When was Ensquare born?

Question

Professor Matlogic posed this question to his smartest math student:

"Famous mathematician Ensquare was born on this day, this month and this year AD at this time (hour and minutes) pm. True to his name, all those five numbers (the day, month, year, hour and also minutes-seperate from hour number-) were square numbers. So can you tell me when Ensquare was born exactly?"

"No, I cannot" said the student.

"OK then. What if I tell you that those 5 numbers are made up of digits 1 to 9,with each digit used exactly once. No repeats. Now, can you?"

"Hmm. Let me think."

"No I cannot still" the student after a long thought.

"OK then. He was born on a Wednesday" Professor declared.

"Now I know!" said the student cheerfully.

Do you know? What is the logic?

Answer 1

After the first statement (numbers are square), we can say that :

The month and hours are a square number lower than or equal to 12: The possibilities are 1, 4 and 9. The day is a square number lower than or equal to 31: The possibilities are 1, 4, 9, 16, 25.

The minutes are a square number lower than 60: The possibilities are 1, 4, 9, 16, 25, 36 and 49. We can remove 49, because at least the 4 or 9 will be used to form the month and hours, and this will contradict the second statement. This leaves: 1, 4, 9, 25, 36.

Concerning the year of birth, between 0 and 2021, there is only 1 square number including a 7 and a 8, which must be used here because they cannot be used anywhere else : 784. The month and time will therefore be worth 1 and 9, but it is not yet clear in what order.

Here are the remaining digits :

At this point we are left with the numbers 2, 3, 5 and 6 to compose the day and the minutes.

This leaves as the only possibility for day and minutes :

For the day: 25 and for the minutes: 36

The third statement allows us to

All we have to do is to know which month is the right one, for this the last statement of the teacher will help us. The 25th of January of the year 784 is a Wednesday, and the 25th of September of the year 784 is a Tuesday (according to an online tool). So the correct month is January, which is 1, leaving us with 9 for the hour number.

And so the answer is :

So Professor Ensquare was born on January 25, 784, at 09:36 pm.

(Sorry if my English is not very good, this is a foreign language for me.)

Answer 2

EDIT: I missed that the question asked for the digits 1 through 9 used exactly once. My logic works for a date that has numbers 1 through 9 used no more than once.

Ensqare was born on

April 25, 81 at 9:36

Reasoning:

First let us get which numbers qualify for the month, hours, days, and minutes. That is a count of 7 (1,4,9,16,25,36,49) numbers that need to fill 4 spots.
I then went the long way around and listed all squares from 1 to 2021. First I removed any number that had a repeated digit or a number that contained a 0, this left us with 27 numbers to choose from for the year.

Looking at the 7 numbers for the min/hr/day/mnth spots, we can see that if the year contains a 6 and a 9, that would eliminate 4 of our 7 numbers (9, 16, 36, 49) and we wouldn't have enough left to populate our min/hr/day/mnth spots. Similarly, a year containing a 4 and a 6 would eliminate 4 numbers (4, 16, 36, 49). Similarly (again), if the year contains a 4 and a 9 that would eliminate 3 numbers (4, 9, 49) and then 16 would clash with 1 and 36. This leaves us with 17 years to choose from.

From here we can start checking numbers. If there are less than 4 valid numbers, it is a bust and we can throw it out. I don't know how to make a table, so this is the best I can do.
Year | valid numbers (1, 4, 9, 16, 25, 36)
841 | 9, 25, 36 (bust)
784 | 1, 9, 16, 25, 36
729 | 1, 4, 16, 36 (bust, 36 and 16 clash)
625 | 1, 4, 25, 49 (bust, 4 and 49)
576 | 1, 4, 9, 49 (bust, 4 and 49)
529 | 1, 4, 16, 36 (bust, 1 and 16)
361 | 4, 9, 25, 49 (bust 4, and 49)
324 | 1, 4, 9, 16, 49 (bust, 4 and 49/1 and 16)
289 | 1, 4, 16, 36 (bust, 1 and 16)
256 | 1, 4, 9, 49 (bust, 4 and 49)
81 | 4, 9, 25, 36
36 | 4, 9, 25, 49 (bust, 4 and 49)
25 | 1, 4, 9, 16, 36, 49
16 | 4, 9, 25, 49 (bust, 4 and 49)
9 | 1, 4, 16, 25, 36
4 | 1, 9, 16, 25, 36
1 | 4, 9, 25, 36, 49

This leaves us with the years 784, 81, 25, 9, 4, 1 as possible solutions

From there I calculated valid dates starting with the largest year, and then checked their day of the week at timeanddate.

For year 784, the number 36 has to be the minutes (it is too large for month, day, or hour) and 25 has to be the day (too large for month or hour).

year | mth | day | hr | min
784 | 9 | 25 | 1 | 36 (Saturday)
784 | 1 | 25 | 9 | 36 (Sunday)


For year 81, the number 36 has to be the minutes and 25 has to be the day from similar logic as above.

year | mth | day | hr | min
81 | 9 | 25 | 4 | 36 (Thurs)
81 | 4 | 25 | 9 | 36 (Wednesday)

April 25, 81 at 9:36 was the first occurrence of a Wednesday so that is where I stopped

Note: If anyone want's to format my tables to be prettier, I would greatly appreciate it