EACH OF THE NUMBERS 1, 2, 3, 4, 5, 6, 7, 8 AND 9 IS TO BE PLACED I...

1. Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 is to be placed in a different square

of a 3×3 table. We color the largest number in each row, red while the smallest

number in each row, green. Let M be the smallest among the three red numbers,

and m be the largest among the three green numbers. Determine all possible

values of M − m.

【Solution】

We have 3 ≤ M ≤ 7, 3 ≤ m ≤ 7 and M≠m. (5 points)Hence the only possible values for

M−m are −4, −3, −2, −1, 1, 2, 3 and 4. (5 points)The diagrams below shows that

each is possible. (10 points)

7 9 8 6 9 8 5 9 8 5 9 8

6 5 4 7 5 4 7 6 4 7 6 1

3 2 1 3 2 1 3 2 1 4 3 2

M=3，m=7 M=3，m=6 M=3，m=5 M=4，m=5

9 8 5 9 6 5 9 6 4 9 6 3

7 4 2 8 4 2 8 5 2 8 4 2

6 3 1 7 3 1 7 3 1 7 5 1

M=6，m=5 M=7，m=5 M=7，m=4 M=7，m=3

The respective values of M−m are 3−7=−4, 3−6=−3, 3−5=−2, 4−5=−1, 6−5=1, 7−5=2,

7−4=3 and 7−3=4.

ANS: −4, −3, −2, −1, 1, 2, 3 and 4