Tuesday, 31 December 2024

Welcoming in 2025

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

  • 2025=452=272+362
  • 2+0+2+5=9=32
  • It is divisible by the square of the sum of its digits:
    202581=25
  • Omitting the zero still leaves a square 225=152
  • Omitting the first two digits still leaves a square 25=52
  • Adding 1 to the first digit gives the square 3025=552
  • Adding 1 to each digit gives the square 3136=562
  • Adding 4 at the front gives a square 42025=2052
  • 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=42 letters
  • Can be written as a sum of three distinct squares in 9=32 different ways e.g. 42+282+352 ... permalink
  • It is the sum of the first nine numbers squared: (1+2++8+9)2=2025
  • Deleting a zero from its cube (8303765625) gives 833765625=288752
  • 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:12233344445555454545
    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 9×9 multiplication table is 2025:
  • The determinant of the 4×4 matrix whose rows are cyclic permutations of the first 4 composite numbers is equal to 2025.

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