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  If you've ever solved the Rubik cube by some method and now you are disappointed that you don't remember it, because you don't solve it regularly, here is a method which is intuitive: https://www.youtube.com/watch?v=zB8cKBYNTps The video also explains it nicely and there is a chance that you remember it even when you don't practice.

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  You may have heard someone say that cross-products exist only in 3-dim and 7-dim. But of course, this depends on what equations you want your cross-product to satisfy. For the 3-dim and 7-dim cases one usually wants a bilinear product of two vectors, which produces an orthogonal vector and satisfies \(|a\times b|^2=|a|^2|b|^2-(a\cdot b)^2\). The solutions could be to take the imaginary parts (i.e. non-scalar) of quaternions and octonions, and \(a\times b=\frac12(ab-ba)\) and \(a\cdot b=-\frac1...

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  In case you have lost count: https://vibemathed.com/

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  You may have seen pictures of 1-dim cellular automata and some look semi-chaotic, but I was surprised that quite a few interesting, precise facts can be said about CA like **Rule 30**: https://www.youtube.com/watch?v=HcGjCcLEgN4 while other basic questions have not been answered. Another slightly related fun fact is that the **Collatz sequence** is also a rather simple local cellular automaton which displays similar patterns (see image). And the question whether it has a non-trivial cycle boils...

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  I made another visualization of the #Collatz sequence and this time with an actual starting number (one with a high expansion ratio). I'm surprised these cellular automata are not shown more often in relation to Collatz sequences and are probably hidden in papers only. #math

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  Building on my last post of the #Collatz visualization one can see that the **highest bits in binary representation of the Collatz sequence are highly predictable**. Here is how: Look at the reverse odd Collatz sequence in binary representation. Once a number starts with 100 (which is guaranteed to happen within the first 4 odd numbers) the pattern starts: Starting from the next number all other second highest bits of the odd numbers will reproduce exactly one of the two subsequence blocks: * `...

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  Another visualization of the #Collatz sequence. This time highlighting some streak-like structures. Btw, the cellular automaton for the Collatz sequence is just \[ s_{i,j}=(s_{i-1,j}\mathbin{\text{mod}} 2) - (s_{i,j+1}\mathbin{\text{mod}} 3)\] None of that seems to help for the Collatz conjecture, but at least one can easily show that in any long cycles at some point about half of the leading binary digits must be `11000...`. #math