+----+----+----+----+
| | | | |
+----+----+----+----+
| | | | |
+----+----+----+----+
| | | | |
+----+----+----+----+
| | | | |
+----+----+----+----+
An integer is written on each unit square of a 4×4 unit square chessboard given above.
The number in each unit square is equal to the sum of the numbers in all unit squares that share a common edge with that unit square.
What is the largest number of positive integers (out of the 16 numbers) that can be written on this chess board?
To find a 4x4 grid that works, assume there are 4 degrees of freedom and fill the top row with a, b, c, d.
By looking at each previous row, the next rows then must be
a-b, -a+b-c, -b+c-d, -c+d
a-2b+c, -2a+2b-2c+d, a-2b+2c-2d, b-2c+d
2a-3b+3c-d, -3a+5b-4c+3d, 3a-4b+5c-3d, -a+3b-3c+2d
This last row doesn't always work, though. Each number in this row must satisfy the sum rule. This yields four extra constraints:
-4a+6b-6c+4d=0
6a-10b+10c-6d=0
-6a+10b-10c+6d=0
4a-6b+6c-4d=0
Solving these, we find a=d and b=c. Going back to the original then we can proceed row by row as before to make a grid that always works:
+----+----+----+----+
| a | b | b | a |
+----+----+----+----+
| a-b| -a | -a | a-b|
+----+----+----+----+
| a-b| -a | -a | a-b|
+----+----+----+----+
| a | b | b | a |
+----+----+----+----+
The appearance of a and -a gives an upper bound. of 12 positive results. This can be obtained by 0<b<a.
The simplest would be a=2, b=1
+----+----+----+----+
| 2 | 1 | 1 | 2 |
+----+----+----+----+
| 1 | -2 | -2 | 1 |
+----+----+----+----+
| 1 | -2 | -2 | 1 |
+----+----+----+----+
| 2 | 1 | 1 | 2 |
+----+----+----+----+
Which has 12 positive entries.
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Posted by Jer
on 2024-11-19 10:20:59 |
Solution
