,Cathy,Dora}$ are called to wear black or red hats, and none of them can see the colors of their own hats.

They are informed that exactly $2$ of them wear black hats and the others wear red hats. After that, they decide to play hide-and-seek under the light.

Then $\textup{Alice}$ has seen $\textup{Betty's}$ and $\textup{Cathy's}$ hats, $\textup{Betty}$ has seen $\textup{Cathy's}$ hat, and $\textup{Dora}$ has seen $\textup{Alice's}$ hat, but not vise versa.

Suddenly, the light goes out, and $\textup{Alice}$ says: "I still don't know what color my hat is".

If what she says and what they are informed are true, which of the statement is true ?

$A.\ \textup{Betty can know the colors of all four girl's hats.}$

$B.\ \textup{Dora can know the colors of all four girl's hats.}$

$C.\ \textup{Betty can know the color of Dora's hat, and vise versa.}$

$D.\ \textup{Betty, Dora can know the color of their own hats.}$

Have a look at my problem set: SAT 1000 problems

B

C

A

D

1 solution

David Vreken
Jul 16, 2020

If Betty and Cathy had the same color hats, then Alice would know what color hat she had, since there are only two black hats and two red hats. But since she says that she still doesn't know what color her hat is, that must mean Betty and Cathy have different color hats.

Since Betty and Cathy have different color hats, that means Alice and Dora must have different color hats, too.

Betty saw Cathy's hat, so she can conclude that her hat was the other color, and Dora saw Alice's hat, so she can conclude that her hat was the other color.

Therefore, Betty and Dora know the color of their own hats .

SAT1000 - P934

1 solution

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