For over 2 years now, a lot of people (and not just me) have been waiting for 2025. It is for the same reason (or should I say reasons, the reasons being 7). However, there is another reason why I have been looking forward to this year.
The primary reason is its close association with the number 9, a number I have always been entranced by. In addition, 2025, in itself, is a special number, (at least like many other numbers if not more). Let's unwind it a bit more, shall we?
I would skip the obvious ones which most would already know. For example, 2025 being a perfect square (the only one for this century), or it being the sum of the cube of first 9 numbers, or it being the square of sum of numbers from 1 to 9.
I will also skip the very visible aspect of it, that is how by multiplying its component numbers (20 and 25) results in the year 2025 (that is (20+25)x(20+25) = 2025), or how it is the result of the multiplication of two perfect squares (81, square of 9 and 25, square of 5) or the sum of three perfect squares (1600, 400, and 25). It's relation with 3, 6, and 9 (the magical trio, as I believe them to be) is very apparent even from these obvious ones.
Let's explore some actual interesting ones, the less commonly known ones. (Please note the connect with 3, 6, and 9.) First, if we go on writing digits from 1 to 45, the number of times corresponding to their values, that is 1 once, 2 twice, 3 thrice, upto 45 forty-five times (1223334444...45454545 (45 times)), the total number of digits is 2025! And this doesn't happen for any other number greater than 1, except for 45 (and 4+5 =9). Coincidence? I don't think so.
Let's explore some graph theory for the second fact. If we go on to color the sides of a rectangle, using at most 9 colors, distinctly, should we take a guess how many ways there can be? Surprise, surprise... 2025 essentially distinct ways! And if you want more, try finding out how many spanning trees could be formed in a complete bipartite graph with 3+5 vertices.
If we go for figurative numbers, again a personal favorite, and just touch upon a bit of geometry (oh! the elegance), we can visualize 2025 in terms of triangular numbers and centered octagons, as it lies so elegantly in their intersection. We have studied triangular numbers in school, grade 5, if I am not mistaken. 2025 is the sum of two consecutive triangular numbers, T44 and T45. (T44 = (44x45)/2 = 990 and T45 = (45x46)/2 = 1035). For centered octagonal numbers, if we choose 23 levels (22 rings plus 1 central point), we get a total of 2025 points.
The true beauty comes up when we plot T44 and T45 in their triangular positions and rearrange the 2025 points to form the centered octagon, a beautiful match in two worlds! It thus signifies growth and prosperity (for the consecutive triangular numbers), and balance and equilibrium (as a centered octagon)... a year of transition, if you will.
Coming to the whimsical aspects, if we write the results of multiplication tables of 1 through 9 (that is 1x1 through 9x9), essentially a 9x9 matrix, the sum of all results would be 2025. Another silly but interesting aspect is that if we add 1 to each digit of 2025, the result 3136 is still a perfect square. And 2025 is quite possibly the only positive square to have this aspect. Additionally, if we add 1 to just 1 digit also (resulting in 3025), it still remains a perfect square. Interestingly, 55 squared is 3025, and 56 squared is 3136. (Did we notice the 3 and 6 aspects?)
Those were 5 aspects. Let's add 4 quick silly but fun ones to make it 9. For all numbers from 1 through 9999, with the last digit strictly greater than the other digits, there are exactly 2025 numbers. If we cube 2025 and omit the 0 in the result (8303765625 becomes 833765625), the number so left is also a square (square of 28875). Did we observe that the sum of the digits of is 9 and that of 28875 is 3?
Keeping its close association with 9, it can be written as a sum of positive squares in only one way, that is 27 squared + 36 squared. And to end it with how we started this piece, let's see something associated to 7. If we take the set {A, B, C} and find the number of strings of 7 letters (that do not contain AAAA or BBBB as a substring), the number is 2025.
Even though there are a lot other special things about 2025 as a number, and I really am looking forward to discuss it in details... even so, we will pause here... for this already has been way lengthier than I initially expected it to be. However, if you are looking forward to a discussion about this year or the number 2025 (royal, purple), we can always find opportunities. Or we can make our own. Have a happy 2025!
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