Mathverse @_mathverse_

Repeat five moves 1,259 times and 19 of the cube's 20 movable pieces are still out of place. The 1,260th repeat puts every one of them home. The moves are R U2 D' B D'. Nothing special about them. Each group of pieces runs on its own clock. Some come home every 2 repeats, others every 4, 9, 14 or 15. The cube is solved only when all the clocks hit zero together. That is the least common multiple: 2² · 3² · 5 · 7 = 1,260. It is the longest loop any move sequence can make on a cube. That is 6,300 moves, about 105 minutes at one turn per second, inside a space of 43,252,003,274,489,856,000 positions. Every shuffle is a set of clocks. Solving it is waiting for the hands to line up.

@_robtimus_ Usually shorter since one extra twist breaks the sync and the parts settle back much sooner

@Shred_0x Once a gearbox costs pennies the real question becomes how many years it lasts before backlash shows up ?

@StillnessHQ The cube just waits until every hidden clock inside it lines up at once

@Y0urDumble 🤝

This robot knee throws away 99.7% of its motor's speed, and that's exactly why it can lift a body. 6,000 rpm goes in. 20 rpm comes out. The first stage is simple: a 32-tooth gear drives a 96-tooth one, so the speed drops 3 times. Then the harmonic drive divides by another 100. Gear ratios multiply, so the total is 3 × 100 = 300. The idler in the middle looks important, but it cancels out completely: (48/32) · (96/48) = 3. It only flips the direction. Speed isn't lost, it's traded for force. Power is torque times speed, so 300 times slower means roughly 300 times stronger, minus friction. The tooth counts hide one more piece of math. 32 and 96 share a factor, so each motor tooth only ever meets 3 of the 96 teeth on the big gear, and those spots wear out first. Pick counts with no common factor, like 31 and 97, and every tooth meets every other one across 3,007 pairings before the pattern repeats. The motor gives the power, but divisibility decides how long the knee lasts. x.com/shred_0x/statu…

Elon Musk admitted something f*cking scary: Optimus is "the hardest product to scale" and the reason hides in 14 screws nobody talks about → Each of those screws sits inside a leg or arm actuator. A motor spins a nut and rotation turns into a straight push → A ring of rollers walks around the screw like planets. They touch the thread along a full line instead of one point, so a small actuator carries loads that would chew through a ball screw → That is why Tesla put it in the joints that hold the weight of the whole robot → Here is the catch. Every thread has to be ground to microns and the stack has to mesh with almost zero backlash → A roller screw costs several times more than a ball screw and very few factories on Earth can build them in volume. Musk said the supply chain had to be built out entirely, and this is the part he meant → Now do the math. 14 screws per robot on a line built for 1M robots a year means 14 million precision screws every 12 months → Nobody has ever made that many. If Tesla cracks the cost, a $20,000 humanoid becomes real. If it doesn't, the most advanced robot factory ever built will sit waiting for threads

Shredder @Shred_0x

3 months ago

x.com/i/article/2080…

92 144 677 13.4M 814

@Shred_0x @_robtimus_ That's roughly 27 units every minute around the clock with zero days off

It took my laptop 391 random guesses to find the two numbers that make an RSA-2048 key, and you just watched every one of them. The recipe is surprisingly simple. Pick a random 1024-bit odd number and ask whether it's prime. Most of the time a tiny prime like 3, 7 or 1999 divides it, so it dies in the sieve instantly. That's what happened to 333 of the 391 candidates. The 58 survivors face Miller-Rabin, a test that throws twenty "witnesses" at the number. One failure and it's out. The first prime showed up on try 63, the second only on try 328, which is about what you'd expect, since roughly one in every 355 odd numbers this size is prime. Multiply the two and you get n, a number with 617 digits. Anyone can see it, and nobody knows a fast way to split it back into its two primes. From those hidden primes Euclid's algorithm needs just 11 steps to build the secret key d. Then the test: "MATHVERSE" goes in, comes out as 617 digits of noise, and only d turns it back into words. The whole security of the key rests on one asymmetry: multiplying two primes takes a millisecond, and undoing it would take longer than anyone has.

@Shred_0x A key like this needs nearly a million qubits and a week of runtime

@_robtimus_ That's why I used the system generator not a homemade one

@Woolrich_X it's the other way round, real data is full of leading ones and it's the flat even spread that gives fakes away

Open your bank app. Look at the first digit of every payment. About 1 in 3 starts with a 1. Not 1 in 9. One in three. This isn't a coincidence, it's a law. And it catches liars. I doubled 2 two thousand times. 2, 4, 8, 16, 32… all the way to a number with 603 digits. 602 of them start with 1. Only 89 start with 9. Benford predicted exactly this: log₁₀(1 + 1/d). 30.1% for 1. 4.6% for 9. The data hits it to the decimal. The secret is a spinning dial. Every doubling turns it by 0.301. The "1" slice is huge, the "9" slice is tiny. Spin long enough and the dial lands where the space is. Now the dangerous part. When people invent numbers, they spread first digits evenly. Real data never does that. Fake data on this test: 880. Real data: 0.21. That's how fraud gets spotted in spreadsheets nobody thought to check. So go back to your bank app. Count the 1s.

@mathemetica And the series never stops: as n₂ grows the lines pile up toward 4/R ≈ 364.6 nm, so there are infinitely many, just past what the eye can see

@matheorems Cut at x = 1/3 and all three shares come out at exactly 2/3, so the cutter's greed lands on a perfectly fair split

A drunk man always finds his way home. A drunk bird only does about 34% of the time. That's not a joke, it's a theorem. Pólya proved it in 1921. Let someone wander randomly, one step at a time, to a random neighbour. On a line, they come back to where they started with probability 100%. On a flat grid, also 100%. Add one more dimension and it breaks. In 3D the chance of ever coming home is 34.05%. We ran 360 walkers in each world for 16,000 steps. On the line, 99.4% were already back. On the plane, 75.3%, still climbing slowly. In space, 36.7%, and it barely moves. Why? In 3D the odds of standing at home after 2n steps shrink like 1/n^1.5. Add them all up and the total stays finite: 1.5164 visits on average, counting the start. So the chance of a second visit is 1 − 1/1.5164. One extra direction to get lost in, and two thirds of the walkers never come back. Everything else > the spinning cube, the pink dots, the sound of each return > is just that sum refusing to reach infinity.

Mathverse @_mathverse_

a week ago

x.com/i/article/2110…

0 0 16 3K 11