6. The numbers $112,121,123,153,243,313$, and 322 are among the rows, columns, and diagonals of a $3 \times 3$ square grid of digits (rows and diagonals read left-to-right, and columns read top-to-bottom). What 3-digit number completes the list?

Solution:
524
The center digit is the middle digit of 4 numbers (hence at least 3 members of the above list), so it must be 2 . The top-left digit begins at least 2 members of the above list, so it must be 1 or 3 . If it is 3 , then after placing 313 we see that we need three more numbers starting with 3 , impossible; hence, it is 1 . So 243 and 313 must (in some order) be the last row and the last column, and now it is easy to complete the grid as shown; the answer is 524 .