2021 colored kangaroos are arranged in a row and are numbered from 1 to 2021. each kangaroo is colored either red, grey, or blue. amongst any three consecutive kangaroos, there are always of all three colors. badr guesses the colors of five kangaroos. these are his guesses: kangaroo no. 2 is grey kangaroo no. 20 is blue kangaroo no. 202 is red kangaroo no. 1002 is blue kangaroo no. 2021 is grey Only one of his guesses is wrong, which one?

The pattern of colors must repeat every three kangaroos because there are always all three colors amongst any three consecutive kangaroos. Therefore, the color of kangaroo number n will be the same as the color of kangaroo number n+3k, where k is any integer.

Let's assign the colors to numbers for simplicity: red = 1, grey = 2, blue = 3.

If kangaroo no. 2 is grey (2), then kangaroo no. 20, which is 18 kangaroos away and a multiple of 3, should also be grey (2). But Badr guessed it to be blue (3), so this guess is wrong.

All the other guesses are consistent with the pattern. For example, kangaroo no. 202 is 200 kangaroos away from kangaroo no. 2, and 200 is a multiple of 3, so it should be the same color as kangaroo no. 2, which is grey (2). Badr guessed it to be red (1), which is consistent with the pattern because red (1) is one color before grey (2) in our numbering system.

Similarly, kangaroo no. 1002 is 1000 kangaroos away from kangaroo no. 2, and 1000 is a multiple of 3, so it should be the same color as kangaroo no. 2, which is grey (2). Badr guessed it to be blue (3), which is consistent with the pattern because blue (3) is one color after grey (2) in our numbering system.

Finally, kangaroo no. 2021 is 2019 kangaroos away from kangaroo no. 2, and 2019 is a multiple of 3, so it should be the same color as kangaroo no. 2, which is grey (2). Badr guessed it to be grey (2), which is consistent with the pattern.

Therefore, the only wrong guess is that kangaroo no. 20 is blue. It should be grey.