Question

Each cell in the 3-by-3 array of squares shown is filled with a digit from 1 to 9,

inclusive. If the four listed conditions are satisfied, what is the resulting
three-digit number when diagonal B is read from upper-left to lower-right?

(1) Each digit from 1 through 9 is used exactly once.
(2) Each row, read from left to right, forms an odd
three-digit multiple of 3 that is not a multiple of 9.
(3) Each column, read from top to bottom, forms an odd
three-digit multiple of 3 that is not a multiple of 9.
(4) Diagonal A, read from lower-left to upper-right,
forms an odd three-digit multiple of 9.

Answer 1

[AI-generated answer — Bot GPT 3.5]

First, let's look at condition (2) and (3). Each row and column must be an odd three-digit multiple of 3 that is not a multiple of 9. The only odd three-digit multiples of 3 are 111, 333, 555, 777, and 999.

Since each digit can only be used once, the only possible combinations for the rows and columns are:

Row 1: 333
Row 2: 555
Row 3: 777

Column 1: 111
Column 2: 777
Column 3: 999

Now, let's look at condition (4). Diagonal A, read from lower-left to upper-right, must be an odd three-digit multiple of 9. The only odd three-digit multiples of 9 are 117, 135, 333, 369, 711, and 999.

Since 333 is already being used in Row 2, we can eliminate it as a possibility for Diagonal A.

That leaves us with the possibilities for Diagonal A:

117 (Row 1, Column 1)
135 (Row 3, Column 1)
369 (Row 1, Column 3)
711 (Row 3, Column 3)
999 (Row 2, Column 2)

Now, let's look at condition (1): Each digit from 1 through 9 is used exactly once.

Since the only remaining digits are 2, 4, 6, and 8, we can determine the placement of these digits:

Row 1: 33328671
Row 2: 55542896
Row 3: 77796814

Now, let's determine the resulting three-digit number when diagonal B is read from upper-left to lower-right.

The digits in Diagonal B are 1, 9, and 6. Therefore, the resulting three-digit number is 196.

So, the answer is 196.