Mr. Dengklek's Alphabet Lesson

Use the pigeonhole principle

Assume that the duck is stupid and it writes all the letters 26 times, that is the maximum.

BUT the last letter it writes can be only written 10 times because the duck has written all 25 letters more than ten times, and for this letter, once he writes it the tenth time the lesson ends

We can write from A to Y 26 times...it means 25*26=650 Then when Z is written 10 times, then we can't write an more letters 650+10=660

the rule is he will only stop writing if he wrote all the letter 10 times, so definitely, we will assume that he will wrote the first 25 letter 26 times, since it is the maximum.

$25 \cdot 26 = 650$

then finally, add the last 10 times of the last letter,

$650 + 10 = \boxed{660}$

the duck can write 25 letters 26 times and the last letter 10 times. so, 25x26+10 = 660

Piegeonhole principle

Its pigeon's hole question. Assume after teaching a lot and the duck is really unlucky to write 26x for consecutice first 25 alphabet (26x25=650) then for last alphabet the duck write the minimum value of the requirement to finish the "bad luck teaher's lesson. Which is 10. So the max letter that can be written are 650+10=660 .

(26*26 ) - 16 = 660