Alice and Bob have just signed the lease on their exciting 1

Alice and Bob have just signed the lease on their exciting 10 foot by 20 foot fully-
furnished NYC studio apartment! They have agreed to share the rent equally (50% each). Fortunately for them, the apartment has 2 doors, one on the East wall and one on the West wall of the apartment. They plan on splitting the apartment by installing a curtain that runs North-South, dividing the apartment into two rectangular ”rooms”.
They were originally planning to put the curtain down the exact middle of the apartment, and give Alice the East room and Bob the West room, but Alice objected to this plan: ”Hey, your side of the apartment has a nice rug! That’s not fair!”. ”Hmm I don’t care about this rug, but I do love the window on this side.” said Bob, ”It would still be unfair even if we switched rooms and I took the East room instead.” ”Too bad our landlord is crazy and won’t let us move any of this stuff” said Alice.
Bob, realizing that square-footage alone is not the only relevant factor, made a proposal: ”Let’s each assign a number of points to the things we value in the apartment, so that we can measure if the curtain placement is fair or not”.
Alice and Bob each separately make a list of their personal preferences by assigning point values to the various features of the apartment (e.g. the square footage, the rug, the window etc.) They both assign a total of exactly 100.00 points to the entire apartment. Then then use these point values to judge potential ”rooms” that are formed with the curtain. You can assume that:
• The point value of the room is the sum of the point values of all the features of the room.
• All the things in the apartment can be divided if the curtain runs over them (e.g. the curtain is allowed to divide the rug so that one piece is on Bob’s side and one piece is on Alice’s side)
• If the curtain divides an object, the point value of that object is divided in the corre-
sponding way (e.g. If Bob assigns 60 points to ”square footage” of the apartment and
his room has 2 of the total area of the apartment, then the point value for the square 3
footage of his room is 2 60 = 40 points.) 3
Problems to solve
1. Explain how its possible that the value of Alice’s room (to Alice) plus the value of Bob’s room (to Bob) could exceed 100.00 points, even though the value of the entire apartment is 100.00 points total for each of them.
2. Use the intermediate value theorem to explain why, no matter what their preference are, there must exist somewhere to put the curtain and a way to decide who gets which room (East or West) so that both Alice and Bob value their room at 50.00 points each.

Solution

Same objects may have different points according to Alice and Bob(eg the rug may be 60 points for bob but 45 points for Alice) as the preference of the object vary from person to person. So, now lets assume the room has been divided in such a way thay:

1. For Alice, her share has total points say 75 and hence bob\'s share is of 25 points only

2. But on the contrary, it may happen that this same divison of room gives Bob\'s share 65 points and Alice\'s share 35 points only from Bob\'s perspective.

3. So Alice is happy with her 75 point share and Bob is happy with his 65 points share while the total of the room becomes 140 points which exceeds the total of 100 points.

Intermediate value theorem states that:

When:

... there must be at least one value c within [a, b] such that f(c) = w

In other words the function y = f(x) at some point must be w = f(c)

So clearly, when a function of alice an bob\' points will be plotted, it will be a continuous function and a point will definitely exist when the points will exceed or be equal to 50. It is a continuous function thanks to the assumption that the objects can be divided in any fraction of points. And so, the theorem is applicable and helpful

Clearly seen that the points shared cannot be below 50 from anyone\'s perspective.