### Brain Teasers

# Red 8's

We wrote down all the 3 digit numbers in an increasing order. We used a red pen for the even numbers digits and a blue pen for the odd numbers digits. How many red 8's are there on the paper?

### Hint

Do it in hundreds. First finds how many 8's there are from 100-199, then 200-299, and so on.### Answer

From 100-109 there is 1 eight (108). From 110-119 there is another 8. So for every 10 numbers there is one number that has an 8. But, the ten digits from 180-189 are different since the tens column now has an eight. There are 5 even numbers in those 10 digits so there must be 5 eights but 188 has 2 eights and it is an even number. So, that means that there are 6 eights in the 180-188 digits. Altogether, the number of eights from 100-199 is 9+6. Since there are 9 hundred numbers (0-99 doesn't count), it would have been multiplied it by 9 but 800-899 is different so only multiply it by 8. Now, figure out how many eights are in the 800-899 numbers. There are more than 50 since there are 50 even numbers where the eight is the hundreds digit. But, 808, 818, etc have 2 eights. So, add 9 more eights because 880-889 is different. Now we have 8(9+6)+50+9 red 8's. There are 5 even numbers in the 880-889 numbers. So, add 5 but 888 has 3 eights so add 1 more. Now we have 8(9+6)+50+9+5+1. Then, simplify:8(9+6)+50+9+5+1

=8(15)+50+9+5+1

=120+50+9+5+1

=185

So, the answer is that there are 185 red eights on the paper.

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