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2D is a multiple of 9, and so its digits add up to a multiple of 9. This means that 4A is a multiple of 9, and its digits add up to a multiple of 9 too.
The digits of 4A can add up to at most 27 (if it's 999) and so must add add up to 9, 18, or 27. These would make 2D equal to 81, 162, or 243 respectively. 2D cannot be 81 (as it's a three-digit number);
it also can't be 243, as this would mean 4A was 999, leading to a clash in the centre square. Therefore the sum of the digits of 4A is 18 and 2D is 162:
The sum of the digits of 4A is 18, so 4A is 369, 468, 567, 666, 765, 864, or 963. Of these, only 765 is a multiple of 5, and so 4A must be 765:
3D is greater than 2D, so the first digit of 3D is at least 2. If the first digit of 3D was 3 or higher, then 5A would be greater than 700, meaning 1D would be greater than 1000 and not a three-digit number.
Therefore:
The last digits of 3D and 5A are the same, and so the last digit of 1D (and the first digit of 5A) must be even. 5A is greater than 500, and so it's first digit must be 6 or 8;
it cannot be 8 as this would make 1D greater than 100. And so:
This only leaves one option for the last two digits to make 1D's clue correct:
Today's number is 812.
