the basic structrual elements of an amusement device formed in accordance with the present invention are hexaflexagons . fig1 show a strip element 10 of ten equilateral triangle elements 11 with two end triangles 13 , and 13 and the triangles are hingedly connected at adjacent edges 12 and each element 11 has an edge 14 that is not used as a hinge edge . progress to fig2 which shows the first fold 16 of one of the hinges 12 . fig2 shows two other hinges 12 which still need to be folded to make the trihexaflexagon element . fig3 now shows that two folds 16 have been made . fig4 shows the final fold 16 at a fold hinge 12 along with the other two folds 16 and 16 that have already been made . finally fig5 shows how the tenth end triangle element 13 is glued or fixedly attached by other means such as staples , weld or tape to the other end triangle element 13 to form the single ninth triangle element 19 . the strucutre illustrate in fig5 is the loop element 15 and is a trihexaflexagon . the loop element 15 is a twisted continuous loop of hingedly connected triangles where the twist is obtained by the way the hinges are folded before the ends of the strip are finally connected to form the continuous loop element 15 . in order to build any hexaflexagon loop element it is necessary that enough twist exist in the loop to restrict the loop to a hexagon shape of 6 triangles on both sides when it is laid out flat . higher order hexaflexagons , such as hexahexaflexagons made from longer strips of hingedly connected triangles , may also be used to build embodiments of my invention , and these kinds of higher loop elements must also satisfy the requirement of just the right amount of twist . for instance trihexaflexagons may be linked to hexahexaflexagons in many different ways , and hexahexaflexagons may be linked to other hexaflexagons to create puzzles of great difficulty and interest . fig5 also shows three arrows leading away from three of the inner vertices of the trihexaflexagon loop element 15 . these three vertices , indicated by the arrows in fig5 are to be separated by first bringing the vertices 40 downward and together to form a radially pinched structure shown in fig6 . in fig6 the three arrows are again shown and the three folded hinges 16 are also shown along with the vertices 40 . finally , fig7 shows how the pinched structure in fig6 has been opened out as indicated by the arrows in fig6 and once again forms into a hexagonal shape but has acquired the three new folded hinges 16 whereas the previous folded hinges 16 in fig6 have now become unfolded hinges 12 in fig7 . the trihexaflexagon loop element 15 can now be linked to another loop element 15 by inserting a strip 10 between one of the folds 16 &# 39 ; in fig8 and then folding the inserted strip 10 into a new loop elements 15 as explained previously for fig5 . fig8 shows the first fold 16 being made upon the inserted strip 10 . fig9 shows the linked 2 - loop structure 30 . the two linked loop elements 15 and 15 have three folds 16 in one element 15 and three folds 16 &# 39 ; in the other element 15 . fig9 also shows a slipagon diagram that represents 30 indicated by the node element 18 and the link line element 17 . the node elements 18 represent loop elements and the link line elements 17 represent the way the loops are linked together . other figures in this specification will also contain slipagon diagrams where appropriate to show how the elements 15 are linked at a glance and to provide a convenient means for discussion of methods of creating slipagons of many different kinds . fig9 represent the simplest slipagon possible since it is formed on only two linked loop elements 15 . fig9 also shows two arrow indicia 20 fixedly marked on each of the two loop elements 15 . this arrow indicia in fig9 can be moved about by sliping or sliding the elements 15 by choosing one of the slide directions shown by the dotted arrows in fig9 . the arrow indicia in fig9 can also be moved by flexing either one or both of the loop elements 15 as previously described by the description of the fig5 , and 7 . a puzzle that is not too difficult is to get the arrow indicia back as shown in fig9 after they are moved about with respect to one another by several slips and folds of the elements 15 . most slipagon puzzles can be either slipped or flexed or both . there may be several positions that a slipagon can be gotten into where either slipping or flexing becomes highly restricted . getting the slipagon back to a normal position may then become extremely difficult . fig1 is a chain slipagon puzzle 31 of four linked elements 15 and also shows the slipagon diagram for these four linked elements . the four arrows in fig1 are indicia fixedly marked upon each of the four loop elements 15 . these arrows can be moved about in many ways with respect to each other by flexing and or slipping the four loop elements 15 in fig1 . it is a pleasant puzzle figuring out how to restore the indicia to their original positions after moving them about at random by flexing and or slipping the four loop elements 15 in fig1 . fig1 illustrates a loop puzzle 32 of four linked loop elements 15 and its slipagon diagram appears with it as four nodes connected by four line links to form a square . fig1 shows the puzzle 32 as a regular tetrahedron but the puzzle 32 can assume many other forms by the operations of slipping and flexing as previously explained , and it can be exceedingly difficult to restore it to the tetrahedral form if it is gotten into a much different form . the three arrow indicia in fig1 are to be complemented by a fourth arrow behind the middle arrow in fig1 and on the opposite side of the puzzle 32 . fig1 is a perspective sketch of another loop puzzle 33 of four linked loop elements 15 along with its slipagon diagram . the slipagon diagram in fig1 appears identical to the slipagon diagram in fig1 but the puzzle 33 pictured in fig1 is a distinctly different structure from the puzzle 32 shown in fig1 . the reason for this is that a different twist was given to the chain of four linked loop elements 15 before it was connected into a loop in the same manner that different twists may be given to moebius bands before connecting them into loops . the structure 33 illustrated in fig1 is flexible , and the four arrow indicia may be mixed up and then restored . the puzzle 33 in fig1 also has several forms different from the octahedral cap illustrated in fig1 , and some of these forms can be very difficult to return from , to the octahedral cap form . fig1 is an overhead sketch view of a loop 34 of six linked loop elements 15 along with it &# 39 ; s slipagon diagram . the six arrow indicia shown on the loop elements 15 of the loop puzzle 34 can be thoroughly mixed up and then restored but restoration can be very difficult . the puzzle 34 has a great many different forms . loops of six linked loop elements 15 can be connected in several distinctly different ways not shown in any of the figures . fig1 shows five different slipagon diagrams of other structures that can be made by linking the elements 15 as indicated by the diagrams a , b , c , d and e . the diagram a in fig1 shows a basic branched slipagon structure . the diagram b in fig1 shows the maximum number of branches possible at a single loop element 15 by linking the loop elements 15 to it . the diagram c in fig1 shows a cubic method of linking the loop elements 15 . the structure indicated by diagram c has been made by the present inventor and it can be collapsed and then opened back out to a three dimensional form in a new position . a deceptive puzzle could be made by eliminating one of the connecting link lines in diagram c thereby obtaining a structure that could be gotten into many positions that the puzzle of diagram c could not be gotten into and making it appear that the structure was still the same as that of diagram c . a slipagon puzzle with the slipagon diagram of fig1 d was made by the present inventor . it has proven to be one of the most difficult slipagon puzzles so far discovered and it can be gotten into positions from which it may take a person hours to get back to some simple starting position . the structure of diagram e shows how many other slipagons can be made simply by designing diagrams and then checking out all possible ways of building the structures represented by a single diagram of linking and looping the loop elements 15 , or by linking and looping more complicated hexaflexagon loop elements . fig1 shows a perspective view of a portion of a self hinging plastic strip with triangle elements 11 &# 39 ; and hinge elements 12 &# 39 ;. strips of this kind can easily be made with appropriate press rollers having dies for creating the hinge depressions as the plastic strip is pressed and rolled between the rollers . several very good , well known , readily available , cheap , self hinging plastics exist and would work well to build the many embodiments of my invention . fig1 shows a plan view of the two ends of a strip of ten equilateral triangle elements 11 &# 34 ; with hinge elements 12 &# 34 ; and end triangles 13 &# 39 ;. the end triangles 13 &# 39 ; in fig1 illustrate how a self hinging plastic strip or other , suitable kind of strip of triangle elements 11 &# 34 ; could be provided with notch elements 21 &# 39 ; in one end element 13 &# 39 ; to mate with notch elements 21 in the other end element 13 &# 39 ; so as to provide a loop element 15 , as previously described , that can be quickly disassembled and unlinked from other loop elements 15 and then reassembled and linked in new ways to other loop elements 15 to create many different kinds of slipagon structures . fig1 shows a simple way to manufacture hinged strips of triangles by means of adhesive tape elements 22 and 22 &# 39 ;, rollers 23 and 23 &# 39 ;, press roller 24 , hopper 26 containing triangles 11 &# 34 ; in alternate orientatiions , and roller 27 under the hopper 26 . the apparatus works by pulling the tape band 25 under the hopper 26 causing one edge of the triangle elements 11 &# 34 ; to fall to the adhesive surface of the tape band 25 as can be seen in the partial frontal section of the hopper bottom in fig1 . after the tape band 25 passes under the hopper 26 and collects triangle elements 11 &# 34 ; with hinge elements 12 &# 34 ; it passes through tape roller 23 &# 39 ; and press roller 24 and recieves another adhesive tape band 25 &# 39 ; on its top from tape roll 22 &# 39 ; completing the assembly of a long strip of hingedly connected equilateral triangles . the hinges are made extra strong from the contact of the adhesive surfaces of the two tape bands 25 and 25 &# 39 ;. the width of the hinge elements 12 &# 34 ; can be varied by changing the speed at which the tape band passes under the hopper 26 . the finished strip of hingedly connected triangles may then be cut into smaller strips , which smaller strips are then used to build the puzzles described by the present invention . while certain specific embodiments of the present invention have been disclosed as typical , the invention is of course not limited to these particular forms , but rather is broadly applicable to all such variations as fall within the scope of the appended claims .
puzzles are comprised of linked loops , where the loops are made by first making flat strips of equilateral triangles by hinging the triangles together at their edges and then folding the strips at the hinges and then connecting the end triangles together to form a twisted loop that has the overall form of a flattened hexagon known in the literature of recreational mathematics as a hexaflexagon . the linked loops are linked hexaflexagons and the loops can be shifted and folded into many different 2 and three dimensional positions with respect to the loops they are linked to . the resulting linked loop puzzles have been now named as slipagons since they can be shifted by sliding loops with respect to the loops they are linked to as well as by folding . the loops can be linked in many ways to form puzzles of greatly varying difficulty where the object of the puzzle can be to get from one shifted geometric form to another or to get a figure drawn on the loops of the puzzle into some given arrangement .