The pattern Sam is using is "Blue Tile, White Tile, Black Tile," which consists of 3 tiles in total: 1 blue, 1 white, and 1 black.
Since the pattern repeats every 3 tiles, we can determine how many complete sets of 3 tiles fit into 85 tiles:
\[ \text{Number of complete sets} = \frac{85}{3} = 28 \text{ complete sets} \text{ with a remainder of } 1 \]
Each complete set of 3 tiles contains 1 black tile. Therefore, the number of black tiles from the complete sets is:
\[ 28 \text{ (sets)} \times 1 \text{ (black tile per set)} = 28 \text{ black tiles} \]
Now, we look at the remainder. Since there is a remainder of 1, that means there is one additional tile beyond the complete sets. In the repeating pattern, the first tile is blue. Thus, this extra tile does not contribute any additional black tiles.
So, the total number of black tiles used is:
\[ \text{Total black tiles} = 28 \text{ (from complete sets)} + 0 \text{ (from remainder)} = 28 \]
Therefore, the number of black tiles Sam will use is:
\[ \boxed{28} \]
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