An anti magic square is an N×N
grid containing the integers from
1 through N^2, such that the sum of
each row, column, and main diagonal
is a different number, and those
sums form a consecutive sequence of
integers.
There are no 3×3 anti magic squares, but there are some that are close in that the sequence of sums is consecutive except for one missing value.
Determine a 3×3 near anti magic square in which the three digit number represented by the top left (hundreds position) to bottom right (units position) main diagonal is a minimum. Present your answer as a 3×3 grid containing the digits 1 through 9.
To show that Fermat's Last Theorem applies only to sums of two terms, Fred asked his friends, Alice, Bob, Carol and Diane, to list three perfect cubes that added up to another perfect cube. Each came up with his or her own list, different from the others'.
All except Diane limited their lists of three cubes to the first 12 cubes. Alice and Bob had two of the same numbers in their lists, but Carol's list had no numbers in common with either of those lists.
Diane, not limiting herself to the first 12 cubes, did use in her list two of the sums from among the three sums of cubes found by Alice, Bob and Carol.
What were Carol's and Diane's lists of cubes?
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