Problem 12: Handshakes

Bill and Hillary go to dinner with two other couples. Some handshakes are exchanged, but no one shakes hands with their spouse, with themselves, or with anyone else more than once. When Bill asked each person how many handshakes they had exchanged, they reported 0, 1, 2, 3, and 4. How many handshakes had Hillary exchanged?
(Due April 18, 2008)

Problem 11 winners are ...

Christopher P. and Ho Yi L.

Congratulations!!

Problem 11: Easter Egg Hunt

Ms. Jabbour's rectangular yard is to be used for the Easter egg hunt this year. The yard is 30 feet wide, 40 feet long and has two 3-foot wide cement paths as shown in the diagram. Her dog Franky is tied to one corner with a 6-foot long leash (assume that Franky cannot reach the cement paths). What is the maximum possible area in which the eggs can be placed if they cannot be placed on the cement walkways or where Franky can reach? (Due March 28, 2008)

Problem 10 winner and solution

Congratulations to Manjit from the main office for submitting a correct solution!

Solution:
Let's first list all possible sets of three cards with different numbers in increasing order having a sum of 13:

(1, 2, 10)(1, 3, 9)(1, 4, 8)(1, 5, 7)(2, 3, 8)(2, 4, 7)(2, 5, 6)(3, 4, 6)

When Alyssa looked at the card on the left but could not tell what the other two cards were, we knew she was not looking at “3” since 3-4-6 is a unique solution and she would know the other two cards in that case. So we can rule out the last case.

(1, 2, 10)(1, 3, 9)(1, 4, 8)(1, 5, 7)(2, 3, 8)(2, 4, 7)(2, 5, 6)

Brandon then looked at the rightmost card and said he could not tell what the other two cards were. By the same reasoning, we can rule out 1-2-10, 1-3-9, and 2-5-6 because each has a unique rightmost card.

(1, 4, 8)(1, 5, 7)(2, 3, 8)(2, 4, 7)

When Christina looked at the middle card and said she could not tell what the other two cards were, by the same reasoning, we can rule out 1-5-7 and 2-3-8 since 5 and 3 are unique numbers in the middle. That leaves us 4 as the only possible middle number. So the answer is 4.

(1, 4, 8)(2, 4, 7)

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)