Albert remarked that even when it is turned upside down, the number plate of his car still made sense.

He observed that this would however increase the registration number on it by 8505.

Given, all of the four digits in the number is different, find out what was his car license number?

First, only 5 digits can be turned upside down and still be a digit -- 0, 1, 6, 8, 9.

Since the upside-down version (U) is 8505 greater, the plate number (N) must be 1494 or less (9999-8505); otherwise U would be more than 4 digits. Furthermore, since every digit is unique, the minimum N is really 0123, the maximum U is 9876, and therefore the range of N is really 0123 to 1371 (9876-8505).

So we know the first digit of N (N1) is 0 or 1. Since each of these possibilities equals itself when upside-down, that means the last digit of U (U4) = N1. In the equation NNNN + 8505 = UUUU, N4 must be one of {0,1,6,8,9); therefore U4 can only be 1 (since for U4 to be 0, N4 would have to be 5, which is not one of the "invertable" digits).

Knowing that U4 and N1 are both 1: 1NNN + 8505 = UUU1, so we can calculate that N4 = 6, and therefore U1 = 9 (6 inverted).

Now we have 1NN6 + 8505 = 9UU1 and only {0,8,9} available for N2 and N3. Going back to 0123
≤ N ≤ 1371, we see that N2 can only be 0. By inversion, U3 is also 0.

So, 10N6 + 8505 = 9U01, with N3 in {8,9} and U3 in {6,8}. N3 = 8 won't work, because 8 + 0 + 1 (carried from 4th position) = 9, which is not available for U2. Therefore, N3 = 9 and U2 = 6.

1096 + 8505 = 9601

Posted by Rick on 2003-02-06 12:16:20