### CAT 1990 Question 34

Instructions

The following questions relate to a game to be played by you and your friend. The game consists of a 4 x 4 board (see below) where each cell contains a positive integer. You and your friend make moves alternately. A move by any of the players consists of splitting the current board configuration into two equal halves and retaining one of them. In your moves you are allowed to split the board only vertically and to decide to retain either the left or the right half. Your friend, in his/her moves, can split the board only horizontally and can retain either the lower or the upper half. After two moves by each player a single cell will remain which can no longer be split and the number in that cell will be treated as the gain (in rupees) of the person who has started the game. A sample game is shown below. So your gain is Re.1. With the same initial board configuration as above and assuming that you have to make the first move, answer the following questions.
Â  Initial Board Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

Â

After your move (retain left)

Â

After your friends move (retain upper)

After your move (retain right)

After your friends move (retain lower)

Â  Â Â

Question 34

# If your first move is (retain right), then whatever moves your friend may select you can always force a gain of no less than

Solution

After retaining right in first move, second person will have 2 quarters to choose in which minimum value is 2 and 3.
If he retains upper half, i will retain right as to maximize my profit of either 4 or 7.
If he retains lower half, i will retain left as to maximize my profit of either 3 or 8.
Hence whatever he retains, i won't have a profit less than 3.

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