Math Quiz Seven

I was recently reminded of a problem I saw published in 88 or 89 in the soviet Math journal (forgot its name ). The original was in Russian alphabet which I cannot type, so I'll transliterate:

 KYOTO +
  KYTO
 -----
 TOKYO

The nice thing in Russian alphabet that all 3 words are names of big cities (KYTO stands for Quito). In Latin writing, I hope there's a KYTO somewhere.

Is this a trick question? Or is there a legitimate (non-zero) solution? Or is discovering the answer to THAT question the whole point of the problem?

The point of math quizzes is that when you find valid, interesting questions about the puzzle, you should answer yourself. Obviously there's a non-zero solution. Take letters to represent distinct digits.

Solution:

(The numbers are to base 7.)

Wow, that was fast.

Curse you, Red Baron! You beat me to the punch by a couple of minutes! Nevertheless, well done!! You have discovered the solution in the lowest base where there is a solution (base 7). Here is a solution in base 11:

This is, I believe the next lowest base solution. There are undoubtedly infinitely many correct solutions. -- AnonymousDonor

The T does not match in the above. The solution first posted is unique and obviously now is spoiled, but the fact that it was solved in maybe less than 1 hour could be a good indication of the IQ level still kicking on C2.

Yes, it is possible to demonstrate that the solution first posted is unique in the sense that no other solution in any *conventional* number system (ordinary base b numbers) exists. I struggled for a while attempting to find another solution in some *exotic* number system, such as the integers in base -7, with digits {0,1,2,3,a,b,c}, where a=-1, b=-2, c=-3. (It is possible to represent all the integers uniquely in base -7. Counting begins: 1, 2, 3, ac, ab, aa, a0, a1, a2, a3, bc, bb, ba, b0. Those are the numbers 1 through 14. Note that the positions in a base -7 number representing odd powers of 7, (-7)^1, (-7)^3, (-7)^5, etc. have a negative positional factor, so that, for example, c2 = ((-3) * (-7)^1) + 2 = 21 + 2 = 23. And in base -7, 123 = (1 * (-7 ^ 2)) + (2 * (-7 ^ 1)) + 3 = 49 - 14 + 3 = 38.) Well, it turns out that that no solution exists in base -7, or in any other *exotic* number system, either, as far as I can determine.

Well, something kept bothering me about the uniqueness of the solution. It just felt intuitively as though the problem should be, or could be, made a special case of a more general problem, a problem with a more general solution, or set of solutions. Then it struck me in a flash! If the problem is reformulated as

 KYOTO +
 AKYTO
 -----
 TOKYO

one can produce a schema (or algorithm) which generates an infinite number of solutions, of which the solution first posted is a special case. (Of course, AKYTO does not seem to have any meaning; but that's neither here nor there.)

Can someone please confirm that this solution-generating schema (or algorithm) actually exists for the revised quiz, and perhaps even write it down? Just so that I can be sure that I am not hallucinating again, as when I claimed a solution in base 11 for the original puzzle!

-- AnonymousDonor

OK, let's try this. First, let us adopt the notation Z(n) for a digit with value n (for 0 thru 9, these are the ordinary decimal digits; for digits greater than 9, which is to say for bases greater than 10, we need to devise these additional digits, of course.) Now, for all n > 2, specify base (4n-1), and

 O = 0;
 T = Z(3n-1);
 Y = Z(2n-1);
 K = Z(2n);
 A = Z(n-2);

This seems to satisfy the general condition and yield a general solution.

For n = 2, there is a special case. The base is 7; the values of T = 5, Y = 3, K = 4 are correctly generated, as in the first solution above; however, instead of A = 0 (which would conflict with O = 0), in this case A = spaces; the white space in the leading position being the equivalent of zero.

-- JohnReynoldsTheStudent

Hence if the unknown digits have to be between 0 and 9 inclusive, there is just one additional non-exotic solution (corresponding to n=3), as shown below, which uses a base of 11.

 65080 +
 16580
 -----
 80650

[How come JRTS added his reply at the same time as AnonymousDonor's italicized comment and question?]