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Problem of the Day #426: Honey Drops January 2, 2013

Posted by Saketh in : potd , add a comment

Two bears are sharing some honey they found by playing a little game. They get two pots, and divide the honey (not necessarily evenly) between the pots. Let the first pot contain $a$ drops of honey, and let the second contain $b$ drops of honey.

The game proceeds as follows. Each turn, one of the bears takes $2k$ drops of honey from one of the pots, eats $k$ of the drops, and places the other $k$ drops in the other pot. The bears alternate until no valid move is available, at which point the last bear to move wins.

For which values of $a$ and $b$ does the bear that goes first have a winning strategy?

Problem of the Day #279: Elves, Elves, Everywhere! December 23, 2011

Posted by Saketh in : potd , add a comment

Sohail and Kevin are playing with some elves. They take turns presenting one coin to any elf that is still awake. Each elf will accept a total of $5$ coins before dispensing a present and going to sleep.

If there are $2011$ elves present, and Sohail gets to go first, how many presents is he guaranteed to win? Both Sohail and Kevin play optimally.

Problem of the Day #56: Spring Cleaning May 14, 2011

Posted by Aziz in : potd , add a comment

Five people, Sreenath, Mitchell, Alex, Albert and Sam, live in the same house and need to clean up before their parents come visit.

Each person gets his own room. A person will clean only their own room. Each person’s own room has a certain mess level, greater values requiring more cleaning. A cycle consists of everyone cleaning a certain amount of their own room and dumping the rest of their mess into another person’s room. Each person is required to clean between $5$ and $20$ of their mess each cycle. Whatever is left uncleaned is dumped into a randomly chosen other person’s room. Whenever someone dumps their remaining mess, the person receiving it will gain an additional 5 mess units as a cost for dumping.

Mess Factor of:
Sreenath — 150
Mitchell — 80
Alex — 50
Albert — 10
Sam — 70

The goal is to minimize the number of cycles it takes to clean the house.

Determine the expected number of cycles it would take if everyone acted independently and if they came up with and followed an optimal plan before beginning the cleanup.

A Two-Player Board Game March 20, 2011

Posted by Saketh in : bonus , add a comment

On a 5×5 board, two players, Albert and Billy, alternately mark numbers on empty cells. The first player, Albert, always marks 1′s, the second, Billy, 0′s. One number is marked per turn, until the board is filled. For each of the nine 3 x 3 squares the sum of the nine numbers on its cells is computed. How large can the Albert force the maximum of these sums to be regardless of Billy’s behavior?