Problem 10: Three Cards

Three cards are drawn from a regular deck of 52 cards and are placed face-down on a table. Alyssa, Brandon, and Christina are told that

a. the numbers are all different (Ace = 1, Jack = 11, Queen = 12, King = 13)
b. the sum of the three numbers is 13, and
c. they are in increasing order, left to right

First Alyssa looks at the number on the leftmost card and says, “I don’t have enough information to determine the other two numbers.” Then Brandon looks at the number on the rightmost card and says, “I don’t have enough information to determine the other two numbers.” Finally, Christina looks at the number on the middle card and says, “I don’t have enough information to determine the other two numbers.” Assume that each person knows that the other two reason perfectly and hears their comments, what is the number on the middle card? (Due March 7, 2008)

Problem 9 Solution:

This is a variation of the famous Monty Hall problem or Monty Hall paradox. The original problem was based on the game show Let's Make a Deal where you are given a choice of three doors where behind one door is a new car, and behind the other two, goats. No matter which door you choose, the host, knowing what is behind each door, always opens one of the remaining doors that has the goat, and asks if you want to switch. Is it to your advantage to switch?

The answer to this problem is counterintuitive and has generated many heated debates. You can find more information about this interesting problem by searching for “Monty Hall” on the internet. Here is an explanation of why switching is better:

Below are the three possible scenarios:

Door1 Door2 Door3
Goat Goat Car
Goat Car Goat
Car Goat Goat

If the contestant picks door 3, and the host opens a door with the goat (either door 1 or 2 in case 1, door 1 in case 2, and door 2 in case 3), then in two of the three cases shown above, he will win the car by switching. So the probability of winning by switching is 2/3. Thus it is advantageous to switch.

Problem 9: Valentine's Day Gift

Suppose for a Valentine's Day gift you are asked to choose only one of three identical looking boxes of chocolates, but you are told that inside one of the boxes are chocolates AND a diamond ring! You pick a box, say the first on the left, and your Valentine, who knows what's inside the boxes, opens another box, say the middle one, which has only chocolates. He then says to you, "Do you want to pick the box on the right?" Is it to your advantage to switch your choice? Please explain.
(Due February 15, 2008)

Problem 8 winner is ...

Vivian! Good job.

This is an altered version of the famous "Census Taker Problem". There are exactly two triplets with a product of 72 and equal sums. This is necessary in order to confuse the census taker -- even though he sees the house number, he still cannot tell which triplet is the answer, until he finds out that there is an "oldest child". Can you find another number that has this property and can be used in a similar problem?