Question 150
An interesting property of the number 2013 is that 3 is the last digit of the sum $2+0+1$. Repeating this process, that is, writing to the right the last digit of the sum of the previous three digits, we get a sequence:
$$
2,0,1,3,4,8,5,7 \ldots
$$
a) Prove that, starting with the sequence 2,0,1, in this order, we can also find the three consecutive numbers $1,2,2$, in this order.
b) Observe that if a triple of three consecutive numbers appears again in the same order, then the whole sequence "repeats" successively. For example, the sequence below is not the sequence of the statement, but it repeats every four numbers
$$
\ldots \underline{124312431243 \ldots 12431243 \ldots}
$$
Verify that some triple of digits repeats in the sequence of the statement.
c) Suppose that at the first appearance of "$a, b, c$" in the sequence, the number immediately before is $x$, and at its second appearance it is $y$, that is, in the sequence we find the numbers in the following order:
$$
\ldots, x, a, b, c, \ldots, y, a, b, c \ldots
$$
Show that $x=y$.
d) Given that 1,2,2 appeared in the sequence, in this order, show that the sequence of digits $2,0,1$ will eventually appear again, also in this order.
$$
2,0,1,3,4,8,5,7 \ldots
$$
a) Prove that, starting with the sequence 2,0,1, in this order, we can also find the three consecutive numbers $1,2,2$, in this order.
b) Observe that if a triple of three consecutive numbers appears again in the same order, then the whole sequence "repeats" successively. For example, the sequence below is not the sequence of the statement, but it repeats every four numbers
$$
\ldots \underline{124312431243 \ldots 12431243 \ldots}
$$
Verify that some triple of digits repeats in the sequence of the statement.
c) Suppose that at the first appearance of "$a, b, c$" in the sequence, the number immediately before is $x$, and at its second appearance it is $y$, that is, in the sequence we find the numbers in the following order:
$$
\ldots, x, a, b, c, \ldots, y, a, b, c \ldots
$$
Show that $x=y$.
d) Given that 1,2,2 appeared in the sequence, in this order, show that the sequence of digits $2,0,1$ will eventually appear again, also in this order.