## Task 3 - 210913

Elsa claims:

"One can represent the number 1981 as follows: Write down the natural numbers from 1 to a suitable number $n$ in sequence and place exactly one of the four operation symbols $+,-, \cdot, :$ between each of these numbers. The same operation symbol does not have to be chosen everywhere; not all four operation symbols need to appear. The order of the natural numbers must not be changed; parentheses and additional arithmetic symbols should not occur. The result of the arithmetic expression written in this way is 1981."

Is Elsa's claim true?
Elsa's claim is true. To prove this, one example suffices. Such examples can be found, for instance, through the following consideration:

There is a number $m$ such that the sum of the natural numbers from 1 to $m$ just exceeds the number 1981. This is the case for $m=63$; because it holds that

$$
1+2+3+\ldots+63=\frac{63 \cdot 64}{2}=2016
$$

Since $2016-1981=35$, this sum must be reduced by 35 to reach 1981. This can be achieved, for example, by alternately adding and subtracting the next 70 natural numbers following 63, i.e.,

$$
1+2+3+\ldots+63+64-65+66-67+-\ldots+132-133=1981
$$

Another way is to insert -17 instead of +17, which reduces the sum by 34. This results in

$$
1+2+3+\ldots+17-17+18+\ldots+64=1982
$$

and by adding +64 and -65, the desired number is reached.

Another example is:

$1+2 \cdot 3 \cdot 4 \cdot 5: 6+7 \cdot 8 \cdot 9 \cdot 10 \cdot 11: 12-13 \cdot 14 \cdot 15-16 \cdot 17+18 \cdot 19=1+20+4620-2870-272+342=1981$