Find the number n such that the following alphanumeric equation:

   KYOTO
   KYOTO
 + KYOTO
   TOKYO

has a solution in the base-n number system.

(Each letter in the equation denotes a digit in this system, and different letters denote different digits)

Firstly, I have to alert you all that this is the SECOND part of a full solution that I post for this problem. The first part is in a previous comment (which unfortunately has 'no subject' in the heading due to troubles with my browser - *%&*$ spyware!)

So, we have shown that letter O must be equal to 0 (see part 1 of this solution in previous comment). Now, notice that with O=0, the given problem separates nicely into 3 components; column 5 (which gives us no further information and can be henceforth disregarded), columns 3 and 4, and columns 1 and 2. We can separate the first pair of columns from the second because of the O's in column 3.

As O=0, columns 3 and 4 together tell us that T+T+T=KY, where KY here represents positional notation in the base n (that is, KY does not represent K times Y, but rather nK+Y). Notice here that we cannot be carrying a digit from column 3 into column 2 - in fact, the digit K is itself just the amount that we carried from column 4 into column 3 and then added to the 3 zeros. (It is the fact that there is NO CARRY from column 3 into column 2 which makes us able to separate the problem into the 2 pairs of columns now)

Now here's the neat part. Notice that columns 1 and 2 together tell us that KY+KY+KY=TO (again using positional notation in the base n). So, 3(KY)=TO. But columns 3 and 4 told us that 3(T)=KY. Hence, 3(3(T))=TO, or 9(T)=TO. Remembering now that O=0, we can see that in our base n

9(T)=T0
9(T)=n(T)+0
9(T)=n(T)
n=9

So, the base that we're looking for has to be 9! At this point we can go on to search for possible ways to number the letters so that the summation works in base 9, although the problem doesn't ask us to do this. It turns out that there are only 4 possible ways to number the letters; these were given in the first comment posted for this problem. I leave it to the interested reader to show that KYOTO can only be one of these 4 base 9 numbers; 13040, 16050, 23070 or 26080.

I hope that this solution was clear and not too long-winded for you! This problem was very interesting - again, thanks DJ for keeping them coming.

-John

Posted by John Reid on 2003-09-21 15:53:58