From the introduction, a different number 19 is in each cell of Number
Box 1. By clue 2, the three numbers in the last column sum to 8.
The possible combinations summing to 8 are 1, 2, 5 and 1, 3, 4. If the 1
were in either of the corner cells of column 3, by clue 3, the other three
numbers in the corners would add to 22. Combinations possible are 9, 7, 6
and 9, 8, 5. The first combination conflicts with clue 2, where neither a
6 or 7 can be in column 3. The second possibility conflicts with clue 4,
since there is no way to get that difference given 9, 8, 5, and 1 as corner
possibilities. Therefore, the 1 in the two possible combinations for column
3 must be at the end of the middle row. If 3 and 4 were the other two numbers
in the last column, by clue 3, the number in the upper left cell of the box
would be a 7 or 9. However, there is no way for clue 4 to work. So, 2 and 5
are the other numbers in the corners of column 3 with a 9 and 7 then in the
leftmost corners. By clue 4, the 7 must be in the upper left corner with 5
in the lower right corner; 2 is in the upper right and 9 in the lower left
corner. The remaining numbers are 3, 4, 6, and 8. By clue 1, the 8 must
be in the center of the top row and the 3 must be in the center of the bottom
row. Finally, 6 is in the leftmost cell of the middle row with 4 in the
center (clue 5). Number Box 1 is as follows:


