Next future date such that the all 8 digits of date format DD/MM/YYYY is all different and the product of DD, MM and YYYY is a square number

This puzzle is inspired from here.

When is the future date such that the all 8 digits of date format DD/MM/YYYY is all different and the product of DD, MM and YYYY is a square number?

Note that "the product of DD, MM, YYYY" means DD × MM × YYYY.

• I assume it is intended to be done without a computer, can you add the tag if I am correct? Apr 24 at 16:25

13/09/2548

First notice that

If 0 does not appear in MM then MM=12 and 0 must appear in DD.
Also, either 1 or 2 must appear in either DD or MM.

Furthermore

If YYYY contains a "loose" (with an odd power) prime factor greater than 31 (or equal to 11 bearing in mind repetition) then there is no way for DD x MM x YYYY to be a square. Also, if YYYY has a second highest loose prime factor greater than 10 then we can also not form a square.

The first future YYYY that satisfies this is 2349 although this would make DD=29 which is not allowed by repetition.
The next YYYY that satisfies this constraint is 2375 which is $$5^3 \times 19$$. This forces DD=19 and MM=05 but we have used 5 twice.
The next YYYY that works is 2394 which forces DD=19 and 9 is repeated.
The next YYYY=2436 but DD=29 and 2 is repeated.
The next YYYY=2457 which puts DD=13 but then MM must be divisible by 3 and 7.
The next YYYY=2496 which factorizes as $$2^6 \times 3 \times 13$$. This forces DD=13 (to not get repetition) but then MM=03 and we repeat 3.
The next YYYY=2548 which factorizes as $$2^2 \times 7^2 \times 13$$. This forces DD=13 and MM to be a square. The only MM which avoids repetition is MM=09 so we arrive at the answer.

This method is slightly tedious but, bear in mind, for each possibility we only need to check divisibility by ten primes (not a complete factorisation).

• I don't understand why you only consider DD in your first deduction, why can't the digit appear in YYYY? Apr 25 at 14:32
• @MatthieuM If MM=12 what are the possible values of DD that do not include '1' or '2'? Apr 25 at 14:49
• Ah! Of course. I was focusing on why YYYY couldn't include the digit, instead of why DD couldn't exist without the digit. Thanks! Apr 25 at 15:03