### Brain Teasers

# Postman

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 100,000 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?

The day he retired, he noticed that he had delivered a total of more than 100,000 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?

### Hint

The number of days he had worked also has its digits arranged in ascending order.### 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 123456 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 134689=367^2.

Hence, he had worked for 367 days, and had delivered 134689 letters.

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## Comments

Good teaser - I had to work it out the long way round.

Must be nice to retire after only 367 days work!

Must be nice to retire after only 367 days work!

You mean I got one right? Yipppee keep these coming. As I am not so good at math, but learning

There is a MUCH simpler way to get the answer - and I dont mean "Click here for answer". If you square the digits 1-9, you see the letters MUST end in 9 and days MUST end in 7 (cannot end in 3). Days greater than 351. Try 357, 367, 457, 467 - they are the ONLY possibilities.

Thanks radguy, I missed it because I was showing how to solve this teaser WITHOUT the hint.

Good Teaser. Had to put a bit of thought into it before figuring it out. Good Job!

Wow! That was a tough one! But a good one too!

WOW, hard, but not too hard, keep them coming

Dec 21, 2005

There is a flaw in the logic in that the mailman delivered more than 100,000 letters but there is not a limit on how much more...so the numbers ascending cound be in the range of the given answer up to and including 456789 This would give you a different count for the number of days he worked (and let's not forget that postmen do not work on federal holidays)

Then you can try to correct it.

my brain hurts !

Nice teaser!

No need to "correct" the teaser. 134689 is the largest square number with ascending digits.

No need to "correct" the teaser. 134689 is the largest square number with ascending digits.

"There is a MUCH simpler way to get the answer - and I dont mean "Click here for answer". If you square the digits 1-9, you see the letters MUST end in 9 and days MUST end in 7 (cannot end in 3). Days greater than 351. Try 357, 367, 457, 467 - they are the ONLY possibilities."

That's exactly what I did :-D

That's exactly what I did :-D

Wow ! This teaser is really good. But plz tell me the answer is the largest square number in ascending order .

Thank you so much. I got first prize with this teaser in sub district level . Nice question .keep it up . Special thanks to braingle.....

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