60 children stand in a row. Each child has a high-visibility vest and a backpack. The colours of their high-visibility vests always alternate: yellow, green, yellow, green, ... The colours of their backpacks make the following pattern: red, brown, purple, red, brown, purple, ... How many children have a yellow safety vest and a purple backpack?

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Let's understand this pattern problem. We have 60 children in a row, each wearing a vest and carrying a backpack. The vest colors alternate between yellow and green, while backpack colors follow a three-color pattern: red, brown, purple. We need to find how many children have both a yellow vest and a purple backpack. Now let's analyze the vest pattern. The vests alternate between yellow and green, starting with yellow. This means yellow vests appear at odd positions: 1, 3, 5, 7, and so on. Green vests appear at even positions: 2, 4, 6, 8, and so on. We can express yellow vest positions mathematically as 2k minus 1, where k is any positive integer. Now let's examine the backpack pattern. The backpacks cycle through three colors: red, brown, purple, repeating every three positions. Purple backpacks appear at positions 3, 6, 9, 12, and so on. These are all multiples of 3. We can express purple backpack positions as 3k, where k is any positive integer. To find children with both yellow vests and purple backpacks, we need positions that are both odd numbers and multiples of 3. These are the odd multiples of 3. Looking at multiples of 3 up to 60, we find that positions 3, 9, 15, 21, 27, 33, 39, 45, 51, and 57 are odd. These are exactly the positions where children have both a yellow vest and a purple backpack. Let's summarize our solution. We found that yellow vests appear at odd positions, and purple backpacks appear at multiples of 3. The intersection of these two conditions gives us the odd multiples of 3 within the range of 1 to 60. These positions are: 3, 9, 15, 21, 27, 33, 39, 45, 51, and 57. Counting these positions, we get exactly 10 children who have both a yellow vest and a purple backpack. Therefore, the answer is 10.