=============================================================================== Notation =============================================================================== magic-x,y: configuration in row x, column y of table magic27 GRAY: configurations with vanishing invariants, but which we do not know if they are constructible. MISTAKE: error in table magic27 DUPLICATE: duplicate configuration in table magic27 hinge equivalent: can be obtained from each other just by rotation around hinging sides. appendix with constructions not included in magic27 =============================================================================== Table magic27 =============================================================================== magic-1,1 aka 011-013-031-033-101-103-141-143s: (initial puzzle configuration) hinge equivalent to (1,3), (1,12), (3,1), (3,2), (3,3), (3,5), (3,11), (5,1), (5,2), (6,7) hinge equivalent to two of (6,12) magic-1,2 aka 011-013-110-114-310-314-411-413s: one move from (1,1), cylinder, constructible hinge equivalent to (1,13), (2,8), (5,13) magic-1,3 aka 011-013-031-033-101-103-211-213s -> (1,1) magic-1,4 aka 011-013-031-103-141-213-231-233s: one configuration, constructible hinge equivalent to (1,9) (*) magic-1,5 aka 011-013-031-033-101-103-110-130s: two configurations, both constructible hinge equivalent to each other hinge equivalent to missing 2x2x2+4 configuration magic-1,6 aka 011-013-101-114-301-314-411-413s: two configurations, both constructible from L-1 and L-5 of [Nourse] magic-1,7 aka 011-013-031-033-101-141-211-231s: two configurations, both constructible from L-1 and L-6 of [Nourse] hinge equivalent to (1,11) and some of (6,11) magic-1,8 aka 011-013-031-033-101-103-110-114s: two configurations, both constructible from S-1 and S-3 of [Nourse] hinge equivalent to (2,1) and some of (6,11) magic-1,9 aka 011-013-101-123-301-323-411-413s -> (1,4) (*) magic-1,10 aka 011-013-031-114-132-211-213-231s: one configuration, constructible hinge equivalent to (5,3), (5,6), (5,14) magic-1,11 aka 011-013-110-114-211-213-301-303s -> (1,7) magic-1,12 aka 011-013-101-103-121-123-231-233s -> (1,1) magic-1,13 aka 011-013-110-114-211-312-314-413s -> (1,2) magic-1,14 aka 011-013-110-112-310-312-411-413s: one configuration, constructible hinge equivalent to (6,8) magic-1,15 aka 011-013-031-033-101-103-112-132s: one configuration, constructible hinge equivalent to (5,4), (4,3) magic-1,16 aka 011-013-101-114-211-303-312-314s: two configurations, both constructible from L-1 and L-7 of [Nourse] hinge equivalent to (2,10), (3,8), (4,9) magic-1,17 aka 011-013-101-112-114-213-303-314s: two configurations, one constructible RRmRUmRmDvRmUm: constructible RRmRDmRmUvRmDm: GRAY magic-2,1 aka 011-013-110-114-121-123-231-233s -> (1,8) magic-2,2 aka 011-013-101-112-301-312-411-413s: two configurations, both constructible hinge equivalent to (6,9) magic-2,3 aka 011-013-031-033-101-112-132-141s: four configurations RRmUmRDRUmRm: constructible RRmUmRvDRvUmRm -> previous (RRmUmRDRUmRm) RRmDmRURDmRm: constructible, from L-3 of [Nourse] RRmDmRvURvDmRm -> previous hinge equivalent to (4,8), (3,7) magic-2,4 aka 011-013-031-033-101-112-123-134s: two configurations, both constructible directly from the puzzle in the solved position magic-2,5 aka 011-013-031-033-112-132-211-231s: four configurations. RURLvLmULmLv and RURLvLvULvLv are hinge equivalent and constructible RURLmLmULmLm and RURLmLvULvLm are hinge equivalent and constructible hinge equivalent to (2,6), (3,9), (4,4), (5,5), (6,5) magic-2,6 aka 011-013-031-033-101-103-121-123s -> (2,5) magic-2,7 aka 011-013-031-033-101-110-112-121s: two configurations, one constructible RURmDvRvUvLvUm: constructible. (see also (4,16)). Start from the boomerang solution shape, place it with the missing corner at up-right position and the lower-left tile of "slash" type. turn up 180 degrees the lowest row of three tiles Open in 3D in a way similar to the "obtain the cube" move Turn two tiles in the final shape RURmDvRvUvLvUm: GRAY magic-2,8 aka 011-101-121-211-231-321-341-431s -> (1,2) magic-2,9 aka 011-033-103-112-114-121-123-132s: two configurations, both constructible magic-2,10 aka 011-033-101-112-121-123-132-134s -> (1,16) magic-2,11 aka 011-101-110-112-121-123-132-211s: three configurations (the cube) RmRmUmLmLmDvDmDv: constructible, info in Koeller's page RmRmUmLmLvRmLvDm: GRAY RmRmDmLmLvRmLvUm: constructible, from (1,14) magic-2,12 aka 011-013-110-112-211-213-310-312s: one configuration, constructible from one of (3,4) magic-2,13 aka 011-013-110-112-114-121-123-132s: one configuration, constructible from (1,14) magic-2,14 aka 011-101-112-121-123-132-143-233s: one configuration, constructible hinge equivalent to (5,10) magic-2,15: probably picture with a mistake magic-3,1 aka 011-013-031-033-101-103-301-303s -> (1,1) magic-3,2 aka 011-013-031-114-132-233-314-334s -> (1,1) magic-3,3 aka 011-013-031-114-132-233-314-334s = (3,2) [DUPLICATE] magic-3,4 aka 011-013-031-101-103-121-211-301s: two configurations, both constructible one is near (5,10), the other is near figure 16b of [Pao] hinge equivalent to (3,12) magic-3,5 aka 011-013-121-123-321-323-431-433s -> (1,1) magic-3,6 aka 011-013-101-103-121-211-231-321s: two configurations, both constructible from L-3 and L-8 of [Nourse] magic-3,7 aka 011-013-110-123-132-231-310-321s -> (2,3) magic-3,8 aka 011-013-110-123-233-310-321-332s -> (1,16) magic-3,9 aka 011-013-031-033-112-132-312-332s -> (2,5) magic-3,10 aka 011-013-031-123-132-323-332-433s: two configurations, both constructible magic-3,11 aka 011-013-121-123-231-233-341-343s -> (1,1) magic-3,12 aka 011-013-110-112-132-211-231-310s -> (3,4) magic-3,13 aka 011-101-121-123-132-211-213-312s: two configurations, both constructible RmRmRmUvUmRvUmUv: constructible from L5 of [Nourse], open into (4,10) RmRmRmDvDmRvDmDv: constructible from one of (5,5) magic-3,14 aka 011-112-121-123-321-323-332-433s: two configurations, one constructible RRmUmRvDUmLmUv: constructible RRmDmRvUDmLmDv: GRAY hinge equivalent to (5,7) magic-3,15 aka 011-112-121-123-321-323-332-433s = (3,14) [DUPLICATE] magic-4,1 aka 011-101-121-132-211-233-312-323s -> (2,9) magic-4,2 aka 011-101-121-123-211-213-233-323s: two configurations, both constructible one starting from S-2, the other starting from S-3 [Nourse] magic-4,3 aka 011-013-121-123-132-321-323-332s -> (1,15) magic-4,4 aka 011-013-121-123-231-233-321-323s -> (2,5) magic-4,5 aka 011-112-121-213-233-321-332-431s: one configuration, constructible magic-4,6 aka 011-101-121-123-211-213-233-323s = (4,2) [DUPLICATE] magic-4,7 aka 011-033-112-121-123-132-312-323s: two configurations, both hinge equivalent to (6,4) and constructible magic-4,8 aka 011-112-121-123-231-233-332-341s -> (2,3) magic-4,9 aka 011-112-121-213-231-314-323-332s -> (1,16) magic-4,10 aka 011-101-121-211-231-233-321-323s: two configurations, both constructible from L-2 and L-5 of [Nourse] magic-4,11 aka 011-112-121-123-321-323-332-433s = (3,14) [DUPLICATE] magic-4,12 aka 112-121-123-132-312-321-323-332s: four configurations, all constructible RUmRUvRUmRUv f=0 symcount=4 typeinv=yes, constructible RUmUUvRUvUUm f=4 symcount=2 typeinv=yes, constructible RUmUvUmRUmUvUm f=4 symcount=4 typeinv=no, constructible RUvUmUvRUvUmUv f=4 symcount=4 typeinv=no, constructible magic-4,13 aka 112-121-123-231-233-312-321-323s: two configurations, one constructible RUUmLUmLvULv: constructible RmUvUmLvRmUvUmLv GRAY (four flaps) magic-4,14 aka 112-121-123-132-211-233-312-323s: ten configurations, six constructible RUmRLvUmLvUmLv: constructible RUmRLvUvLmUvLv: GRAY RUmRmLUvLmUvLv: GRAY RUmRvUmRmLUvLv: constructible RUvRLmUmLvUmLm: constructible (hinge equivalent to previous) RUvRLmUvLmUvLm: constructible (hinge equivalent to previous) RmUmRvUmRvUmRmLv: constructible RmUmUvUmRmLvUmLv: GRAY (four flaps) RmUvRmUvRvLmRvUv: GRAY RmUvRmLvUvUmUvLv: constructible (four flaps) magic-4,15 impossible, it has 9 tiles!!!!: magic-4,16 aka 112-121-123-211-233-321-323-332s: four configurations, all constructible RUUmLmUvLmUvLv: constructible (symcount=1 typeing=yes) RUmRvUvULmUvLv: constructible (symcount=2 typeinv=no) RUvRmUmULvUmLm: constructible (hinge equivalent to previous) RmUmUvLmRvUvUmLv: constructible (four flaps). From (2,7): turn 180 degrees diagonally the tile at bottom of box to obtain a 2x2 square with two opposite piles of three superposed tiles. lift four flaps on opposite sides to obtain the final shape. magic-4,17 aka 112-121-123-132-211-213-312-332s: two configurations, one constructibe RURmUvRvUmRmUv: constructible RURmLRvUmUUv (four flaps) GRAY magic-5,1 aka 011-013-031-033-114-134-215-235s -> (1,1) magic-5,2 aka 011-013-015-031-033-035-110-130s -> (1,1) magic-5,3 aka 011-013-015-031-035-132-134-233s -> (1,10) magic-5,4 aka 011-013-015-031-033-035-121-123s -> (1,15) magic-5,5 aka 011-013-015-031-033-035-112-132s -> (2,5) magic-5,6 aka 011-013-015-121-125-132-134-143s -> (1,10) magic-5,7 aka 011-013-114-121-125-132-233-235s -> (3,14) magic-5,8 aka 011-013-110-114-211-213-215-314s: one configuration, constructible magic-5,9 aka 013-112-114-211-213-215-312-314s: one configuration, constructible magic-5,10 aka 011-013-015-101-125-211-213-215s -> (2,14) magic-5,11 aka 011-013-101-103-123-125-213-215s: one configuration, constructible magic-5,12 aka 011-013-015-101-103-105-121-211s: two configuration, both constructible from L-3 and L-5 of [Nourse] magic-5,13 aka 011-013-015-110-116-211-213-215s -> (1,2) magic-5,14 aka 011-013-015-101-105-112-114-123s -> (1,10) magic-6,1 aka 112-121-123-132-312-321-512-521s: two configurations, both constructible one from L-2, the other from L-8 of [Nourse] magic-6,2 aka 112-121-312-321-323-332-523-532s: one configuration, constructible from S-2 of [Nourse] magic-6,3 aka 112-121-213-215-231-323-325-332s -> (4,7) magic-6,4 aka 112-121-213-215-233-235-321-332s: one configuration, constructible (from (1,4)) hinge equivalent to (4,7) [both configurations] magic-6,5 aka 112-121-123-125-231-233-235-332s -> (2,5) magic-6,6 aka 112-121-312-321-323-332-512-523s: one configuration, constructible magic-6,7 aka 011-013-015-017-101-103-105-107s -> (1,1) magic-6,8 aka 011-013-015-017-112-116-213-215s -> (1,14) magic-6,9 aka 011-013-015-017-103-105-112-116s -> (2,2) magic-6,10 aka 011-013-101-110-114-121-211-213s: two configurations, both constructible magic-6,11 aka 011-013-101-103-110-114-121-123s: four configuration, all constructible two are hinge equivalent to (1,7), two are hinge equivalent to (1,8) magic-6,12 aka 011-013-101-103-121-123-211-213s: 3 distinct puzzle configurations: RUmDmUmRUmDmUm and RUvDvUvRUvDvUv are hinge equivalent to (1,1) RUmRDmRUmRDm: serpentine, constructible. magic-6,13 aka 011-013-101-103-110-112-121-123s: two configurations, both constructible RRmDmUmRmLUmDm: constructible RRmDmLmUUmRmDm: constructible (from \sla E_2 N_3 E W_2 S_1 N_1 W S of [Pao], Figure 17(a)) magic-6,14 aka 112-121-123-132-134-143-145-154s: one configuration, constructible =============================================================================== Configurations with vanishing invariants not in magic27 =============================================================================== armchair aka 011-013-101-110-112-123-211-213s: one configuration, constructible RRmDmLmURmUmLm: constructible (from n.4 of cubechair) (*) chairtable aka 011-013-101-112-123-211-213-312s: one configuration, constructible Start from configuration (1,4) oriented such that the upper-right-front tile is of type backslash. rotate the upper-right-front tile around its right vertical side (together with its lower neighbour) against the upper-right lateral tile. It becomes a flap, to be lowered in horizontal position. That flap is the internal horizontal tile of the final configuration, moving it into position will free another flap that becomes the external horizontal tile. (*) cubechair aka 011-013-101-103-110-112-121-211s: four configurations, three constructible RRmRmUmLmLmRDm: constructible (f=1) Start from L1 of [Nourse] - by hinging about the top and bottom horizontal sides slide up of one position all four tiles on top of each stack together with the two 'south' tiles in the bottom of their stack - turn the resulting (flat) shape face-down to obtain a shape with three stacks of two tiles and two single tiles (on the left). - open it in 3D in a manner similar to the 'cube' construction - lower an internal flap to obtain the final shape. RRmUmDmRmUmRDm: constructible (f=0) Start from (5,3) with the lower-left tile of type slash. Then: - fold down the "bridge" with all three flaps stacked in a single pile and the left vertical tile ending up on top. - fold right the two leftmost tiles (valley fold) ending up with a 2x2 square - proceed as in the next configuration of "cubechair" RRmDmUmRmDmRUm: constructible (f=0) from L3 [Nourse], turn down 180 degrees the two south tiles on top of the two stacks and arrange the two superposed tiles originally in the north-most position such that in the end we have a 2x2 square with stacks of 2 tiles (left) and 1 tile (right) in the north row and 4 tiles (left) and 1 tile (right) in the south row. now open in 3D with the following hinging: - the two stacked tiles in the north-left position are hinged along the north edge - the upper two stacked tiles in the south-left position are hinged along the left edge - the two middle stacked tiles (SL) hinged right - the two lower stacked tiles (SL) hinged south RUmLmDmUmLmRDm: constructible (f=1, from (ns) shape of Figure 10 of [Pao]) (*) cubeflaps aka 011-013-103-105-112-114-123-213s: one configuration, constructible hinge equivalent to "cubeflaps2" cubeflaps2 aka 011-013-101-110-112-121-211-231s -> cubeflaps openbox aka 011-013-031-123-132-134-143-233s: two configurations, both constructible periscope aka 011-013-101-110-114-123-211-213s: two configurations, one constructible RRmUmLmDUmLmDm: constructible Start from the boomerang solution shape, place it with the missing corner at up-right position and the lower-left tile of "backslash" type. turn down 180 degrees the upper row of two tiles (valley fold) turn left 180 degrees the right column of two tiles (valley fold). We obtain a 2x2 square with three stacked tiles in the upper-right position and two pair of two stacked tile in the two adjacent positions. Switch the hinging side of these two piles of two stacked tiles and open carefully RRmDmLmUDmLmUm: GRAY sleeve aka 011-013-015-101-103-123-125-213s: two configurations, both constructible RRUmRDmUmRDm: constructible RRUvRDvUvRDv: constructible (from openbox...) rot90 aka 112-121-123-132-211-233-312-332s: one configuration, constructible RUmUvLmRLvUmUv: from (4,16) four flaps. mensola aka 112-121-123-211-213-231-323-332s: eight configurations (all with 2 flaps): RURmUvRUUmLv: GRAY RUmRmLUmUvRmUv: constructible RUmRvUUmLmUvLv: constructible RUmRvUmRvUUmLv: constructible RUmRvUmRvUvULv: constructible RUmRvUmUvLRvUv: constructible RUmULvUmLvUmLm: constructible RmUmRvUvUmLvUmLv: constructible =============================================================================== polyominoids missing from magic27 table but hinge equivalent: 011-013-015-017-031-033-035-037s: box=0x2x4+8 -> (1,1) (plane starting configuration) 011-013-015-031-033-035-051-053s: box=0x3x3+8 -> (1,5) (plane solved configuration) 011-013-121-123-130-321-323-330s: box=2x2x2+4 -> (1,5) (hinge equivalent to plane solved configuration) 011-013-031-114-132-215-233-235s: box=1x2x3+6 -> (1,1) (hinge equivalent to plane starting configuration) 011-013-015-110-114-211-213-312s: box=1x2x3+4 -> (1,14) 011-013-110-112-310-312-411-512s: box=1x2x3+4 -> (1,14) 011-013-112-114-215-312-314-413s: box=1x2x3+3 -> (1,14) 011-013-112-114-211-213-215-312s: box=1x2x3+2 -> (2,12) 112-121-123-125-132-134-143-152s: box=1x3x3+0 -> (2,12)