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:
  \( \dfrac{2025}{81}=25\)
• 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: \( (1+2+ \dots +8+9)^2=2025\)
• 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 \(-2025\).