Problem 1 (The Three-Cap Problem)

One cold winter morning in Maine a mother insists that her three daughters, Ann, Betty, and Cindy wear caps to school and so she randomly picks 3 caps from a pile of 5 caps, three of which are white and two are red. The girls stand in line, one behind the other, waiting for the mother to put on their caps. Cindy stands in the back and can see both of the other girls; Betty stands in the middle and can see Ann who stands in the front; but Ann standing in the front cannot see either of the other girls. The mother then puts a cap on each of the girl's heads. For fun, the mother asks the girls if they can guess the color of their own caps. (The girls know their mother picked the caps from the 5 caps, three white and two red.) Cindy, who is in the back, says she has no idea the color of her own cap. Betty, hearing what Cindy says, also says she has now idea the color of her cap. But Ann, hearing what Cindi and Betty say, says she does knows the color of her cap. How is Ann able to determine the color of her cap ? What is the color of Ann's cap ?

Solution

If there is one thing useful in solving complicated problems, it is pictures. Let's make a simple drawing illustrating the three girls standing in line with Cindi in the back, Betty in the middle, and Ann in the front, keeping in mind that none of the girls can see their own caps, but Cindi can see the caps of Betty and Ann, Cindi can see the cap of Ann, and Ann cannot see any of the caps. If we now call R = RED CAP and W = WHITE CAP, the possible colors of the caps on the girl's heads are :

Cindi-Betty-Ann (initial possible colors)

1. WWW
2. RRW
3. RWR
4. RWW
5. WRR
6. WRW
7. WWR

Now, believe it or not, the fact Cindy (in the back) has no idea the color of her own cap tells us a lot since if Betty and Ann both wore red hats, then Cindi would know her cap was white. Hence, we can eliminate the possibilities that both B and C wear red caps and are left with five possibilities:

Cindi-Betty-Ann (possible colors after Cindi's announcement)

• RRW
• RWR
• RWW
• WRW
• WWR
• WWW

But now, Betty starts to think ! (These girls are smart.) Upon hearing what Cindi says, Betty knows that both she and Ann do not both wear red, and so if Ann wore red, Betty would announce she was wearing white. But Betty announces she doesn't know her own color and so Betty's announcement tells Ann (in fact it tells all three girls!) that Ann must be wearing white! In other words, we throw out possibilities 3 and 7, leaving the following possibilities for the cap colors:

Cindi-Betty-Ann (possible colors after Betty's announcement)

• RRW
• RWW
• WRW
• WWW

Hence, we know that Ann must be wearing a white cap, but do not know the colors of the caps of Cindi or Betty.

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Last modified on Friday, February 12, 1999