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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: $25(111+333+555+777+999) = 69
375$