Question

Study this matrix.
In this game there are two players. The first player can split the matrix vertically into two equal halves and choose one half for further play. The next move on this half is by the other player who will be split it only horizontally and choose one half for further play. The game will continue in this manner. At the end, the last number left is the first player's gain. If you start the game, retain the right half and, again right half after your opponent's move, then how should your opponent play to minimize your gain?

6	2	5	1
3	1	4	7
4	1	9	5
3	1	2	4

• A. Retain upper, retain lower
• B. Retain upper, retain upper
• C. Retain lower, retain upper
• D. Retain lower, retain lower

Answer 1

The Correct Option is B

To determine the optimal strategy for the opponent to minimize the first player's gain, we analyze the matrix splitting process: 

6	2	5	1
3	1	4	7
4	1	9	5
3	1	2	4

The starting matrix is 4x4. The first player splits vertically and retains the right half:

5	1
4	7
9	5
2	4

Now, the opponent splits this horizontally. To minimize the first player's gain, consider the values in each half.

Upper half:

5	1
4	7

Summation: 5+1+4+7=17

Lower half:

9	5
2	4

Summation: 9+5+2+4=20

The opponent should retain the upper half as its sum is lower.

The resulting matrix is:

5	1
4	7

The first player retains the right half again:

Remaining values:

1

7

The last value chosen for the first player would be 7, which is minimized. Intrinsically, the opponent should thus "Retain upper, retain upper" to minimize the first player's gain.