“I tell you space is more plentiful than you think but it is far less substantial.” — David Duncan, “Occam’sRazor”
H. J. KAMACK490 Stamford DriveNewark, DE 19711
T. R. KEANE332 Spalding Rd. Wilmington, DE 19803
With the aid of a small computer belonging to one of us, we have worked out many properties of the four-dimensional counterpart of Rubik’s Cube, which we call the Rubik Tesseract because it is based on the samegeometrical principles that Rubik realized in his famous cube. It will be obvious that we have been guidedby Dr. Singmaster’s analysis of the Rubik cube and that our analysis could not have been made withoutstudying his “Notes on Rubik’s Magic Cube”(13).
A tesseract is a four–dimensional hypercube(7). It is bounded by eight cubical “faces” which meet at 16corners, 32 edges, and 24 plane surfaces that we will call “squares”, reserving the word “faces” for theboundary cubes. If we slice the tesseract into 81 equal smaller tesseracts, which we will call “tessies”, andpermit each of the eight “outer” layers of 27 tessies to rotate as a rigid body in a manner that preserves theshape of the main tesseract, then we have a Rubik tesseract. (The rotations will be defined more preciselyin the next section.)
One tessie is at the center of the main tesseract, and one of them lies at the center of each of the eightrotatable layers. These nine tessies retain their positions in a rotation; the other 72 can be moved from oneposition to another. As with the Rubik cube, the re-arrangements can be very complex.
Each intersection of two or more faces of the main tesseract is the site of a movable tessie, and the (cubical)boundaries of that tessie which lie in the intersecting faces are its “facets”. Since four faces meet at eachcorner of the tesseract, there are sixteen 4-faceted tessies, which we call “tetrads”. Three faces meet at eachedge, so there are 32 3-faceted tessies which we call “triads”; and two faces meet at each square, so thereare 24 2-faceted tessies that we call “dyads”. Altogether the 72 movable tessies have 208 facets.
When the tesseract is sliced into tessies, each of its cubical faces is divided into 27 smaller cubes like the“cubies” of a Rubik cube (if the cubies were real cubes instead of only having the external appearance ofcubes). One of these cubes is at the center of the face and the other 26 are facets of some of the movabletessies. That is, if we think of a face as a Rubik cube, the eight corner cubies are facets of tetrads, the twelveedge cubies are facets of triads, and the six axial cubies are facets of dyads. Since each facet belongs to adifferent tessie, there are 26 movable tessies in the layer adjacent to a given face, and rotations of this layerwill involve only these 26 tessies.
Many people find it difficult, perhaps impossible, to visualize a tesseract (although it is not different inprinciple from trying to visualize a three-dimensional machine from blueprints of it). But its faces can bevisualized in three dimensions by rotating them into a common 3-space, in much the same manner that thesurface of a cube can be developed onto a plane surface. Salvador Dali used this idea in his famous painting“Corpus Hypercubus” (See Ref. 7) and we show such a representation in Figure 1. To avoid confusiononly the facets of four tetradic positions are shown. Facets belonging to the same position are numberedconsecutively; e.g. 145, 146, 147, 148. Further details in this Figure will be explained in the subsequentdiscussion.
We number the faces of the tesseract 1 to 8 so that opposite faces are congruent, mod 4. (5 is opposite to
1, etc. It is sometimes convenient to denote the face opposite to 1 by 1 , etc.) Then the position of a tessiecan be designated by the faces in which its facets lie; for example, the tetradic position whose facets wehave numbered 145-148 in Figure 1 is designated 1234. A triad will have a 3-digit number; for example, 127designates a triad with facets in faces 1, 2 and 7. It could also be designated 123 . Similarly, 12 designatesa dyad.
The tesseract has four principal axes that meet orthogonally at its center and are perpendicular to the facesthey intersect. The directed axes passing through faces 1 to 4 are considered positive and are numbered 1to 4 also. The negative axes are 5 to 8, 5 being the negative 1-axis, etc. We sometimes call these axes -1to -4. In Figure 1 we show the positive axes in each face, as they are rotated along with the face into the“3-space” of the picture.
In 2-space, a rotation occurs around a center (a point), in 3-space around an axis (a line). In 4-space it occursaround a plane that we will call the “plane of rotation” (not to be confused with the planes, perpendicular toit, in which points move through arcs under the rotation). For the Rubik tesseract, the permitted rotationsare 90 degree rotations (or multiples thereof) of a layer of tessies around any of the planes of rotation formedby a pair of principal axes. The principal axis that “points to” the layer being turned is called the “primaryaxis”; it may be any one of the eight axes, positive or negative. The other principal axis that defines theplane of a rotation is called the “secondary axis” of the rotation.
The face of the tesseract to which the primary axis of a rotation points (and which bears the same number)turns as a whole, in its own 3-space, around an axis through its center parallel to the secondary axis ofrotation. (Similarly, a rotation of a Rubik cube causes one of its faces to rotate in its own plane, about itscenter point.) If the rotation of this face is clockwise, as seen by an observer in the same 3–space who looksbackward along the positive (translated) secondary axis from outside the cube, we call the Rubik rotationpositive. A counterclockwise rotation is called negative. There are, therefore, three different positive Rubikrotations for each face of the tesseract, corresponding to the three axes about which the face can be rotatedin its own 3–space.
A positive 90 degree rotation about a primary axis, i, and a secondary axis, j, will be called a basic rotationof the Rubik tesseract and will be denoted by Rij. The inverse, or negative rotation, will be denoted by R .
Now i can have any value from 1 to 8 and j any value from 1 to 4, but j and j + 4 must be different from i. So there are 24 basic rotations of the Rubik tesseract.
3. Rotations Expressed as Permutations of Tessies
Each basic rotation changes the positions of tessies having facets lying in the face of the primary axis, andthe rotation can be described as a product of cyclic permutations of the tessie positions. It is evident fromthe manner in which the face rotates about the translated secondary axis that each basic rotation will involvetwo 4–cycles of tetrads, three 4–cycles of triads, and one 4–cycle of dyads, all disjoint. The other two dyadswith facets in the face are in the plane of rotation (their facets in the face are on the translated secondaryaxis) and so they turn but do not move to new positions under the rotation. Altogether, 24 tessies and 76facets change position under each rotation.
In Figure 1 we show the motion of the tetrad whose “Start” position (indicated by the colors of the faces)is 1234, under the action of the rotation R12. The successive positions of the tetrad can be described by asingle tessie cycle or by four cycles of facets, as shown on the Figure.
To obtain the 24 basic permutations we wrote a computer program which calculated the permutations of the208 facets and converted them into permutations of tessies. Each facet was assigned a number from 1 to 208and its location identified by a coordinate vector. The i-th coordinates of the facets lying in Face i (i = 1 to4) were assigned the value of 2, and -2 for the facets in Face i + 4, or −i. The other coordinates of the facetsin the same face were given values of 1, –1, or zero according to their locations in the tri–axial coordinateframe of the face. Thus, dyadic facets have two zero coordinates, triadic facets have one, and tetradic facetshave none. For instance, Facet No. 160 in Figure 1 will be seen to have the coordinate vector (1, 1, −1, −2).
Each rotation was described by a 4x4 rotation matrix(12). For instance, the rotation R12 has the matrix
for facets whose 1–component is positive, and the unit matrix for all other facets. Then the new locationand facet number after a rotation was found by pre–multiplying the old coordinate vector by the rotationmatrix. Thus, multiplying (1, 1, −1, −2) by R12 gives the vector (1, 1, −2, 1), which Figure 1 shows to be thelocation of Facet 155.
The 24 basic rotations expressed as cyclic permutations are listed in Table 1. The digits in the first positionof each cycle can be written in any order, but the subsequent positions must follow in corresponding orderto maintain the proper orientations of the tessies.
Our nomenclature is similar to Singmaster’s. We could, for example write
However, we find that the numerical system is preferable for computer operation and for visualizing moves,and especially for turning the tesseract around in 4–space. (See Section 6a).
The foregoing discussion has been mainly a matter of definitions, preparatory to our main purpose, which isto analyze the moves on the Rubik tesseract in the manner that Singmaster analyzed the moves on the Rubikcube; that is to say, to determine what permutations of the tessies are possible, and to find “useful moves”that would permit any possible permutation to be accomplished. A “move” is any sequence of the basicrotations; a useful move is a move that involves only a small number of tessies, from which more complexpermutations can be built up.
Since we cannot build a physical Rubik tesseract, we needed two tools to look for useful moves. The firstwas a device for visualizing the tesseract and suggesting useful moves; the second was a computer modelto calculate the permutations resulting from the moves. Our visualizing device is simple: the diagram ofFigure 2 mounted on a pin–board. This diagram is the “development” of the faces shown in Figure 1 withthe faces moved apart so that they don’t overlap. The locations of the facets of any tessie are easily foundfrom the coordinate axes and marked with pins. A basic rotation Rij is followed by looking at face i andvisualizing it turning clockwise around axis j. Thus the new position of any facets in that face can be foundand marked, and the other facets of the same tessie in other faces can be found because they have the sameset of face numbers. The pin–board diagram is especially useful for finding conjugating moves.
The computer model of the tesseract permits us to carry out moves and see the results. It contains an array(M) that describes the current arrangement of the tesseract. The model will perform the following routineson command:
Re–arrange the M–array in accordance with an input move, that consists of basic rotations andstored combinations of basic rotations. The M–array can also be reset to the “Start” arrangement.
Calculate and display the tessie cycles that produce the current arrangement from the “Start”arrangement
A program listing is included in Appendix B. The programming language is “Tiny Pascal”(5), which weselected for its suitability to the problem and its availability to us. For the final phases of this work, we
purchased a version of Tiny Pascal which operates under CP/M (the most widely used operating system for8–bit microcomputers). The conversion of the program to standard Pascal is relatively straightforward.
The model contains the 24 basic rotations in a data bank, in the form of the facet permutations that werecalculated by the program described in Section 3. In the data bank, dyadic facets are numbered from 1to 48, and facets of the same dyad are numbered consecutively; that is, facets 1 and 2 are on the samedyad, 3 and 4 are on the same dyad, etc. The triadic facets are numbered from 49 to 144, with the samesystem of consecutive numbering. The tetradic facets are numbered from 145 to 208, again with consecutivenumbering. These values are stored in the array M, which is a vector of 208 elements. Initially, M[J] containsthe value J, which means that each facet, and hence each tessie, is at the initial, or “Start” position.
After a move has been carried out, M[J] will contain the original location of the facet at location J. Forexample, a tesseract in the initial configuration operated on by the permutation (1, 5, 7, 11) results inM[1]=11, M[5]=1, M[7]=5, and M[11]=7. The M vector, then, is a complete description of the state of thetesseract.
Each basic rotation moves 24 tessies with their 76 facets to new locations in the tesseract. These 76 facetsare permuted in 19 four-cycles. This permutation is carried out by PROC Z, which accesses the data bankand carries out each 4-cycle, using PROC PERMUTE to modify the M-vector. All of the basic rotations arein the CASE statement at the beginning of PROC Z. For a basic rotation Rij, the argument N of PROCZ is the number ij. For rotations involving “negative” faces, two numbers are accepted: −ij, or (i + 4)j. For example, Z(12) denotes a basic rotation of Face 1, with secondary axis 2. Z(-12) or Z(52) are equallyacceptable means of indicating a basic rotation of the opposite face.
As a convenience, useful moves were added to PROC Z as they were found. The moves Q1 to Q36 (seefollowing Sections and Appendix A) are called by using Z(101) to Z(136). Other moves are included witharguments of 200 − 425; these are not intended to be used directly, but are required to carry out the Qimoves. All of these moves call on basic rotations, or call on other moves that ultimately use basic rotations. The recursion capability of Tiny Pascal is very useful in programming these operations.
At any point, the effect of the moves that have been performed on the tesseract can be determined by enteringthe value 99. This causes the REKAP variable to be set TRUE, which causes the procedure GETCYCLESto execute, first for the dyads, then for the triads, and finally for the tetrads. The cycles are printed for eachtype of tessie. GETCYCLES is described below.
After the cycles are obtained, more moves may be made, and the cycles can be printed out for the tesseractin its new state. Or, the tesseract may be brought back to the initial state by entering 999, and a new setof operations begun.
This procedure takes advantage of the fact that the facets of any single tessie are numbered consecutively. Itbegins by determining if any facet of a given tessie has been moved from the “Start” position. The variableACTIVITY is set TRUE if this is the case. Then the permutations are traced out by using the procedureLOOKFOR, which recursively “chases” the original occupant of M[J] until it finds it, retaining along theway the locations it has traversed. When all the permutations of facets of a given tessie have been traced,the cycles are printed out. The facet numbers are not printed; instead the face numbers of the tesseractare used, so that the tessie notation described in Section 1 is the result. When orientation changes occur,it often happens that cycles of one facet differ in length from another facet of the same tessie. This is dealtwith by revising the cycle length to a length equal to the least common multiple of all the cycles. Each cycleis then simply repeated the required number of times.
GETCYCLES starts with the lowest numbered facet, and works up. When a facet J has been involved in a
permutation cycle, the variable FLAG[J] is set FALSE, and this facet is considered no further. This avoidsrepetition of permutation cycles for each tessie involved in the cycle.
The variable OK is used to indicate whether a legal move has been attempted. If a move is attempted whichis not in the repertoire, an appropriate message is printed, and the user is prompted for another input move.
We use a computer with a Z80 processor, with a 4 MHz clock speed, and 56K bytes of programmablememory. The program in its present state carries out approximately 45 basic rotations per second. Thelongest move in PROC Z is Q22, which consists of 14616 basic rotations - 323 seconds is required for thismove. The program is operated in a 48K CP/M system, with the data bank occupying 2032 bytes above48K (C000 hex). We have not attempted to operate with smaller storage than this. The Tiny Pascal systemwas purchased from Supersoft, Inc. They describe the compiler as requiring a minimum 36K CP/M system.
Even with the computer model and the pin-board, making moves on the tesseract is more cumbersome thanmaking moves on a Rubik cube. But this disadvantage seems to be offset by the greater flexibility for makingmoves in four dimensions. At any rate, we were able to find all the moves necessary to generate any possiblepermutation, as detailed in Appendix A.
We first consider permutations of position. (Permutations of orientation are discussed in the next section.)From the cycle structure of the basic rotations, all such permutations of tetrads must be even. The positionalpermutations of triads and dyads considered by themselves can be even or odd, but taken together they mustbe even. The simplest possible permutation of tetrads, triads, or dyads by themselves is, therefore, a pair ofinterchanges.
By using the same idea that Singmaster suggested for the cube we were able to find very quickly a move, Q1,that involved only two interchanges of dyads and two of triads. (It is rather remarkable that from this onemove we were able to generate systematically all the moves, both positional permutations and orientationpermutations, that are necessary to define the group of the Rubik tesseract.) From Q1 it was fairly simpleto find two interchanges of triads alone:
Finding two interchanges of tetrads was more involved but fairly straightforward. We arrived at:
(The details of these moves are given in Appendix A.)
A move involving one dyadic and one triadic interchange can also be developed. (See Appendix A). In factevery possible permutation of positions can be obtained from the above three moves.
Although these moves are “simple” in terms of the permutations involved, they are not short. The last resultabove, for example, requires 1812 basic rotations. Nevertheless it is built up from Q1 in only about ten shortsteps, and since the computer does the calculations in reasonable times, there is no great incentive to keepthe moves short in terms of the number of basic rotations.
Permutations of orientation on the tesseract are even more interesting than those on the Rubik cube. Theyalso come in a greater variety of types, so that Singmaster’s terminology is not adequate to describe them.
We have adopted a system in which a twisted cycle is indicated by letters a, b, c, d. For instance, abc meansthat the first 3 facets of each tessie permute cyclically, the first to the position of the second, etc. Thus:
(123, 567)abc = (123, 567, 312, 756, 231, 675)
etc. We will call a permutation involving two letters a “flip” or a “reflection”, one involving three letters a“twist”, and one involving four letters, such as (ab)(cd), a “cross”.
Before discussing these permutations it is necessary to look more closely at the coordinate system for thetesseract. It is evident that we could have numbered the coordinate axes in numerous ways (4! × 24 to beexact). These fall into two sets such that any two coordinate frames in the same set are superposable byturning the frame around in 4–space; but each frame in one set is an enantiomorph of one in the other set. [For more discussion of enantiomorphism see Ref. 6] In order to study changes in orientation we need tohave a rule to know when two frames are congruous (i.e., superposable by rotations in 4-space). The rule isthat the number of transpositions of positive axes plus the number of reversals of axes must be even. Forinstance, referring back to Section 3, the following frames are all congruous:
Thus, for example, it is possible to move a tetrad from 1234 to 1283 (R12 does it) but impossible to movefrom 1234 to 1238. The same rule applies in three dimensions. (This is why a corner cubie can be twistedbut not reflected.) In three dimensions it is not necessary to think about the rule because, when dealing witha physical object, it is impossible to violate it; but for four dimensions we must be aware of it. (Incidentally,an examination of the basic rotations in Table 1 shows that each tetradic move transposes one pair of axesand reverses one axis, in accordance with this rule. That is, each cycle is written congruously, as it must be. The two tetradic cycles of a rotation happen to be written incongruously to each other, but this does notmatter.).
An application of this rule is the operation that we call “transpose” in Appendix A, which we employ to deriveuseful moves. The scheme is based on the fact that the digits in the tessie positions of a permutation arerelated to the subscripts on the rotations that generate the permutation. Thus we can get a new permutationmerely by changing the subscripts of the rotation. For instance, the permutation (13, 14). . . can be changedto (23, 24). . . by interchanging 1 and 2 on the subscripts of the rotations involved. But, according tothe above rule, if we make only one transposition we are looking at a “mirror image” of the tesseract, inwhich all rotations are reversed in direction. Thus, when we transpose two digits in the permutation, wemust also invert all the rotations involved in the move (whether or not their subscripts are affected by thetransposition).
We now consider the orientation-permutations for dyads, triads, and tetrads.
Dyads have only two orientations. A reversal of orientation is called a flip, and since this is an odd permu-tation, it follows that flips must occur in pairs. In Appendix A we derive a pair of flips:
Thus the situation for dyads is much the same as on the Rubik cube.
c. Triadic Twists and Reflections - Gene Splicing
Triads have six permutations of orientation corresponding to the S3 group (symmetric group of permutationson three letters). Three of them are twists (abc, acb, and I=identity) and three are reflections (ab, ac, andbc). On the Rubik cube, only twists of corner cubies are possible because the reflections are 3-dimensionalenantiomorphs. But just as two-dimensional enantiomorphs can be superposed in three dimensions, so3-dimensional enantiomorphs, which the triadic reflections are, can be superposed in four dimensions. Areflection is an odd permutation, so they must occur in pairs. A twist is an even permutation, so theycan occur in isolation. Such an isolated twist is derived, in Appendix A, by use of the transposition devicedescribed in Section 6a:
We did not find any direct way to generate a pair of reflections, so we used a device we call “gene splicing”based on a rather far-fetched analogy with the recently developed techniques of genetic transformations. First, we found a long complex permutation (the “chromosome”) that contains the desired reflection (the“gene”). We splice onto this chromosome another “gene”, one of the isolated twists found above. Then wemake a permutation that “mutates” the reflection into the twist and vice-versa, on the chromosome. Finally,by reacting this chromosome with the original one, we isolate the two genes, obtaining:
There are 24 permutations of the four facets of a tetrad, comprising the S4 group. Four of them are crosses[(ab)(cd), (ac)(bd), (ad)(bc), and I], eight of them are twists, six are 2-cycles (e.g. ab, etc.), and six are4-cycles (e.g. abcd, etc.). Two-cycles and 4-cycles cannot occur in 4-space because they violate the rulegiven above in Section 6a. That is, they involve an odd number of transpositions of axes and no reversals,so they are 4-dimensional enantiomorphs. For instance
and this permutation cannot occur. Likewise, a 4–cycle is equivalent to an odd number of transpositionsand can’t occur either.
The even permutations, which form the alternating group (A4), are all possible. [See Ref. 3 for explanationof the group-theoretic terminology used in the following analysis.] The crosses are a normal subgroup of thealternating group, that we shall call N . That is:
The cosets of N , which we call S and Z, are composed of the twists. Specifically
The sets N ,S, and Z form the quotient-group of A4 by N :
in which N acts as the identity element. The group multiplication table is:
meaning that the product of an element from N and one from S is in S, the product of two elements of Z isin S; the product of an element of N and one from Z, or of two elements from S, is in Z; and the productof an element from S and one from Z, or of two elements from N , is in N .
The quotient-group is cyclic and isomorphic with the group of residue classes, mod 3. If we count each S-twist as +1, each Z-twist as −1, and each cross as zero, then the above table states that in any interactionsamong the tetradic orientations the sum of all three types is invariant, mod 3. Since initially the sum iszero, the sum is always congruent to zero, mod 3. More simply put, the sum of S-twists (+1) and Z-twists(−1) must be congruent to zero, mod 3, but there is no constraint on the crosses. This is the equivalent,for the tetrads, of the rule for corner-cubies on the Rubik cube: that the sum of the clockwise (+1) andcounterclockwise (−1) twists is congruent to zero, mod 3. But whereas the Rubik cube has only one twist ofeach type, the tesseract has four, not to mention the crosses; and they can be combined in any way subjectto this constraint.
Therefore, an isolated cross is possible, but the minimum number of twists is two, of which one must be anS-twist and the other a Z-twist. Examples of both cases were found (see Appendix A). Thus:
In Q18 we have another example of the rule we stated in Section 6a. For, (1283)adc = (1238)acd. If wewrote Q18 in the form (1234)acd(1238)acd we would seem to violate the S − Z addition rule. But this is onlybecause 1238 is on the “opposite side of the mirror” from 1234. It is exactly as if we twisted two corners ofa Rubik cube and then looked at one of them directly and the other in a mirror; we would see two twists inthe same direction. So, in expressions like Q18, we ought to write the tessies congruously.
There are 32 possible pairs of twists on 1234 and 1283, with one being an S-twist and the other a Z-twist,and all of them can be obtained as shown in Appendix A. Similarly, the other two crosses on 1234 can beobtained.
The group of the Rubik tesseract is composed of all those permutations that can be generated by the basicrotations in Table 1. We have shown that the constraints on this group are:
Tetradic positional permutations must be even;
Dyadic plus triadic positional permutations must be even;
Dyadic flips and triadic reflections must be even;
Tetradic S - Z twists must be congruent to zero, mod 3
We have also shown that all possible permutations within these constraints can actually be generated.
The number of permutations of position is therefore (24! 32! /2) × (16! /2). The number of dyadic flips is224/2. With respect to the triads, any of the six orientations is possible for the first 31, but if the number of
reflections to that point is even, the last orientation must be one of the three twists; and if it is odd, the lastorientation must be one of the three reflections. So the total number of possibilities is 632/2. Similarly, thefirst fifteen tetrads can have any of 12 orientations independently, but if the S – Z sum at that point is zero,the last orientation must be one of the four crosses (including I), or if it is +1 the last orientation must bea Z-twist, or if it is -1, an S-twist. In each case, there are four possibilities for the last tetrad instead of 12,so the number of combinations is 1216/3.
So the total number of permutations, the order of the Rubik tesseract group, is (24! 32! 16! /4) × (224/2) ×(632/2) × (1216/3) or approximately 1.76 × 10120, which is about the same as the number of ways to play agame of chess(2).
We have dealt only with very simple permutations and short cycles (except for the “chromosomes”) butvery long cycles and complex permutations are obviously possible. There are elements of order 12,432,420(= 22 × 33 × 5 × 7 × 11 × 13 × 23), consisting of an 11-cycle and a 13-cycle of dyads, a 23-cycle and a twisted9-cycle of triads, and a 5-cycle, a 7-cycle, and two crossed 2-cycles of tetrads. From cursory examination ofthe possibilities, this is probably the maximum order of an element in the group, but it might not be.
“In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not entertherein?” E. A. Abbott - “Flatland”
The foregoing ideas can obviously be extended to spaces of higher dimensions. For brevity, we shall call anN -dimensional regular orthogonal polytope an N -tope. The Rubik N -tope is obtained by slicing an N -topeinto 3N equal smaller N -topes, called “topies”, and permitting any outer layer of 3N−1 topies to rotaterigidly in a manner analogous to the rotations of the Rubik cube and tesseract. The structural features ofthe Rubik N -tope are listed in Table 2.
The “axes” of a rotation of a Rubik N -tope are (N-2)-dimensional Euclidean spaces. For the 5-tope, forexample, they are 3-spaces and there are six of them for each face, so that the Rubik 5-tope has sixty basicrotations.
The permutations of topies generated by the basic rotations form the group of the Rubik N -tope. The orderof the group is calculated as follows. The basic rotations are always odd permutations of dyads and triadsand are even permutations for n-ads of n > 3. (See at bottom of Table 2). So, the number of permutationsof position is
where aN,n is the number of n-ads (see Table 2).
For n < N , the n-ads can have any orientations of the Sn group (symmetric group on n letters), providedthat the total number of permutations of orientation is even; hence there are (1/2)(n! )aN,n permutations oforientation of n-ads for n < N .
For the N -ads, the orientations are limited to the even permutations, that is, to the alternating group AN ,because odd permutions are enantiomorphic. The number of possible permutations depends on the numbertN of orbits in the action of the N -tope group on the set of patterns of orientations of N -ads.
There are (N ! /2)aN,N such patterns. So the total number of permutations of orientation is:
and the order of the Rubik N-tope group is PpPo. For N equal to 3 or 4, the number of orbits, tN , is equalto 3. We have seen in Section 6d that the reason there are three orbits for the case N = 4 is that the
alternating group on four letters has a normal subgroup. For N > 4, AN has no normal subgroup; Buhleret al give a proof that tN = 1 for N > 4 (Ref. 4).
For example, the order of the Rubik 5-tope group is
40! 80! 80! 32! 23468024806032 ≈ 7.017 × 10560
It should be noted that this calculation gives only the maximum possible order for the Rubik N -tope group,since we have not actually demonstrated, as we did for N = 4, that all the permutations can actually bereached. But on the basis of the results for the tesseract we feel strongly that this is the actual order.
Like Mr. Square of Flatland(1), after this brief glimpse we must now descend from these insubstantial spaces.
9. The Rubik Cube Group as a Subgroup of the Rubik Tesseract Group
The Rubik cube group is a tiny subgroup of the Rubik tesseract group, considerably smaller, in terms of therelative numbers of their elements, than a proton is to the entire universe we know. Nevertheless, because ofrecent interest in the Rubik cube among a small part of humanity, we should take a look at this subgroup. After all, the structure of a proton is not without interest, at least to an even smaller part of humanity.
We have already noted that each face of a Rubik tesseract corresponds to a Rubik cube in the sense that eachfacet of the face is a cubie: tetradic facets are corner cubies, triadic facets are edge cubies, and dyadic facetsare axial cubies. Now consider a particular face, say Face 4, and the rotations R14, R24, R34, R54, R64, R74of the six faces adjoining it. The permutations generated by these rotations form a group in which no facetsenter or leave Face 4, and their effects within Face 4 are exactly the same as the six basic rotations of theRubik cube, except that they are left-handed. In fact each of the above rotations has one triadic and onetetradic cycle involving Face 4, and if we drop the 4 (which is is invariant) from each position of these cyclesand neglect the other cycles not involving Face 4, and also drop the subscript 4 from the R’s, we have
R1 = (12, 17, 16, 13)(123, 172, 167, 136)
R2 = (12, 32, 52, 72)(123, 325, 527, 721)
R3 = (13, 63, 53, 23)(123, 613, 563, 253)
R5 = (52, 53, 56, 57)(523, 536, 567, 572)
R6 = (16, 76, 56, 36)(163, 761, 567, 365)
R7 = (17, 27, 57, 67)(127, 257, 567, 617)
This is a set of basic (left-handed) rotations for the Rubik cube group. Additionally we note that therotations R41, R42, and R43 cause Face 4 to rotate about the secondary axis, in its own 3-space; that is, theyturn the Rubik cube around. The rotation R41, for example, turns the cube 90 degrees about Axis 1. Theeffect on the other axes is given by the dyadic cycle of R41 : (24, 34, 64, 74). So Axis 2 goes to Axis 3, Axis3 to Axis 6, etc. and we see that these are also left-handed turns.
Viewing the Rubik cube as just a face of a Rubik tesseract throws a new light on the problem of “quarkisolation” discussed by D. R. Hofstadter (10). In his article he describes an analogy (attributed to S. W. Golomb — see also Ref. 9) between corner–cubie twists and quarks. Like quarks, a twist cannot occur inisolation on a Rubik cube, but only in pairs of opposite twist or in three’s with the same twist.
From our present vantage point, we see that the corner–cubie twists are mere shadows of tetradic twists thata four–dimensional hypercubist can generate. For instance, he can generate a pair of twists such as
which to a Rubik cubist in the space of Face 4 appears as an isolated corner twist (a clockwise twist of theURF cubie); for he does not know that in another 3–space parallel to his, there is another Rubik cube (Face8) on which a counterclockwise twist has appeared simultaneously.
The hypercubist can play many mystifying tricks on the poor cubist. When the latter has made an ordinarypair of twists, such as (123)abc(253)acb - which are really (1234)abc(2534)acb — the invisible hypercubist cansnatch one of them away by conjugating with R2 , which moves (2534)
a single edge-cubie, and even stranger, perhaps, he can reflect a single corner-cubie, by (1234)(ab)(cd).
Analogously, the quarks may be shadows of hyperquarks, like the shadows of the puppets on the wall ofPlato’s cave, that can be produced or snatched away by an invisible puppeteer*. Some physicists believe that“wormholes” exist connecting “two distinct but asymptotically flat universes” and perhaps that particlescan pass through such wormholes from one universe to the other (11). Perhaps this is the way to isolate aquark, like the hypercubist moves a twist from one face to another.
Finally, we would note that we have three types of tetrads, N , S, and Z, corresponding to the three colors ofquarks, and each type has four “flavors”. If the physicists had stopped when they had four flavors of quarkswe would perhaps have a better analogy; but now that they believe in five or maybe six flavors (8) we findwe have reached or surpassed the limits of our analogical powers.
* Behold! human beings living in an underground den,. . . here they have been since their childhood, . . .
chained so that they cannot move, and can only see before them. . . . Behind them a fire is blazing at adistance, and between the fire and the prisoners there is a raised way; and . . . a low wall . . . like thescreen which marionette players [use to] show their puppets. . .
You have shown me a strange image, and they are strange prisoners.
Like ourselves, I replied; and they see only their own shadows, or the shadows of one another, which the firethrows on the opposite wall of the cave?. . . And of the objects which are being carried [by the puppeteers]in like manner they would only see the shadows?
To them, I said, the truth would be literally nothing but the shadows of the images.
Assuredly not, he said; I have hardly ever known a mathematician who was capable of reasoning.
1. E. A. Abbott “Flatland”, Blackwell (Oxford) 1944
2. Jeremy Bernstein, “The Analytical Engine”, William Morrow & Co., N.Y. pp 102–103. Reports the
calculation by Claude Shannon that there are about 10120 possible chess games.
3. G. Birkhoff and S. Maclane “A Survey of Modern Algebra”, Chapter VI, Macmillan, N.Y. 1949.
Chapter VI gives a clear explanation of the group–theoretical terminology used in this paper.
4. Joe Buhler, Brad Jackson, Dave Sibley, “An N–Dimensional Rubik Cube”, preprint. This paper
gives a formal mathematical derivation of the maximum order of the Rubik N-tope group and alsoconcludes that tN = 1 for N > 4.
5. K-M Chung and H. Yuen “A Tiny Pascal Compiler” BYTE, September, October, November 1978
6. M. Gardner “The Ambidextrous Universe”, Chapter 17, “The Fourth Dimension”, Scribner 1964.
7. Martin Gardner, “Mathematical Games: Is It Possible to Visualize a Four–Dimensional Figure?”,
Scientific American, November 1966, pp 138-143.
8. H. Georgi, “A Unified Theory of Elementary Particles and Forces”, Scientific American, April 1981.
9. Solomon W. Golomb, “Rubik’s Cube and a Model of Quark Confinement”, American Journal of
10. Douglas Hofstadter, Scientific American, March 1981, p. 20, “Metamagical Themas”
11. C.W. Misner, K.S. Thorne, J.A. Wheeler, “Gravitation”, Chapter 31, pp 836-840.
12. O. Schreier and E. Sperner “Modern Algebra and Matrix Theory” Chapter V par 24 “Orthogonal
13. David Singmaster, “Notes on Rubik’s Magic Cube” - 5th Ed. 1980, Mathematical Sciences and
Computing Polytechnic of the South Bank, London SE1 OAA, England. A more recent elabora-tion on the “Notes” is “Handbook of Cubik Math” by Alexander H. Frey and Singmaster, EnslowPublishers, 1982.
(145, 149, 157, 153)(146, 150, 158, 154)(147, 152, 159, 156)(148, 151, 160, 155)
The large cubes are faces of the tesseract that have been rotated into the 3–space of Face 4,which is hidden behind Face 3 and below Face 1. The small cubes are the facets of a tetradas it moves through the cycle (1234, 1283, 1278, 1247) under the action of rotation R12. The“start” position of this tetrad, as indicated by the colors of its facets, is 1234.
The coordinate axesof each face have beenrotated with the faceinto the 3–space ofFace 4.
R12 =(13, 18, 17, 14)(123, 128, 127, 124)(134, 183, 178, 147)(163, 168, 167, 164)
(1234, 1283, 1278, 1247)(1634, 1683, 1678, 1647)
R13 =(12, 14, 16, 18)(123, 143, 163, 183)(124, 146, 168, 182)(127, 147, 167, 187)
(1234, 1436, 1638, 1832)(1274, 1476, 1678, 1872)
R14 =(12, 17, 16, 13)(123, 172, 167, 136)(124, 174, 164, 134)(128, 178, 168, 138)
(1234, 1724, 1674, 1364)(1238, 1728, 1678, 1368)
R21 =(23, 24, 27, 28)(123, 124, 127, 128)(523, 524, 527, 528)(234, 247, 278, 283)
(1234, 1247, 1278, 1283)(5234, 5247, 5278, 5283)
R23 =(12, 82, 52, 42)(123, 823, 523, 423)(124, 821, 528, 425)(127, 827, 527, 427)
(1234, 8231, 5238, 4235)(1274, 8271, 5278, 4275)
R24 =(12, 32, 52, 72)(123, 325, 527, 721)(124, 324, 524, 724)(128, 328, 528, 728)
(1234, 3254, 5274, 7214)(1238, 3258, 5278, 7218)
R31 =(23, 83, 63, 43)(123, 183, 163, 143)(523, 583, 563, 543)(234, 832, 638, 436)
(1234, 1832, 1638, 1436)(5234, 5832, 5638, 5436)
R32 =(13, 43, 53, 83)(123, 423, 523, 823)(134, 435, 538, 831)(163, 463, 563, 863)
(1234, 4235, 5238, 8231)(1634, 4635, 5638, 8631)
R34 =(13, 63, 53, 23)(123, 613, 563, 253)(134, 634, 534, 234)(138, 638, 538, 238)
(1234, 6134, 5634, 2534)(1238, 6138, 5638, 2538)
R41 =(24, 34, 64, 74)(124, 134, 164, 174)(524, 534, 564, 574)(234, 364, 674, 724)
(1234, 1364, 1674, 1724)(5234, 5364, 5674, 5724)
R42 =(14, 74, 54, 34)(124, 724, 524, 324)(134, 714, 574, 354)(164, 764, 564, 364)
(1234, 7214, 5274, 3254)(1634, 7614, 5674, 3654)
R43 =(14, 24, 54, 64)(124, 254, 564, 614)(134, 234, 534, 634)(174, 274, 574, 674)
(1234, 2534, 5634, 6134)(1274, 2574, 5674, 6174)
R52 =(53, 54, 57, 58)(523, 524, 527, 528)(534, 547, 578, 583)(563, 564, 567, 568)
(5234, 5247, 5278, 5283)(5634, 5647, 5678, 5683)
R53 =(52, 58, 56, 54)(523, 583, 563, 543)(524, 582, 568, 546)(527, 587, 567, 547)
(5234, 5832, 5638, 5436)(5274, 5872, 5678, 5476)
R54 =(52, 53, 56, 57)(523, 536, 567, 572)(524, 534, 564, 574)(528, 538, 568, 578)
(5234, 5364, 5674, 5724)(5238, 5368, 5678, 5728)
R61 =(63, 68, 67, 64)(163, 168, 167, 164)(563, 568, 567, 564)(634, 683, 678, 647)
(1634, 1683, 1678, 1647)(5634, 5683, 5678, 5647)
R63 =(16, 46, 56, 86)(163, 463, 563, 863)(164, 465, 568, 861)(167, 467, 567, 867)
(1634, 4635, 5638, 8631)(1674, 4675, 5678, 8671)
R64 =(16, 76, 56, 36)(163, 761, 567, 365)(164, 764, 564, 364)(168, 768, 568, 368)
(1634, 7614, 5674, 3654)(1638, 7618, 5678, 3658)
R71 =(27, 47, 67, 87)(127, 147, 167, 187)(527, 547, 567, 587)(274, 476, 678, 872)
(1274, 1476, 1678, 1872)(5274, 5476, 5678, 5872)
R72 =(17, 87, 57, 47)(127, 827, 527, 427)(167, 867, 567, 467)(174, 871, 578, 475)
(1274, 8271, 5278, 4275)(1674, 8671, 5678, 4675)
R74 =(17, 27, 57, 67)(127, 257, 567, 617)(174, 274, 574, 674)(178, 278, 578, 678)
(1274, 2574, 5674, 6174)(1278, 2578, 5678, 6178)
R81 =(28, 78, 68, 38)(128, 178, 168, 138)(528, 578, 568, 538)(238, 728, 678, 368)
(1238, 1728, 1678, 1368)(5238, 5728, 5678, 5368)
R82 =(18, 38, 58, 78)(128, 328, 528, 728)(138, 358, 578, 718)(168, 368, 568, 768)
(1238, 3258, 5278, 7218)(1638, 3658, 5678, 7618)
R83 =(18, 68, 58, 28)(128, 618, 568, 258)(138, 638, 538, 238)(178, 678, 578, 278)
(1238, 6138, 5638, 2538)(1278, 6178, 5678, 2578)
Rotation “Axes” per Face = (1/2)(N − 1)(N − 2)
Number of Basic Rotations = N (N − 1)(N − 2)
Number of Movable “Topies” = 3N − 2N − 1
Number of n-ads per face that do not movein a specific rotation of that face
The basic rotations Rij are listed in Table 1. Useful moves are designated Q1, Q2, etc. The subscripts onthe Q’s have no significance as coordinates. Other moves needed temporarily are denoted by other capitalletters. Moves are written in order from left to right; e.g., R12R13 means that R12 is performed first, followedby R13. The following notation is used for brevity:
Inverse : The inverses of Rij, Qi, etc. are R , Q , etc.
Transpose : Q[ij]k means that subscripts i and j are interchanged on all the rotations that comprise Qk,
and that all the rotations are inverted. The effect of this is to transpose i and j in thepermutation produced by Qk.
Qk = R52R34 = (53, 54, 57, 58, 23, 13, 63) etc.,
34 = (63, 64, 67, 68, 13, 23, 53) etc.
Note that in the permutation, 53 is treated as 1 3 which changes to 2 3 = 63 etc. Similarly,in the subscripts on the R’s, 52 = −12 changes to −21 or 61, etc. Note also that therotations are inverted even if their subscripts are not involved in the transposition, as R34above. Transposition is a way of turning the tesseract around in 4-space.
We began looking for useful moves by following D. Singmaster’s advice for the Rubik cube, looking at somesimple moves. Almost at once we found two which led immediately to our first useful move, Q1. These were
(1234, 1278)(1238, 1274)(5234, 5278)(5238, 5274)
(R2 R2 )2 =(123, 127)(124, 128)(523, 527)(524, 528)
(1234, 1278)(1238, 1274)(5234, 5278)(5238, 5274)
Neither of these moves is very useful in itself, but they are obviously alike in most cycles, so we take theirproduct:
(R2 R2 )2 = (23, 27)(24, 28)(234, 278)(238, 274)
Q1 is an interchange of two pairs of dyads and two pairs of triads. It turns out that from Q1 we can generatea series of useful moves, including all those necessary to prove the order of the group of the Rubik tesseract. Some simple moves related to Q1 are:
Q2 = (Q1R32)c = (123, 827, 423)(523, 427, 823)
Then, letting X = (R32)aR14(R32) we obtain a single 3-cycle of triads:
Now we reduce Q1 to a pair of triadic interchanges by the following strategy. The conjugating factorX = (R32)aR2 (R
interchanges 238 and 274 with 674 and 638, which are not affected by Q
the opposite side of the tesseract. So XQ1X = (23, 27)(24, 28)(234, 278)(638, 674) and (since X = X):
This is the minimum number of triadic interchanges, since a single interchange would be an odd permutation. Next we reduce Q1 to a pair of dyadic interchanges, which is the minimum possible for the same reason. Let
Q5 = Q[24]1 = (43, 47)(42, 46)(432, 476)(436, 472)
Q1Q5 = (23, 27)(43, 47)(24, 28, 64)(234, 278, 674)(238, 634, 274)
We eliminate the 3–cycles by cubing. Thus
The process for obtaining a pair of tetradic interchanges is longer, but it is fairly straightforward because wecan treat it as a problem on the Rubik cube. If we think of Face 2 as a Rubik cube, we see that the effect ofQ1 is to rotate the entire middle layer of the cube by 180 degrees around axis 1. Similarly, the effect of R221is to rotate the entire cube 180 degrees around the same axis. Therefore R2 Q
the top and bottom layers 180 degrees without affecting the middle layer. That is, R2 Q
opposite edge cubies and the diagonally opposite corner cubies, top and bottom. To cancel out the edge-cubieinterchanges we need triadic interchanges involving Face 2. We get these by conjugations on Q4, as follows:
Q12 and Q13 are the desired moves which have the effect in Face 2 of swapping the opposite edge-cubies onthe top and bottom. So
1)Q12Q13 = (1234, 1278)(1247, 1283)(5234, 5278)(5247, 5283)
We now have, in Q14, an interchange of four pairs of tetrads. We next set out to reduce this to two pairs bythe same strategy that we got Q4, but it is difficult to conjugate one of these pairs of tetrads to the otherside of the tesseract without affecting any of the others.
We first rearrange the interchanging pairs, as follows:
= (1234, 1283)(1278, 7214)(5234, 5283)(5278, 7254)
We remark that the two tetrads, 1247 and 5247, have been re-oriented. We now move two pairs to theopposite side of the tesseract:
= (1346, 1836)(1647, 7681)(5234, 5283)(5278, 7254)
(Actually, this move was unnecessary. We could have proceeded to the next step from Q15.) We apply thecommutator move:
64 c = (1346, 1836)(5678, 7654)(1876)abc(1746)acb
By cubing Q17 we have, finally, two pairs of tetrads:
Now we have found two pairs of interchanges on dyads (Q6), on triads (Q4), and on tetrads (Q3 ). By
conjugation we can obtain such transpositions on any two pairs of dyads, triads, or tetrads. Since anypermutation can be expressed as a product of transpositions, we can build up any positional permutationthat is even for dyads, triads, and tetrads. In the case of a permutation that is odd for dyads and triads(and necessarily even for tetrads) we can multiply it by any basic rotation and the product will be even fordyads, triads, and tetrads, as illustrated below.
To obtain an interchange of one pair of dyads and one pair of triads, as, for example:
B = (123, 124, 127, 128)(523, 524, 527, 528)
C = (1234, 1247, 1278, 1283)(5234, 5247, 5278, 5283)
We can build up A from conjugations of Q6 and Q12, i.e.:
Likewise, B can be built up from Q9 and Q11:
The number of basic rotations to get Q36 is 11,919.
In making the tetradic conjugations above it is helpful to note that the proper orientations can be producedmost easily if the desired interchange changes the same number of faces as the original interchange. In thecase above, for example, we wish to conjugate from Q3 to C
1. Each interchange in C1 holds two faces fixed.
In Q3 , the interchange (1346, 1836) also holds two faces fixed (1 and 6) but the interchange (5678, 7654)
changes three faces. Therefore we conjugate in two steps using only the first interchange of Q3 . That is, we
first get C0 from (1346, 1836), leaving (5678, 7654) alone. Then we get
again leaving the second interchange alone. Then the product of these gives C1.
As a bonus from Q17 we have a pair of twists:
For convenience we move them to the other side of the tesseract:
We now proceed to obtain the other permutations of orientation of tetrad 1234. We first make the sametwist on a third tetrad:
Next we obtain a different twist by transposition:
Q[12]19 = (2134)acd(2178)adc = (1234)bcd(1278)bdc
Combining Q18 and Q[12]19 gives a cross on 1234:
Q18Q[12]19 = (1234)(bcd)(acd)(1283)adc(1278)bdc
Q[12]19Q18 = (1234)(ad)(bc)(1283)adc(1278)bdc
(Note that the permutations (acd) and (bcd) multiply in reverse order to the moves Q18 and Q[12]19.) Sowe get
We can also obtain Q22 in half as many steps from:
Q22 = Q18Q[12]19(Q[12]19Q18) = (Q18Q[12]19)c
By multiplying Q18 in turn by each of the crosses we obtain all of the S-twists on tetrad 1234:
and by squaring these moves (including Q18) we get all the Z-twists on 1234. We can easily move the crossesto tetrad 1283 [R Q
20R12 = (1283)(ac)(bd), etc.] and generate all eight twists on that tetrad. In this way
we can generate all of the 32 possible pairs of 1234 and 1283 with an S-twist on one of them and a Z-twiston the other. We can easily move the crosses and twist-pairs anywhere on the tesseract, and thus obtain allpossible permutations of tetradic orientations.
Dyads have only two orientations, and flips must occur in pairs. It is easy to find a representative pair bytransposition. We start with
Q24 = (Q6Q[12]6)(Q23Q[12]23) = (12, 43)(21, 43) = (12)ab(34)ab
A triadic twist can also be found by use of transpositions. These twists can occur singly. We start with Q7,which we move to
Q[23]25 = (132, 234)(172, 274) = (123, 243)(172, 274)
Q25Q[23]25 = (123)abc(234)abc(163, 364)(172, 274)
To reduce this to a single twist we make another transposition:
The reverse twist is of course Q2 = (123)
These triadic twists and the dyadic flip-pair can obviously be moved anywhere on the tesseract.
Triadic Reflections — “Gene Splicing”
The final type of permutation of orientation we need to find is a pair of triadic reflections (they cannot occursingly), which we will do by a method we call “gene splicing” for reasons that will be obvious. We first lookfor a move that includes such a reflection as part of the permutation. Such moves are easy to find. Forinstance
The remainder of the permutation, denoted by X, is a long and complicated sequence of cycles which involveall but seven of the 31 other triads. We call Q30 the “chromosome”. We do not need to concern ourselveswith the details of X except to identify the “neutral” triads that are not affected by it, some of which wewish to use for conjugating purposes. The four we use are 275, 872, 576, and 678, all of which have facets inFace 7. We start with Q4 = (238, 274)(638, 674) and convert it to
in which all the triads also have facets in Face 7. We can therefore move A to the four neutral triads by aRubik-cube move; for instance
[That is, if we regard Face 7 as a Rubik cube, A is a pair of edge-swaps : Right-Front with Right-Back andLeft-Front with Left-Back; while Q31 swaps Right-Down with Right-Back and Left-Down with Left-Back. This conjugation is easily found on the Rubik cube.]
The next step is to move 134 to one of the neutral positions without affecting the other three positions:
The third step is to twist triad 275, which we do by conjugating Q29:
Step 4 is to splice this gene into the chromosome:
Q32 is a “reagent” that will react with the chromosome Q33Q30 only at the “gene sites” 275 and 134. Carrying out this reaction, we mutate gene 134 from type bc to type abc and we mutate gene 275 in thereverse manner:
Finally, mixing the two chromosomes Q30 and Q34, we isolate the genes 134 and 275:
34 = (134)(abc)(bc)(275)(bc)X X = (134)ac(275)bc
From Q35 and Q33 we can get the other two reflections on triad 275; that is:
Similarly we can generate the other two reflections on 134. Thus all nine combinations of reflections of 134and 275 can be found, and we can move these anywhere on the tesseract.
In summary, we have shown how to generate any permutation of the Rubik tesseract group.
1 PROGRAM TESS2X (TDATA); {Search for Useful Moves on Rubik Tesseract}
OK, DONE, RECAP, FACE, ANS, NMOVES : INTEGER;
SAVE:=M[K1]; M[K1]:=M[K4]; M[K4]:=M[K3]; M[K3]:=M[K2]; M[K2]:=SAVE
Z(-12); Z(24); Z(24); Z(24); Z(42); Z(42);
Z(42); Z(42); Z(24); Z(-12); Z(-12); Z(-12)
Z(42); Z(-12); Z(-12); Z(32); Z(13); Z(13)
Z(13); Z(13); Z(-21); Z(-21); Z(117); Z(117);
Z(219); Z(118); Z(219); Z(118); Z(219); Z(118)
Z(14); Z(-32); Z(-32); Z(-32); Z(106); Z(-32); Z(14); Z(14); Z(14)
Z(12); Z(82); Z(107); Z(82); Z(82); Z(82); Z(12); Z(12); Z(12)
Z(53); Z(53); Z(53); Z(43); Z(43); Z(53); Z(53); Z(43);
Z(43); Z(43); Z(43); Z(53); Z(53); Z(43); Z(43); Z(53)
Z(14); Z(14); Z(13); Z(13); Z(13); Z(53); Z(23);
Z(23); Z(23); Z(23); Z(53); Z(53); Z(53); Z(13); Z(14); Z(14)
Z(41); Z(41); Z(41); Z(12); Z(31); Z(31); Z(31); Z(134)
Z(34); Z(21); Z(21); Z(21); Z(34); Z(34); Z(34); Z(106);
Z(74); Z(21); Z(21); Z(21); Z(74); Z(74); Z(74); Z(106);
Z(42); Z(12); Z(12); Z(12); Z(32); Z(112); Z(32); Z(32); Z(32);
Z(82); Z(12); Z(12); Z(12); Z(72); Z(112); Z(72); Z(72); Z(72);
Z(32); Z(12); Z(12); Z(12); Z(52); Z(32); Z(32); Z(32); Z(109);
Z(111); Z(32); Z(52); Z(52); Z(52); Z(12); Z(32); Z(32); Z(32);
Z(32); Z(32); Z(32); Z(12); Z(52); Z(52); Z(52); Z(32); Z(109);
Z(111); Z(32); Z(32); Z(32); Z(52); Z(12); Z(12); Z(12); Z(32);
Z(31); Z(31); Z(21); Z(21); Z(21); Z(31); Z(31); Z(140);
Z(12); Z(12); Z(12); Z(14); Z(14); Z(117); Z(117); Z(117);
Z(12); Z(12); Z(12); Z(14); Z(14); Z(117); Z(117); Z(117);
Z(21); Z(21); Z(21); Z(-21); Z(12); Z(12); Z(-21); Z(-21); Z(-21);
Z(21); Z(21); Z(13); Z(13); Z(21); Z(21); Z(13); Z(13)
Z(201); Z(31); Z(31); Z(31); Z(-31); Z(-31); Z(-31);
Z(201); Z(31); Z(31); Z(31); Z(-31); Z(-31); Z(-31);
Z(24); Z(24); Z(24); Z(-24); Z(-24); Z(-24); Z(42); Z(42);
Z(24);Z(-24);Z(24);Z(24);Z(43); Z(43);Z(24);Z(24);Z(43);Z(43)
Z(201); Z(205); Z(201); Z(205); Z(201); Z(205)
Z(12); Z(12); Z(201); Z(208); Z(209); Z(210); Z(211)
Z(41); Z(41); Z(41); Z(-21); Z(-21); Z(31); Z(31); Z(31)
Z(216); Z(-14); Z(216); Z(-14); Z(-14); Z(-14)
Z(23); Z(23); Z(-12); Z(-12); Z(217); Z(217);
Z(24); Z(24); Z(24); Z(-31); Z(206); Z(-31); Z(-31); Z(-31); Z(24)
Z(21); Z(21); Z(21); Z(-41); Z(-41); Z(-41); Z(207); Z(-41); Z(21)
Z(13); Z(13); Z(13); Z(-13); Z(31); Z(31); Z(-13); Z(-13); Z(-13);
Z(13); Z(13); Z(13); Z(32); Z(32); Z(13); Z(13); Z(32); Z(32)
Z(301); Z(23); Z(23); Z(23); Z(-23); Z(-23); Z(-23); Z(14); Z(14);
Z(301); Z(23); Z(23); Z(23); Z(-23); Z(-23); Z(-23); Z(14); Z(14);
Z(13); Z(13); Z(13); Z(83); Z(83); Z(83); Z(307); Z(83); Z(13)
Z(23); Z(-23); Z(-23); Z(-23); Z(32); Z(32); Z(-23); Z(23); Z(23);
Z(23); Z(23); Z(23); Z(31); Z(31); Z(23); Z(23); Z(31); Z(31)
Z(401); Z(13); Z(-13); Z(24); Z(24); Z(13); Z(13); Z(13); Z(-13);
Z(401); Z(13); Z(-13); Z(24); Z(24); Z(13); Z(13); Z(13); Z(-13);
Z(23); Z(-43); Z(407); Z(-43); Z(-43); Z(-43); Z(23); Z(23); Z(23)
PERMUTE(MEM[S],MEM[S+1],MEM[S+2],MEM[S+3])
IF NOT OK THEN WRITE(N#,’ not understood - no action taken’,13,10)
VAR N,PTR,ACTIVITY,J,K, LINEPOS, JJ, JK, CYLENGTH, NPOS, TAG: INTEGER;
ACTIVITY:=ACTIVITY OR ((M[K]<>K) AND FLAG[K])

Chemistry on the fringes of the former USSRFollowing admission to the European Union, are the Baltic states on course for economic reform? Bea Perks reports unsolved,’ said Lukevics. ‘Twelve of world’s largest producers of oil shale. anticancer drug on the Japanese Finland (the countries’ capital cities, and cardiovascular and infectious disease. considerably since Latvia was rel

BROCHURE ON RESEARCH ON MONEY AND FINANCE IN IGIDR Foreword From its founding in 1986, IGIDR had hoped to work in the areasof money and finance. It was felt that IGIDR should havecomparative advantage in these areas as it is established byReserve Bank and is located in Mumbai, India’s financial capital. Research in these areas was slow to start as good facultymembers are hard to find. H