Welcoming in 2025

$2025$ is a very "squarish" number as the following properties will illustrate:

• $2025={45}^2={27}^2+{36}^2$
• $2+0+2+5=9=3^2$
• It is divisible by the square of the sum of its digits:
  ${\displaystyle \frac{2025}{81}}=25$
• Omitting the zero still leaves a square $225={15}^2$
• Omitting the first two digits still leaves a square $25=5^2$
• Adding $1$ to the first digit gives the square $3025={55}^2$
• Adding $1$ to each digit gives the square $3136={56}^2$
• Adding $4$ at the front gives a square $42025={205}^2$
• $2025$ is the smallest square that can be formed from $20$ by adding one or more digits
• Square that can be seen on a digital clock as in $20:25$
• Written as "twenty twenty five" it has $16=4^2$ letters
• Can be written as a sum of three distinct squares in $9=3^2$ different ways e.g. $4^2+{28}^2+{35}^2$ ... permalink
• It is the sum of the first nine numbers squared: $(1+2+\cdots +8+9{)}^2=2025$
• Deleting a zero from its cube ($8303765625$) gives $833765625={28875}^2$
• Imagine writing down the number $1$ once, the number $2$ twice, the number $3$ three times, and so on up to the number $45$ forty-five times, like this:$$12233344445555\dots 454545$$The total number of digits is $2025$, which is the square of $45$. This coincidence does not occur for any other number greater than $1$.
• The sum of entries (in red, below) of a $9\times 9$ multiplication table is $2025$:

• The determinant of the $4\times 4$ matrix whose rows are cyclic permutations of the first $4$ composite numbers is equal to $-2025$.