Difficult puzzle
A set of duples in an ordered list is generated by a certain symmetrical function in steps from 1 to 14.
{1 1} is the zeroth member.
The list starts:
{{1 1}.{2 3},{9 18},{40 107},{178 617},{752 3329},{2989 16712},{11019 75845},{38049 290024},{120169 629249},...}
Calculate the next 5 duples in the list; describe the step function and what it does.
...
{1 1} is the zeroth member.
The list starts:
{{1 1}.{2 3},{9 18},{40 107},{178 617},{752 3329},{2989 16712},{11019 75845},{38049 290024},{120169 629249},...}
Calculate the next 5 duples in the list; describe the step function and what it does.
...
Replies

skipperclue #1: the list is a set of permutations;
clue #2: there are prime factors in one or both members of each ordered pair;
clue #3: the second value in each duple is zero, for steps 12,13,14, only the first member has a value after step 11. 
skipperok, clue #4: the numbers in each pair in the list are ordered by n, the number of quarter or full (through pi/2 or pi) turns of a 2slice group, which is found on the Pocket 2x2x2 cube. This has 6 colors but the list is for a 3color map on the "2cube" and Rubik's rotation group of cube products.
So that I just copied a table of numbers from the wikipedia page for the Pocket Cube and divided them by 3, except the first, trivial pair of 1s, which should really be {1/3 1/3} to be consistent I suppose. Then, my 1/3 map or list is for the 3x3x3 and up, since higher numbers of slices all have the 2x2x2 cube in each corner, and each subgroup has 3 colors. Alternately each subgroup has 3 more colors 'opposite' each corner so that a 3x3x3 is 6x2x2x2 pockets, and so on.
Trying to derive an ideal formula for the permutations in the 2slice cube and map this to the shallowslices on the icosahedron of the 4color 'toy' I bought a while ago.
These Rubik's puzzles and the various Platonic solids, sliced deeply or otherwise, represent another look at ancient problems and stuff like why there are a finite number of regular solids. The 4color sphere/icosahedrally sectioned number puzzle is a dual space for the 3/6 colored cube, since the "missing red" means the hole left behind has up to 3 colors around it, this maps directly to the 3 colors on corner pieces of the sliced cube, colored up to 6.
It's a toughie, but there are a finite number of permutations per n slices, starting with 3 or 6, or some combination as in bandaged and void cube puzzles. Since the cube is less symmetrical than the sphere, a shallow sliced cube with movable pieces would also require a removable piece like the Magic Number sphere (The Brain Racker) has. On the cube, pieces would have to be able to go "around a corner", so that slicing the whole cube is actually easier, unless maybe you used deformable sections or could roll and unroll them. An electronic cubetype puzzle would work too: make one section blank and allow one of the surrounding sections to fill it.
The sectioned cube and sphere  shallow cut  are like 3d versions of the square movingnumber puzzles; deep slices allow color permutations which are partitioned by n the number of rotations; each group depends on the previous group having been generated by n, but contains independent primes.
For some reason the first prime appears after 3 full turns and is 107 (i.e. invariant) for 3 or 6 colors. What if there are 4 colors, or 2 (a 2polytope)? What happens when the number of colors changes, IOW?
Since, the 3x3x3 has 9 separate pieces per face, you can color each face up to 9?
You are reading an archived discussion.
Related Posts
Hello Everybody, James here from USA
Hi:
I'm not an engineer (or even close to it); I'm a marketing/advertising consultantabout as far as you can get!!
My husband and I are (hopefully) retiring to an island...
Lifehacker  Lifehacker Pack 2009: Our List of Essential Free Windows Downloads  Lifehacker Pack 2009 (Windows)We feature downloads of all kinds every day at Lifehacker. Today, however, we're...
More...
Can anyone tell me good innovating ideas for final year project in Mechanical engineering?
Following might be useful to those with wordpress powered blog 
1. Create a favicon. You may use any of the following free services 
Dynamic Drive FavIcon Generator
favicon.ico...