| From the introduction, a different number 1-9 is in each cell of Number
Square 1. By clue 1, the upper-left to lower-right diagonal numbers sum
to 24. The only way to get 24 is for the numbers 7, 8, and 9 to be in the
cells in some order. By clue 2, the numbers in the four corners of the
square add to 24; while by clue 3, the lower-left to upper-right diagonal
sums to 14. If the 7 and 8 were in corners of the square, with 9 in the
middle cell, the other two corners would sum to 9 (clue 2)--but the lower-left
to upper-right diagonal would equal 18, no (3). If the 7 and 9 were in
corners of the square, with 8 in the middle cell, the other two corners
would sum to 8 (2)--but the lower-left to upper-right diagonal would equal
16, again no (3). So, the 7 must be in the center cell; and the lower-left
and upper-right corners add to 7 (3). We now try the 8 in the upper-left
and the 9 in the lower-right corner. We let the lower-left corner cell be
X and the middle cell in the rightmost column be Y. By clue 3,
the upper-right corner number would be 7-X. Then by clue 5, the
rightmost column would give 7-X + Y + 9 = 16, or
16 + Y -X = 16. Solving, X = Y, meaning the two
cells would have the same number--impossible. Therefore, the 8 cannot be in
the upper-left corner; the 9 is there, with the 8 in the lower-right corner
cell of Number Square 1. Since the bottom row of the square adds to
18 (4), the lower-left corner must contain a 4 or higher; a 1, 2, or 3 in
that cell would necessitate a 9, 8, or 7 in the middle cell of the bottom
row. If the lower-left corner number were 5, the middle cell in the bottom
row would also be 5 (4). If the lower-left corner number were 6, the
upper-right corner one would be 1 (3)--but the middle cell in the rightmost
column would also be 7 (5). So, the lower-left corner number is 4, the
upper-right corner one is 3 (3), and the middle cell in the rightmost column
is 5 (5). The middle cell in the bottom row equals 6 (4). By clue 6, the 2
is in the middle cell of the top row, with 1 in the middle cell of the
leftmost column. Number Square 1 contains the digits 1-9 as follows:
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