Enigma 1710

Spoiler for New Scientist Enigma 1710:

Since the second world war, car registrations in my country have consisted of three parts: the first is the four-figure year of manufacture, the second is made up of two letters, and the third consists of a number which, in my case, has three digits. However, we are now switching to all-digit car registrations, consisting of three parts. Existing registrations such as mine will be adapted by changing the two letters to a number by replacing A by 1, B by 2, and so on, so that, for example, GL would become 712.

The three numbers making up the three parts of my car’s registration will then all be multiples of 18, but none will be a perfect square. And overall the registration will use each of the digits 0 to 9 exactly once.

What is my car’s registration before the changes?

Okay, so the first number is four digits; the last is three digits. The middle number (after translating letters) could be two, three, or four digits, except we’re told each digit from 0 to 9 is used exactly once, so it must be three digits. All three numbers are divisible by 18.

The first must be a year since (the start of? end of? it doesn’t matter) World War II; the possibilities are:

1944 1962 1980 1998

But repeated digits are not allowed, so throw out the first and last:

1944 1962 1980 1998

None of these is a perfect square, so we can’t reject any on those grounds.

The possibilities for the second and third numbers are:

108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 378 396 414 432 450 468 486 504 522 540 558 576 594 612 630 648 666 684 702 720 738 756 774 792 810 828 846 864 882 900 918 936 954 972 990

But we can throw out any of these that are squares:

108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 378 396 414 432 450 468 486 504 522 540 558 576 594 612 630 648 666 684 702 720 738 756 774 792 810 828 846 864 882 900 918 936 954 972 990

or that have repeated digits:

108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 378 396 414 432 450 468 486 504 522 540 558 576 594 612 630 648 666 684 702 720 738 756 774 792 810 828 846 864 882 900 918 936 954 972 990

Since the first number must contain 1 and 9, the other two must not contain either:

108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 378 396 414 432 450 468 486 504 522 540 558 576 594 612 630 648 666 684 702 720 738 756 774 792 810 828 846 864 882 900 918 936 954 972 990

The second number must be a concatenation of two numbers, one less than 10 and the other greater than 19 but less than 27. Either the first or the second digit must be a 2, and the digit after the 2 must be less than 7:

108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 378 396 414 432 450 468 486 504 522 540 558 576 594 612 630 648 666 684 702 720 738 756 774 792 810 828 846 864 882 900 918 936 954 972 990

But 720 is not compatible with either 1962 or 1980, so the second number must be 234; then the first must be 1980. The remaining digits are 5, 6, and 7. 576 has already been ruled out since it’s the square of 24, so the third number must be 756.

The original license plate must be 1980 WD 756.