Problem number 30 states, 2021 balls are arranged in a row and are numbered from 1 to 2021. Each ball is colored in one of four colors, green, red, yellow or blue. Among any five consecutive balls, there is exactly one red, one yellow and one blue ball. After any red ball, the next ball is yellow. The balls numbered 2, 20 and 202 are green. What color is the ball numbered 2021? Wow, this is quite the problem and it sounds impossible at first, but let's take it one step at a time and review what the problem told us. So there are 2021 balls numbered 1 through 2021. In any group of five balls, there exists exactly one blue, one red and one yellow. So no more, no less, exactly one of each color. Any red ball must be followed by a yellow ball and the balls 2, 20 and 202 are green. With nowhere to start, let's just start coloring some balls in the middle of the set of balls. Let's consider these five for example. We know that exactly one must be blue, one must be red and one must be yellow. This means that the remaining two must be green. Keep in mind, we are completely arbitrarily choosing which of these balls are which color and the pattern here is likely wrong. But what we notice is that if we shift our view over one ball, is that there is no blue ball in the view anymore. So the next ball in the view must be blue. Shifting our view again would result in a red ball getting pushed out, so the next ball must be red. Same with the next ball on yellow. Shifting it again, we would have all of the three required balls, so the next ball will be green. Same with the next one. So what we may notice here is that we have a pattern that repeats every five balls. If we can figure out what this pattern is, we can figure out what the 2021st element is. So let's restart again, but this time only consider five balls, since we know that the pattern repeats. Let's call the last ball 5x, where x is going to be some integer. So this will be the 5th, 10th, 15th, and so on ball. All the balls before it we can label as 5x minus whichever ball it is before. So the one directly before it will be 5x-1, the one before that 5x-2. So for example, when x equals 1, the balls would be valued 1, 2, 3, and 4. If x would be equal to 2, then their values would be 6, 7, 8, 9, 10. So now that we have a consistent way of numbering our balls, let's take a look at where 2021 falls. So the number 2021 can be broken up into 5x405-4. So whatever the leftmost ball is, that will be the color of the 2021st ball. Okay, let's start with the colors. We know 220 and 202 are green. Representing them as a product of 5, we get an offset of minus 3 and an offset of minus 0. This means any ball whose number is 5x-0 or 5x-3 will be green. Now let's take a look at our second rule. Our second rule told us that any red ball must be followed by a yellow ball, and we know that there must be exactly 1 red ball and 1 yellow ball in every section of 5 balls. And the only spot where they can slot in is in this opening, in 5x-2 and 5x-1. This is the only spot where there is room for 2 consecutive balls. Now by the first rule, we know that there must be exactly 1 red, blue, and yellow ball present in each sequence of 5. So we know the first ball in the pattern is going to be blue. This was the ball the 2021st ball could be represented as, which means the 2021st ball is blue. So the question asked us, what color is the ball numbered 2021? The answer is blue, letter D.

color pattern

2021 balls

repeating sequence

red yellow blue

ball color challenge