## Task 1 - 080931

Marlies explains to Claus-Peter a method, according to which, she believes, the squares of natural numbers from 26 to 50 can be easily determined if one knows the squares of natural numbers up to 25 by heart.

"If you want to calculate the square of 42, for example, you form the complement of this number to 50 and square it. In this case, it would be 64.

You place the difference between your number and 25 in front of it, in your case, 17.

The number thus formed, 1764 in this case, is already the square of 42."

Check the correctness of this method for all numbers in the specified range!
We consider three cases. Let $x$ always be the number in question.

1. The square of the complement of the number to 50 is a single digit (i.e., $x \in\{47, \ldots, 50\}$). Since $(50-x)^{2}$ is a single digit, the formed number is then $10(x-25)+(50-x)^{2}=x^{2}-90 x+2250$.

This is equal to $x^{2}$ only if $x=25$, but this does not lie in the considered range. Therefore, the method does not work for these numbers.

2. The square of the complement of the number to 50 is a two-digit number (i.e., $x \in\{41, \ldots, 46\}$). Then the method yields the number $100(x-25)+(50-x)^{2}=x^{2}$, so the method works here.
3. The square of the complement of the number to 50 is a three-digit number (i.e., $x \in\{26, \ldots, 40\}$). Then the method yields the number $1000(x-25)+(50-x)^{2}=x^{2}+900 x-22500$, and this is equal to $x^{2}$ only if $x=25$, which does not lie in the considered range.

Therefore, the method works exactly when $x$ is between 41 and 46.