## Task 1 - 100811

Determine the number of all six-digit natural numbers in which the digit sequence 1970 (i.e., the base digits 1, 9, 7, 0 in this order and without any other digits in between) occurs!

What is the smallest and what is the largest of these six-digit numbers?
If the two missing digits of the sought number are denoted by $x$ and $y$, then six-digit numbers of the specified type can only have the following forms:

a) $\overline{x y 1970}$ with $1 \leq x \leq 9$ and $0 \leq y \leq 9$,

b) $\overline{x 1970 y}$ with $1 \leq x \leq 9$ and $0 \leq y \leq 9$,

c) $\overline{1970 x y}$ with $0 \leq x \leq 9$ and $0 \leq y \leq 9$.

All these numbers are distinct from each other; for if two of them belong to the same group denoted by a), b), or c), then the smaller of the two is the one in which the two-digit number $\overline{x y}$ formed from $\bar{x}$ and $\bar{y}$ is also the smaller; if they belong to different groups, they differ (e.g.) in their third digit.

In group a) and b), there are exactly as many numbers of the required type as there are numbers $\overline{x y}$ with $10 \leq \overline{x y} \leq 99$, which are exactly 90 numbers each, and in group c) as many as there are numbers $\overline{x y}$ with $0 \leq \overline{x y} \leq 99$, which are exactly 100 numbers.

Thus, the sought number of such numbers is $2 \cdot 90 + 100 = 280$.

From the above, the smallest and largest numbers in each group are:

in group a) 101970 and 991970,

in group b) 119700 and 919709,

in group c) 197000 and 197099.

Consequently, the smallest of the described numbers is 101970 and the largest is 991970.