Another very easy problem,

Apoor, a four year old child, had to write $1$ to $ n$ numbers as a punishment. If he had made a mistake while writing any number, say $r$, then he had to write that number again $(r-1)$ times, which he always wrote correctly. If he had made mistakes while writing all except one number and the total numbers he wrote is $111$, then find the number that he wrote correctly.

Question

Another very easy problem, 

Apoor, a four year old child, had to write $1$ to $ n$ numbers as a punishment. If he had made a mistake while writing any number, say $r$, then he had to write that number again $(r-1)$ times, which he always wrote correctly. If he had made mistakes while writing all except one number and the total numbers he wrote is $111$, then find the number that he wrote correctly.

anonymous · Answer

I have a feeling that @KingGeorge will solve this one under 2 minutes.

kinggeorge · Answer

I'm gonna say that he wrote 10 correctly.

anonymous · Answer

Haha I knew it! ;)

anonymous · Answer

Very nice :)

kinggeorge · Answer

Since he wrote very number except one incorrectly, we can say he wrote $$1+2+3+4+...+n-(r-1)=111$$numbers total. Where $r$ is the number he wrote correctly. If we look at the triangular numbers at http://oeis.org/A000217 we see that $T_{15}=120$ is the smallest triangular number greater than 111. The next is $T_{16}=136$. If it were $T_{16}$ he would have had to have written 25 down correctly, but he can't have done that since $25>16$. Thus, he wrote the numbers up to 15, and wrote $120-(r-1)=111$ numbers total. Solving this, we get $r=10$.

kinggeorge · Answer

Typo. If it were $T_{16}$ he would have had to written 26 down correctly. Not 25.

kinggeorge · Answer

I think it's worth noting that this problem has a solution for every possible number of "total numbers" he wrote.