## 43. Math Puzzler $12 / 68$

Mr. Dull has nothing to do. He starts counting all the numbers in natural order: $1 +$ $2 + 3 + \cdots$ and so on. The doorbell rings, and Mr. Dull just manages to note down the last intermediate result he found, which is 6328. However, he forgets the last number he added.

Can you help him without being as dull as he is?
The problem relates to the task that young Gauss was supposed to solve. The teacher had assigned the task of adding all numbers from 1 to 100. Gauss was finished in 5 minutes because he noticed that there are 50 pairs of numbers that add up to 101: 100+1, 99+2, 98+3, etc.

Using this trick,

$$
\begin{aligned}
& 1+2+3+\ldots+(x-2)+(x-1)+x=6328 \\
& x+(x-1)+(x-2)+\ldots+3+2+1=6328
\end{aligned}
$$

Adding both sums, we get $x(x+1)=12656$. The quadratic equation has the solution $x=112$. If solving quadratic equations is not yet known, one can use a square table to determine that $\sqrt{12656}$ lies between 112 and 113. A check with 112 confirms the result.