A postman delivered letters every day. On the first day, he delivered one letter; on the second day he delivered three letters; on the third day, he delivered five letters, and so on. Each day from the second on he delivered two letters more than the preceding day.
The day he retired, he noticed that he had delivered a total of more than 1,00,000 letters.
Something fun is that the total number of letters he had delivered has its digits arranged in ascending order.
How many days had he worked, and how many letters had he delivered in total?
Answer
It’s easy to see that the number of letters he had delivered is the square of the number of days he had worked.
Square numbers can’t end in 7 or 8, and 1,23,456 is not a square because it is divisible by 3 but not by 9. Hence the number of letters must end with 9.
Since odd squares have remainder 1 upon division by 8, the number formed by its fourth and fifth digits must be divisible by 4, which can be 48, 56 or 68.
Numbers that have remainder 2 upon division by 3, and those that are divisible by 3 but not by 9 are not squares. By trial and error, we know that the only square number satisfying these conditions is 1,34,689=367^2.
Hence, he had worked for 367 days, and had delivered 1,34,689 letters.
