The cells of a 3×3 table were numbered from 1 to 9, each number appearing exactly once. For each row the cell with the greatest number is colored red and the cell with the smallest number is colored green. Let A be the smallest of the numbers in the red cells and B the greatest of the numbers in the green cells.
a. Show a distribution of the numbers in the table such that A−B=4.
b. Show a distribution of the numbers in the table such that A−B=−3.
c. Is it possible that A=4 and B=3?
Solution
a. For example,
7
4
1
8
5
2
9
6
3
Notice that A=min{7,8,9}=7 and B=max{1,2,3}=3, so A−B=4.
b. For example,
1
2
3
4
5
7
6
8
9
Notice that A=min{3,7,9}=3 and B=max{1,4,6}=6, so A−B=−3.
c. No, it's not possible. Since B=3 is the greatest of the numbers in green cells, which are the smallest ones in their respective rows, the numbers 1, 2 and 3 must be on different rows. Now A=4 means that 4 is the greatest number of a row, but it could only happen if two of the numbers 1, 2, 3 are on the same row as 4, which cannot happen.
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Source: MathNet,
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