Recent content by jaap

And also: https://www.jaapsch.net/puzzles/pgl25.htm

I had the privilege of visiting his home in London twenty or so years ago. I don't remember exactly how that happened - I think I first met him at a puzzle meet in London. What I do remember is his joy at showing me his puzzles, and his large collection of books on recreational mathematics. I...

If you use the solver built into the javascript simulation on my page, you will get an optimal solution (in 'qtm'). The solution it finds is: L-2 R' L2 R' L' R L R-2 L R L R L' R' L (15/18) Like most optimal solutions, this does not really provide any insight.

There is another problem that is rarely mentioned, and which can happen even with random-state scramblers for the 4x4x4, namely the quality of the Random Number Generator. Many browsers for example use the xorshift128+ generator in their javascript, and as the name implies, it uses 128 bits of...

Cool. Anyway, this is really good work. I like the clever reductions due to the rotations of the whole puzzle. Have you tested the neighbours (and neighbours of the inverses) of the three depth 14 positions you found in <U,R,F>? It's unlikely to result in a depth 15 position, but it is worth...

Ok. I was a bit confused by your permutations calculation, as it has to preserve orientations, but presumably then you didn't just use a breadth first search using the orientation preserving generators you listed. Did you do a breadth first search using single moves and just filter from the...

Did you count (R' F R F')3 as 4 moves?

Unfortunately that presentation does not use the moves as the generators, and as far as I can tell, no such presentation is known. There is such a presentation for the 2x2x2 cube.

It is done by examining the cycles: http://www.jaapsch.net/puzzles/theory.htm#order

Way back in 1981 some identities were enumerated. There are essentially only 3 that use 12 quarter turns, but no shorter ones apart from the really trivial stuff, and about a dozen of 14q, and many more longer ones. Here are some historic posts from the cube lovers mailing list where they were...

There are an estimated 490 million. In the comments you'll see that Tom Rokicki tested the neighbours of 735000 known distance-20 positions, and found that most, but not all, were isolated. There were only 652 pairs of neighbouring distance-20 positions, and a few larger clusters. The largest...

Consider a permutation consisting of just a 2-cycle and a 4-cycle. I claim that such a permutation has no square root. Let p = (12)(3456), and suppose that q were its square root, i.e. q2 = p. Then q4 = p2 = (35)(46). So q4 acts as the identity on items 1 and 2, but q2 does not. Therefore...

On my Useful Mathematics page I wrote this: On my Cube subgroups page I list a few other neat move sequences for this kind of thing: D = F2R2D2F2U2R2F2 U F2R2U2F2D2R2F2 ( uses U, F2, R2, D2 ) D2 = F2R2L2B2 U2 F2R2L2B2 ( uses F2, B2, R2, L2, U2 ) U2 = FR' FLFL' F2R2 B'RBR F'R ( uses F, B...

Q8 occurs as a subgroup of the 3x3x3 cube however, using the following three move sequences to represent i, j, and k: i = (UR,UF)+(UL,UB)+ = R2 UR'U'R F R2 URU' F' UR2 j = (UB,UF)+(UR,UL)+ = L'U'B' U2 BLU FU2F' k = (UL,UF)+(UB,UR)+ = B'U'R U2 FRF'R' U' BRU'R' I don't know where I first...

This follows directly from the Linearity of expectation - expected values are linear (i.e. can be added together) regardless of whether the events are independent or not. There are 24 possible c-e pairs. Each individually has 1/24 probability. Therefore the expected number of, say, red-white +...