# Red, yellow and orange numbers

Every day, Little Johnny Red picks a positive integer $m$, arbitrarily permutes the digits of $m$ to a new integer $n$, and then computes the sum $m+n$. The sums that can be produced that way are called red integers.

Every day, Little Annie Yellow picks a positive integer $p$ that entirely consists of $1$s, and then multiplies $p$ by nine. The products that can be produced that way are called yellow integers.

An integer is orange, if it simultaneoulsy belongs to the red and the yellow numbers.

Question: How many integers below one Googolplex are orange?

(Remark: All integers in this puzzle are represented in decimal, and all digits are decimal digits.)

• Wouldn't it be more logical to define yellow integers as a number consisting of only 9s? Oct 1 '15 at 8:04

exactly half a Googol ((10^(100))/2)


Every orange number o has to consist completely of the digit 9 and there has to be a number m which completely consists of pairs that add up to 9 (1+8, 2+7, 3+6, 4+5) and has the same length as o.

Other numbers m are not possible, as we have to get a sum of exactly 9 in every digit.

If we tried to do it with a carryover the sum would have to be 18 in every digit with a carryover of one from the digit below, which is not possible on the smallest digit.

Therefore all numbers m need to have an even number of digits which results in all numbers o having an even number of digits.

The maximum number of digits for numbers below below one Googolplex is exactly Googol and as half of those are an odd number of digits the solution is half a Googol.

• Dang, I was a minute too late :( Oct 1 '15 at 8:45

$5*10^{99}$
Yellow numbers are numbers that consist only of $9$'s. $9, 99, 999, 9999, ...$ are yellow numbers. To find orange numbers we need to check the yellows whether they are also red numbers or not. We can easily see that all yellow numbers with an even number of digits ($99, 9999, 999999, ...$) are red numbers, as you can construct a number $n$ with the digits $9 - x, 9 - x, ... , x, x, ...$. For example $9999 = 1188 + 8811$.
Now what is left are the yellow numbers with an odd number of digits. Those are not red numbers, as for every digit $9-x$ you need a second digit $x$ for them to add up to $9$. This will always leave you with one digit left for which there exists no 'partner'.