When it is given a list of 4 numbers, the kangaroo machine continues the list by typing the smallest non-negative integer that is different from each of the 4 preceding terms, and then repeats this process over and over again. Jacob types the numbers 2, 0, 2, 3 into the machine. What number will be the 2023rd in the list? Let's write out a few more terms of our sequence and see if we can see anything interesting. Starting with 2, 0, 2, 3, the smallest non-negative integer that we haven't written yet is 1. Now we read the next 4, and we have 0, 2, 3, 1. The smallest non-negative integer not yet written is 4. Now we have 2, 3, 1, 4. We get 0. 3, 1, 4, 0 gives 2, and 1, 4, 0, 2 gives 3, 4, 0, 2, 3 gives 1, 0, 2, 3, 1 gives 4, and we continue in this fashion. Notice that we have a repetitive sequence, that being 0, 2, 3, 1, 4. We started with the term 2, and then we have 0, 2, 3, 1, 4, we have another 0, 2, 3, 1, 4, another 0, 2, 3, 1, 4, and it seems like this pattern will continue infinitely. Therefore, when we start from the second digit, this sequence has a period of 5. We're seeking the 2023rd number in our list, and because we have a sequence of 5 that repeats, we know that the 2023rd number in the list will be the same as the 3rd number in our list, because 2023 equals 3 mod 5. The 3rd number in our list, we know, is 2, and therefore our answer is C, 2.

kangaroo machine

sequence generation

repetitive sequence

modular arithmetic

2023rd term