---------------------------------------------------------------- | | | | | | | TTTTTTT H H EEEEEEE | | T H H E | | T H H E | | T HHHHHHH EEEEE | | T H H E | | T H H E | | T H H EEEEEEE | | | | | | PPPPPP Y Y RRRRRR A M M III X X | | P P Y Y R R A A MM MM I X X | | P P Y Y R R A A M M M M I X X | | PPPPPP Y RRRRRR AAAAAAA M M M I X | | P Y R R A A M M I X X | | P Y R R A A M M I X X | | P Y R R A A M M III X X | | | | | | | | | | A Solution to the Pyramix | | | | | | | | | | | | | | u | | | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / Lu | Ru \ | | / | \ | | /________|________\ | | /| /|\ |\ | | / | / | \ | \ | | / | LBR / | \ RLB | \ | | B / | / | \ | \ B | | / | / | \ | \ | | / | / | \ | \ | | / LB | / LR | RL \ | RB \ | | / |/ | \| \ | | /________/________|________\________\ | | /| /| /|\ |\ |\ | | / | / | / | \ | \ | \ | | / | LDB / | LRD / | \ RDL | \ RBD | \ | | / | / | / | \ | \ | \ | | / | / | / | \ | \ | \ | | / | / | / | \ | \ | \ | | / Ll | / LD | / Lf | Rf \ | RD \ | Rr \ | | / |/ |/ | \| \| \ | | /________/________/________|________\________\________\ | | | | l D f D r | | | | | | | | | | | |Written by Jonathan Bowen | | Programming Research Group | | Oxford University Computing Laboratory | | 8-11 Keble Road | | Oxford OX1 3QD | | England | | | | Tel +44-865-273840 | | | |Created January 1982 | |Updated April 1985 | |Issue 1.1 | ---------------------------------------------------------------- ---------------------------------------------------------------- | Notation | |--------------------------------------------------------------| | | |1. Description | | ----------- | | | | The pyramix is a tetrahedral pyramid. Each face is | | divided into nine triangular segments. Any face and | | its adjacent segments on the other faces may be | | rotated relative to the rest of the pyramid. | | Alternatively the first two levels of segments from a | | vertex may be rotated relative to the opposite face. | | Additionally the extreme vertex segments may be | | rotated. | | | |2. Faces | | ----- | | | | The following uppercase letters are used to denote the | | four faces of the pyramid: | | | | B for the back face, | | L for the lefthand face, | | R for the righthand face, | | D for the downward face. | | | | The following lowercase letters are used to denote the | | four vertices of the pyramid: | | | | f for the front vertex, | | l for the lefthand vertex, | | r for the righthand vertex, | | u for the upward vertex. | | | | f l r u denote a clockwise rotation of the | | first two levels through 120 | | degrees about the indicated vertex. | | | | f- l- r- u- denote a similar rotation in an | | anticlockwise direction. | | | |3. Segments | | -------- | | | | Each segment of each face is represented by a string | | of one to three letters. The first letter indicates | | the face on which the segment is situated. The | | following letters indicate the adjacent face or faces | | (the order is not important). Note that the segments | | immediately adjacent to the vertices may be considered | | the same as the segments immediately adjacent on each | | face since the outer vertex segments may simply be | | rotated without affecting the rest of the pyramix. | | However these may be specified by the face (uppercase) | | followed by the vertex (lowercase) if desired. The | | The following are examples of segments: | | | | RD the segment on the righthand face in the | | centre of the edge next to the downward | | face. | | | | LBR or LRB the segment on the lefthand face at the | | touching the back and righthand faces. | | | | Bu the segment on the back face at the | | extreme upper vertex. | | | |4. Operation notation | | ------------------ | | | | Segments in brackets denote movement of the segments | | specified. Elements in the brackets are a cyclic group | | and rotate round one place from right to left when the | | operation is performed. E.g. (RL,LB,BR) is transformed | | to (LB,BR,RL). Note that the first element moves to | | the last position. | | | | | ---------------------------------------------------------------- ---------------------------------------------------------------- | Operation | |--------------------------------------------------------------| | | |1. Rotate each of the extreme vertex segments until they | | match across to the adjacent segments. | | | |2. Make the vertex segments match on one face. First | | match two adjacent vertex segments. If the third | | vertex will not match, rotate the vertices round to | | to the adjacent face. The third vertex may now be | | rotated to match. | | | |3. Complete the face by rotating a vertex on the face, | | rotating the fourth vertex as necessary and then | | rotating the vertex on the face back again. The | | following moves are useful and will leave the lefthand | | face unaffected apart from one specified segment: | | | | (i) (LR,BR,...): f r- f- | | | | (ii) (LR,DR,...): u- r u | | | | (iii) (LR,RL) and ...: f r- f- r u- r u | | | |4. Now position completed face at the bottom and match | | the upper central portion to the bottom segments. | | Rotate the segments on the middle of the three upper | | edges using one of the following moves for a | | clockwise/anticlockwise rotation respectively: | | | | (iv) (RL,LB,BR): f u- f- u- f u- f- | | | | (v) (BR,LB,RL): f u f- u f u f- | | | |5. Flip the segments so that they are orientated | | properly using the following move: | | | | (vi) (RL,LR) and (RB,BR): f r- f- r u- r u r- | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ----------------------------------------------------------------