From the introduction, a different number 19 is in each cell of Number
Square 1. By clue 1, the upperleft to lowerright 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 lowerleft to upperright 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 lowerleft
to upperright 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 lowerleft to upperright diagonal would equal
16, again no (3). So, the 7 must be in the center cell; and the lowerleft
and upperright corners add to 7 (3). We now try the 8 in the upperleft
and the 9 in the lowerright corner. We let the lowerleft corner cell be
X and the middle cell in the rightmost column be Y. By clue 3,
the upperright corner number would be 7X. Then by clue 5, the
rightmost column would give 7X + Y + 9 = 16, or
16 + Y X = 16. Solving, X = Y, meaning the two
cells would have the same numberimpossible. Therefore, the 8 cannot be in
the upperleft corner; the 9 is there, with the 8 in the lowerright corner
cell of Number Square 1. Since the bottom row of the square adds to
18 (4), the lowerleft 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 lowerleft corner number were 5, the middle cell in the bottom
row would also be 5 (4). If the lowerleft corner number were 6, the
upperright corner one would be 1 (3)but the middle cell in the rightmost
column would also be 7 (5). So, the lowerleft corner number is 4, the
upperright 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 19 as follows:


