High school Students btw ;(

I was playing around with number strings and noticed a really strange pattern. If you break consecutive numbers into groups that grow by one each time, a math magic trick happens where Pi (π) suddenly appears.

Here is how the grid starts:

* Group 1: 1

* Group 2: 2, 3

* Group 3: 4, 5, 6

* Group 4: 7, 8, 9, 10

* ...and so on.

Let's take Group 4 (7, 8, 9, 10) as an example. The size of this group is 4 numbers.

Now, pick each number in the group and find the absolute differences between it and all the other numbers in the same group, then add those differences up:

* For 7: The differences to 8, 9, and 10 are 1, 2, and 3. (1 + 2 + 3 = 6)

* For 8: The differences to 7, 9, and 10 are 1, 1, and 2. (1 + 1 + 2 = 4)

* For 9: The differences to 7, 8, and 10 are 2, 1, and 1. (2 + 1 + 1 = 4)

* For 10: The differences to 7, 8, and 9 are 3, 2, and 1. (3 + 2 + 1 = 6)

Next, take the reciprocal (1 divided by the sum) for each number, and add them all together:

* (1/6) + (1/4) + (1/4) + (1/6) = 0.8333

Finally, multiply that total by the size of the group (which is 4):

* 0.8333 × 4 = 3.3333

Here is the crazy part: As you go deeper down the grid and the groups get bigger, this final number locks directly onto Pi (3.14159...)!

* With a group size of 10, the answer is 3.1709

* With a group size of 100, the answer is 3.1418

* With a group size of 1,000, the answer is 3.14159

Holy fuck this book is hard as shit. I am a 2nd y math PhD studying this along with Hida's Geometric Modular Forms and Elliptic Curves rn. Looking for people to form a study group devoted to this. Let me know if you're interested and maybe we can form a gc/discord :)

(This book is "Arithmetic Moduli of Elliptic Curves to be clear")

I genuinely want to learn mathematics, physics, programming, computer science, and AI. I want to build a career in computer science, pursue AI research, and hopefully do a PhD someday. But lately, I've been feeling increasingly demotivated.

The problem is that I rely on AI to learn almost everything. Whenever I have a question, I ask GPT. Whenever I don't understand a concept, I ask AI to explain it. And the more I learn this way, the more I wonder: What's the point of learning anything when AI will always know more than I ever could?

Even the most basic AI models seem to possess more knowledge than I could acquire in a lifetime. And with new models becoming increasingly capable, I keep wondering whether the knowledge I'm working so hard to gain will even matter in the future.

With GPT-6, Claude Opus 5.5, Gemini 4, and other models competing to become more capable, it feels like the gap between human intelligence and AI capabilities will only continue to grow.

I want to understand the mathematics behind AI, learn physics, become a better programmer, and contribute to actual research. But sometimes I question why I'm putting in the effort when AI can explain concepts faster, write code better, and potentially solve problems that I might spend years learning to solve myself.

And even when I genuinely want to learn, I'm confused about what I should prioritize learning and whether any of it will remain valuable by the time I become proficient.

I'm not necessarily saying that learning is useless. I genuinely love the idea of understanding how things work. But I'm struggling to reconcile that curiosity with the feeling that AI might eventually outperform me at almost everything I want to become good at.

For those of you studying computer science, mathematics, physics, or AI, how do you deal with this feeling? How do you stay motivated to learn when AI is advancing so rapidly? Do you think deep technical knowledge will still matter for people pursuing AI research and PhDs, or should we be approaching learning differently now?

So for a while now,I have been seeing not many people being active on this sub as before. You can only find them under a new open problem proof, is this the observation with you guys too or am I missing something?

Also,it would be great if you can share your thoughts on other experiences of yours also related to this sub lately along these lines :)

Just create a cat with Fourier series.
I feel like understanding concept by visualization really helps!

As the title states, also how complex can it get?

A brief explanation of proof assistants

How can I study this course better? I prepared for this exam weeks in advance and am devastated by this grade. Any resources are appreciated. I’m crashing out.

I have seen Kneser's method of extending tetration to real numbers, but that doesn't seem to be right as I can't see how 2^^0.5 is not 1.55961...

As a starting point, we can see that 1+1=2 and 2x0.5=1. Also 1.4142...x1.4142...=2 and 2^0.5=1.4142. Similarly 1.55961...^1.55961...=2 and 2^^0.5 should be 1.55961...

To see integer values for all of them, the smallest number after 4 will be 256 where 256x0.5=128, 256^0.5=16 and as 4^4=256, so 256^^0.5 should be 4

We can get values at 0.5, 1.5, 2.5, ... for all numbers by this method, but there should be some way to make the curve continuous as well. Also a^^b should also satisfy a^(a^^b-1), so 2^^1.5 should be 2^1.55961 which is ≈2.9477... and 256^^1.5 should be 256^4 which is 4294967296

Is there any other way to extend tetration to real numbers

Source: The graph is computed exhaustively. Each state is five digits (a, b, c, d, e), each 0–9, so there are 100,000 states. The rule shifts every value one place left and computes a new last value, a⁴ − (a³ >> 6), kept to one digit (mod 10). Here >> is a bit shift, so a³ >> 6 means a³ divided by 64, rounded down. Every state has exactly one successor, and with finitely many states every starting point must eventually repeat, so each state ends up in a cycle.

Tools: A small open-source web app I wrote (React, Vite, Canvas). It enumerates all states, finds every cycle, and draws each cycle as a ring with the states that lead into it as trees around it. The layout is crossing-free.

How to read it:

• Each connected shape is one cycle plus everything that drains into it.
• Colour is the length of the cycle (legend at the bottom).
• Brightness fades with distance from the cycle.
• For this map there are 340 cycles in total, the longest of length 10.

You can type your own formula and explore: https://exxliang193.github.io/FunctionalGraphExplorer/ Code: https://github.com/ExxLiang193/FunctionalGraphExplorer

I started this after noticing a long time ago that adding the last two digits of a number string and appending the sum always ends up repeating.

Yeah so I have this question as a non math student in commerce but I have friends who have taken applied math and I can understand the reason but the thing is that they were upset because my school didn't give the option of taking core math with commerce and I was confused asf like why do you want to do trigonometry etc what's the point? Can people here tell me the reason 🤔.

Are there any good mathematics discord servers that are more focused on math discussion/research than homework help? I just joined the first math server I could find and was a bit dismayed to discover it is filled with hapless students. Is there anything better?

Learning maths is cool. It’s basically fundamental to EVERYTHING but I’m wondering what are the cons of it that isn’t so talked about

This rather surprising 4x4 magic square features 13 cells that revolve around three constants: 11, 13, and 15. By extension, 24 (11 + 13) and 28 (15 + 13) also appear.

I do not think I have discovered everything in this magic square, which is why I am asking for your help.