Math 115                                             Name:                          Key                                                     

Quiz 3                                                  February 20, 2013                                          


1.  Andy and Terry split a cake by the divider-chooser method.  Andy, the divider, likes chocolate icing twice as much as vanilla.  Show 2 different ways that Andy might split the cake to ensure he gets a “fair” piece. 


One way to divide it fairly is to slice it into 2 identical parts: horizontally, through the middle, so each person will get 2 chocolate and 2 vanilla pieces.

The other fair way to divide it is to make one part contain 3 chocolate pieces, and one part contain all 4 vanilla and 1 chocolate.  As explained in class, this can be seen to be fair in a couple of ways, including: if vanilla pieces are worth 1 (quarter, dollar, or whatever) then chocolate pieces are worth 2 (twice as much), and each part is now worth 6.  Alternatively, if you start with 2 equal parts, as in the first solution, and then swap 1 chocolate piece for 2 vanillas, which is a fair swap to Andy, you wind up with this 2nd solution.

2.  Three players, Ali, Bo, and Caz, divide the 13 items shown by the method of markers. 

1       2      3         4        5       6      7        8        9       10     11    12   13


Which items: go to Ali?   9-13

                      go to Bo?   1-3

          go to Caz?  6-7

                      are (initially) left over?  4,5,8


2b.  According to the rules of this method, how should the leftovers be allocated?  EXPLAIN your answer in a complete English sentence.

It doesn’t matter, as each players has already received a share that they consider fair.  At this point each could receive anything additional (or nothing additional) and it would still be fair.
















3.  3 brothers, Adam, Bob, and Chris, divide some games and some clothes by the method of sealed bids. 

The bids (in $) are as follows: 

a.        Find the value of each brother’s fair share

Adam:  1/3 of $48 = $16

Bob:  1/3 of $51 = $17

Chris: 1/3 of $45 = $15



b.      Describe the final settlement (what each brother gets/pays after the first settlement and any surplus is distributed)


First, Adam gets the clothes (worth $18), and Bob gets the games (worth $36).  Since Adam received $2 more in value than his fair share, he pays $2 in cash.  Bob received $19 in value more than his fair share, so he pays $19 in cash.  Chris, who received nothing, takes his fair share of $15 in cash from the money that Adam and Bob paid.  There is now $6 leftover ($21 paid, $15 claimed), so this can be split evenly, giving each brother $2.  So in the end Adam gets the clothes and neither pays nor receives any money (he paid $2 but then got $2 back); Bob got the games and paid $17 (paid $19 before getting $2 back), and Chris received $17 cash ($15+2).  Notice that the cash payments all exactly offset, so no money is leftover, and money doesn’t magically appear out of thin air.