Desserts abound at Thanksgiving celebrations.
Joe and Bob agree to divide a small cake after the big meal. Each generously wants to provide the other with the largest share possible. Joe cuts the cake into two pieces; then Bob cuts one of those pieces into two pieces.
If Joe must take the largest and smallest pieces, what is the largest amount of cake Bob can provide to Joe?
Easy as Pie
Alice and Beth divide a small Thanksgiving pie. Alice cuts the pie into two pieces; Beth cuts one of those pieces into two pieces. Then Alice cuts one of the three pieces into two pieces. Alice gets the largest and smallest pieces produced; Beth gets the middle two pieces.
What strategy does Alice use to get the largest fraction of pie, and how large is it?
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Solutions to week 114
In Primes and Products, Dick is 77. Pete can be either 71 or 83. In Two Pints of Cider, the contents start with (15,0,0) in the three containers and can proceed to (9,0,6), (9,6,0), (3,6,6), (3,10,2), drink, (3,10,0), (3,4,6), (7,0,6), (7,6,0), (1,6,6), (1,10,2), (11,0,2), (11,2,0), (5,2,6), (5,8,0), (0,8,5). This takes 15 steps in all.
Primes and Products answer explained:
If Pete is six years younger, those primes giving Dick a suitable age are 19, 29, 43, 59, 71, 79, 89, and 109. The only pair twelve years apart is 59 and 71, making Pete 71 and Dick 77.
If Pete is six years older, those primes giving Dick a suitable age are 31, 41, 61, 71, 83, 97, and 101. The only pair twelve years apart is 71 and 83, making Pete 83 and Dick 77.
Dick is 77 either way.
Links to all of the puzzles and solutions are on the Complete Varsity Math page.
Come back next week for answers and more puzzles.