Number Detective

Follow the clues to find the mystery number.

Six Is the Sum

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

(w)holy Numbers

A church hymn book contains 700 hymns. The numbers of the hymns are displayed by combining special small single-digit boards. What is the minimum number of small boards that is needed?

Oddly

Stage: 2 Challenge Level:

I looked at this one as soon as I got on the site and started off by writing down all of the ones up to $199$:
$111$, $113$, $115$, $117$ ... $197$, $199$.
I got the answer to all of them which was $3875$.
I then changed the first digit to $3$ to make it $311$, $313$ ... $399$.
Then I did the same for all the rest.
My final answer when I had added it all up was $69 375$.

Nicely done Adam from Poltair Community School and Sports College, Cornwall!

A big thank you too to Michael for your elegant method of finding the sum of all 3 digit numbers each of whose digits is odd:

I think the answer is $69 375$.
1. I estimated the total c $50 000$.
2. I wrote out all the three-digit numbers $100$-$199$ which had odd digits only, and observed a pattern: value of $100$ occurred $25$ times.
3. Decided that if all those numbers were written out, the values of $100$, $300$, $500$, $700$, and $900$ would each occur $25$ times; the values of $10$, $30$, $50$, $70$, and $90$ would do the same; and the values of $1$, $3$, $5$, $7$, and $9$ similarly.
4. Therefore, the sum can be simplified to: $25(100+300+500+700+900) + 25(10+30+50+70+90) + 25(1+3+5+7+9) = 69 375$
5. This is still very large!
We can simplify it further to: $25(111+333+555+777+999) = 69 375$