SOLUTIONS TO HW 3
2. (a) $2.70
(b) $10.80
(c) (60/180)*$10.80+(40/180)*$2.70=$3.60+$.60=$4.20
8. (a) Abe: s1 & s2 (any slice worth atleast $3.75).
(b) Betty: s1, s2, s3, s4 (any slice worth atleast $4.50).
(c) Cory: s1, s2, s4 (atleast $3.00).
(d) Dana: s1 (atleast $2.50).
(e) Abe: s2, Betty: s3, Cory: s4, Dana: s1.
10. (a) Cut 1, Cut 3, and Cut 5. (Each piece must be worth exactly half of the total worth.)
(b) Cut 1--either piece (they are of equal value to her)
Cut 3--the all chocolate piece is worth: (120/180)*66.66%=44.44% so the larger piece is worth 55.56% which means it is more than half to her eyes. She'd pick the big piece.
Cut 5--The smaller piece is worth: (96/180)*66.66% + (72/180)*33.33% = 48.88%, which isnt quiet half to her, so she'd pick the piece on the right.
12. (a) Answers will vary...basically the cutter will divide the cake into 2 equally valued parts.
(b) The chooser picks the piece that is worth half or more of the whole thing.
17. (a) Chooser 1 gets s2, chooser 2 gets s3, chooser 3 gets s1 and the divider gets s4.
(b) Chooser 1 gets s3, chooser 2 gets s1, chooser 3 gets s2 and the divider gets s4.
(c) No. None of the choosers wanted s4 so none of them can end up with it. S4 must go to the divider.
34. (a) Actually yes. Although P4 ends up with the piece at the end of round 1, it was diminished and so a piece of s went back to be part of the R-piece to be divided in round 2. Someone will end up with the little piece P4 cut off.
(b)P4 because he was the last diminisher.
(c)P1 is always the first to cut a piece in every round until she gets a piece.
(d)P5 because P4 already recieved a piece.
42. A gets farm and pays $212000, B gets painting and $159000, C gets house and $53000
| Item | Andre | Bea | Chad | |
| house | 150000 | 146000 | 175000 | |
| farm | 430000 | 425000 | 428000 | |
| painting | 50000 | 59000 | 57000 | |
| total bids | 630000 | 630000 | 660000 | |
| fair share | 210000 | 210000 | 220000 | |
| recieved | 430000 | 59000 | 175000 | |
| surplus | -220000 | 151000 | 45000 | total surplus 24000 |
| tot. surplus/3 | 8000 | 8000 | 8000 | |
| final | -212000 | 159000 | 53000 |
46.
(a) C gets 1,2; A gets 4,5,6,7,8; B gets 10, 11, 12, 13.
(b) Items 3 and 9 are left over.
56. (a) The total area is 30,000 sq meters and the area of C is only 9,000 sq meters. Since P2 and P3 value the land uniformly, each thinks that a fair share must have atleast 10,000 sq meters.
(b) Since there are 21,000 sq meters left, anything that divides the remaining property in parts of 10,500 sq meters will work.
(c) The cut will be a vertical line 112.5 meters from the left.
(d) The cut will be a horizontal line 35 meters down from Park Place (from the top).
63. Ruth cleans the bathroom and pays $11.67, Sarah cooks and pays $11.67, Tamara does all the rest of the chores and gets paid $23.34
| Item | Ruth | Sarah | Tamara | |
| bathroom | -20 | -30 | -40 | |
| cooking | -50 | -10 | -25 | |
| dishes | -30 | -20 | -15 | |
| lawn | -30 | -20 | -10 | |
| vacuum | -20 | -40 | -15 | |
| total bids | -150 | -120 | -105 | |
| fair share | -50 | -40 | -35 | |
| recieved | -20 | -10 | -40 | |
| surplus | -30 | -30 | 5 | total surplus 55 |
| tot. surplus/3 | 18.33 | 18.33 | 18.34 | |
| final | -11.67 | -11.67 | 23.34 |