He was 7 years old, so it was impossible for him to give consent for anything. His parents gave consent on his behalf.

I didn't realize this was published at the time. Still, I wonder what the current, adult Terence thinks. Whether or not legal recourse is available doesn't change Tao's feelings about it and isn't determinative regarding republishing it now is a good idea.

I didn't see that in the document. What page is it on?

I think they mean at the bottom of p216 (pdf page 4), where he says he doesn't know, r+s=80 but there isn't enough information to solve for r and s.

There's two questions that are intended to be wrong (probably to test confidence). One with insufficient information and where the question itself implies falsehoods.

The questions are on page 215 (3/26) and Tao's answers are on the next page.

There are probably hundreds of people on this site who had the same enthusiasm for math and time dedication as Terence Tao, but lacked his extreme outlier fluid intelligence, processing speed, perfect memory, and even handwriting talent(!). Terence Tao mastered calculus at an age when most future-mathemician geniuses weren't yet strong readers of chapter books.

I mean I can make up any speculation too if you want. I seriously seriously doubt your claim. It’s extremely extremely rare to find “3-4 hours of maths textbook reading” levels of enthusiasm in 7 year olds.

I don’t doubt he’s talented too. But, it’s a chicken and egg problem and I’m sick of people pretending it isn’t.

You are saying something interesting but too esoteric. Can you explain for beginners?

You could get rich by solving ARC 2 tasks yourself instead of forwarding the work to an LLM, given a client willing to pay LLM rate.

Title: "California’s New Bill Requires DOJ-Approved 3D Printers That Report on Themselves"

Actual fact: California’s New Bill Requires that 3D Printers Get DOJ Approval as Firearm-Blocking"

(The "report on themselves" is fiction invented by Adafruit.)

"to be able to get a 3D printer" is implied in the "requires" wording. There's no problem with that part.

Hmm, how many bits has the space of all positive integers?

That's not what the (clumsily written) article is about. It's not about sampling the integers to choose an integer, it's about sampling the prime factors of an integer, as a measure of evenness of distribution of prime factors.

It's measuring the information in the prime factorization, the information in the number as a value.

Here's an intuitive description of the entropy, [log(log(n)) -sum(log(p_i) log(log(p_i)))]:

The entropy of a random integer N is the volume of the gap between how much space N takes up and how much space its internal components take up.

This can be visualized as The City of N, in base 2. (OP used log_e, but that's too hard to draw.)

1. The Foundation (The Factors)

Take a random number N and break it into its prime factors. We write these prime factors in binary, side-by-side, along the bottom of a page.

The total width of this baseline is roughly log_2(N)(the number of bits in N).

2. The Cloud Ceiling (The Potential)

We write down the length of (N written down in base 2) in base 2. (If N = 46 = 101110 (base 2), its length is ~6 = 110 (base 2),

We write that number vertically (110) to set the Maximum Ceiling Height.

Finally, we look at the number N itself.

3. The Buildings (The Structure):

Above each prime factor, we construct a building.

* The Width: The width of the building is simply the length of that prime factor in bits.

* The Height: To determine how tall the building is, we look at its width and write that number down vertically in binary.

To normalize, we zoom our camera so the length (log) of N fills the view.

The Entropy (The Visible Sky):

The Sky: This is the empty space between the tops of the buildings and the top of the picture (cloud ceiling).

The Entropy of N is exactly the total area of the visible sky.

If N is prime, the building is as wide and tall as the whole city and touches the cloud ceiling. No Sky. Zero Entropy.

If N is a random integer, it usually has one wide building (the largest prime) that is almost as tall as the ceiling, and a few tiny huts (small primes) that leave a massive gap of blue sky above them.

Here is the visualization for N = 46. (Binary 101110, length ~6).

          (46)
      1 0 1 1 1 0  (length ~5.5)
     ---------------------------
         |1 1 1 1 1|
         |0 0 0 0 0|  
     |1 1|1 1 1 1 1|
     +---+---------+
     |1 0|1 0 1 1 1|
      (2)    (23)

(Visualization not exact due to rounding of logarithms, and because)

Interpretation:

Building 23 is tall. It reaches Level 3 (101 is length 5). It touches the ceiling (Level 3). There is zero sky above it.

Building 2 is short. It only reaches Level 2 (10 is length 2). There is one unit of sky visible above it.

Total Entropy: The total empty area above the buildings is small (just that gap above factor 2), which matches the math: 46 is "low entropy" because it is dominated by the large factor 23.

A number with High Entropy would look like a row of low, equal-height huts, leaving a massive amount of open sky above the entire city.

This is a very unenlightening exposition that could discourage people from studying this beautiful problem.

A much better way to analyze it geometrically. The 6D problem has 2D trivial symmetries, and is parameterized by 4 polar coordinates: circumcenter distance from origin, circumradius, and 2 internal angles of the triangle. Then the solution is just [expected value of the area of a triangle on unit circle] times the radius integral in the OP article, divided by the π³ volume of the triangle space in rectangular coordinates.

HN as a population skews heavily away from "real-world peer-scheduled activities"

Submitted blog post is from Lichess user (rated ~2150 on Lichess), not Lichess organization itself.