Question

Consider a d × d table A, with d ? 1, where each of its entries contains exactly one element of the set {0, 1, 2}. Index the rows of the table starting from 0 for the top row, and the columns of the table starting from 0 from the left column. Hence, each element of the table is referenced by A_{i,j}, where i is the index of the row the element belongs in, and j is the index of the column the element belongs in. Furthermore, consider the sums of the entries of the table per row (namely R_i = ?_{k=0}^{d-1} A_{i,k} for i = 0, 1, ..., d - 1), the sums of the entries of the table per column (namely C_j = ?_{k=0}^{d-1} A_{k,j} for j = 0, 1, ..., d - 1) and the sums across the diagonals of the table (namely D_1 = ?_{k=0}^{d-1} A_{k,k} and D_2 = ?_{k=0}^{d-1} A_{k,d-1-k}). Show that there are always two sums (out the row, column and diagonal sums described above) that are equal.

          Consider a d × d table A, with d ? 1, where each of its entries contains exactly one element of the set {0, 1, 2}. Index the rows of the table starting from 0 for the top row, and the columns of the table starting from 0 from the left column. Hence, each element of the table is referenced by A_{i,j}, where i is the index of the row the element belongs in, and j is the index of the column the element belongs in.
Furthermore, consider the sums of the entries of the table per row (namely R_i = ?_{k=0}^{d-1} A_{i,k} for i = 0, 1, ..., d - 1), the sums of the entries of the table per column (namely C_j = ?_{k=0}^{d-1} A_{k,j} for j = 0, 1, ..., d - 1) and the sums across the diagonals of the table (namely D_1 = ?_{k=0}^{d-1} A_{k,k} and D_2 = ?_{k=0}^{d-1} A_{k,d-1-k}).
Show that there are always two sums (out the row, column and diagonal sums described above) that are equal.

Consider a d × d table A, with d ? 1, where each of its entries contains exactly one element of the set 0, 1, 2. Index the rows of the table starting from 0 for the top row, and the columns of the table starting from 0 from the left column. Hence, each element of the table is referenced by Ai,j, where i is the index of the row the element belongs in, and j is the index of the column the element belongs in.
Furthermore, consider the sums of the entries of the table per row (namely Ri = ?k=0^d-1 Ai,k for i = 0, 1, ..., d - 1), the sums of the entries of the table per column (namely Cj = ?k=0^d-1 Ak,j for j = 0, 1, ..., d - 1) and the sums across the diagonals of the table (namely D1 = ?k=0^d-1 Ak,k and D2 = ?k=0^d-1 Ak,d-1-k).
Show that there are always two sums (out the row, column and diagonal sums described above) that are equal.

Added by Patricia C.

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