Raj has ten pairs of red, nine pairs of white and eight pairs of black shoes in a box. So, he has 20 red shoes, 18 white shoes, and 16 black shoes.

To find the maximum number of attempts we need to take the worst-case scenario. Let Raj draw all white and black shoes, which amounts to 18 + 16 = 34 shoes.

Now, if he picks any two shoes, they are certainly going to be red. However, we also need to ensure that the red shoes picked by him make a pair, i.e. there should be one red shoe for left foot and one red shoe for right foot. For this to happen we need to pick 11 red shoes.

So, maximum number of attempts to get a red pair of shoes = 34 + 11 = 45