Q: I drive at an average speed of 30 miles per hour to the railroad station each morning and just catch my train. On a particular morning there was a lot of traffic and at the halfway point I found I had averaged only 15 miles per hour. How fast must I drive for the rest of the way to catch my train?
A: It’s impossible.
Let’s say that you need to go 30 miles which would take an hour.
If you were at the halfway point, 15 miles, at 15 miles per hour, you have already been traveling for an hour to go that 15 miles — and you are only halfway there!
Sorry…you will miss your train.
Q: Write down all the integers from 1 through 60 to form the number
Now delete 100 digits from this number.
Without rearranging the digits, what is the largest number possible?
A: Writing down all integers from 1 through 60 gives you a number 111 digits long:
9 numbers at 1 digit apiece: 9 digits
51 numbers at 2 digits apiece: 102 digits
Total: 111 digits.
So deleting 100 digits leaves us with 11 digits. It’s be best to have as many 9’s as we can as far to the left as we can. But there’s only 6 of those, and one of them is already in the hundreds place before we do any deleting.
If we’re not allowed to rearrange the digits, the largest number is:
99 999 785 960.
Q; For what value of n does 3^1998 + 9^999 + 27^n = 3^1999?
(^ = to the power of)
Rewriting each number in base 3:
3^1998 + (3^2)^999 + (3^3)^n = 3^1999
3^1998 + 3^1998 + 3^(3n) = 3^1999
It takes 3 3^1998s to make 3^1999 as 3 x 3^1998 = 3^1999
Therefore 3^(3n) = 3^1998
so 3n = 1998 and n = 666