On certain algorithms in the practice of geometry and the theory of numbers

Pub . Mat . UAB Vol . 29 Ns 1 Abril 1985 ON CERTAIN ALGORITHMS IN THE PRACTICEOF GEOMETRY ANDTHE THEORY OF NUMBERS 0 . Introduction PeterHilton and Jean Pedersen [4e demonstrated in 111 and 131 a systematic method of folding a straightstrip of naper, by whatwe called a pn-címaxcf bold¿nq prcoceduke,  to approximate, to any desired degree of accuracy, a regularconvexs-gon and certainregularstar s-gons, provided that  s E F, the set of 6o1d¡nq numbm .  Here is definéd to be the set of all integers s of the form 2x x 1 , where x > 1, y > 2 . 2_1 Of course, suchnumbers s areodd . By introducincT SecokLdaxq folds on the stripof paper we showed how it is possible to approximate regular 2 k s-gons, whe re  s E F and  k > 1  (and we included, forthe sake of comple teness, the  exact  constructions of the regular  2 k -gons, k >2) . The only remaining numbers > 3 are those of the form 2 k a,  where a is odd,  : P 1  and  not  a folding number and  k > 0 . However, the method for approximating thoseregularpolygons can be described by a seguence of steps as follows (consult [ll for details) . First, sincewe know that, for any odd number a, 2(D(a) = 1 mod a, where fi(a) is the Euler totientfunction, it follows that a is a factor of some element of F, say s, with s = a£ . We canusethe primaryfolding procedure to obtain a strip of paper suitable for approximating a regular s-gon . If we then introduce k' secondary fold lines at each pointthat would havebeen a vertex of the regular s-gon, we, canuse a longer strip of this folded tape to construct a regular 2 k s-áon . We then gluethis 2 k s-gon to a pieceof paper and fold on the lines connectina every 9th vertex to produce the desired 2 k a-gon . In [2] and [3], we introduced an algorithm for finding the optimal sEF such that als . In summary, the above procedures (using primary andsecondary folds) provided us, in conjunction with the algorithm referred to above, with a systematic methodthatcould be used to approximateregular convex s-gons for all s % 3 . The same procedures produced manyregular  Stax  s-gons, where  s E F .  In fact, as discussed and provedin [2), for a given s = (x,y) E F,  the exactnumber of star s-gons produced by the primaryfolding procedure is 2 4'(y)xy . Further, thesecould be explicitly described . In [2] we raised the question as to whether by generalizing in a natural waythe primary folding, we might be able to avoid the gluingstep described above, and also be able to fold a,Pl regular star polygons . In this paper we answer that question, in the affirmative . Given  a,b odd with  a <2 and a prime to b,  we describe in Section 1 a genenatí .zed primary folding procedure which approximates a regular star {a}-gon . There are, then, very obvioussecondary procedures whichallow us to remove the restriction thatboth a and b be odd . Thegeneralization consists in allowinga procedure of arbitrary periodicity . The prócedures in previous papers have all been of period 1 or 2 . An interesting aspect of the content of this paper, and the otherpaperswe refer to, is the way the geometry motivates the number theory, andthe subsequentinteraction between the two topics . Indeed, althouqh the  Quabí-Ondevc Theonem  of Section 2 would stand on its own merits as an interestingpieceof number theory, it is hard to imagine howone would have discovered it without the geometric motivation . Moreover, although our generalized primary folding procedure obviates the needto glue a constructed N-qon to a oieceof paper in order to construct an M-gon, with MIN, the number theory generated by the gluing technique, described in [2l and [3], stands in its own right, and is in no sense superseded by the more sophisticated paper-folding proceduresof this articles, nor subsumed in the numbertheorythat arises from those more sophisticated procedu res . In Section 1 we describe the pacer-foldingprocedure which enables us to construct arbitrary star polygons . We have sought, by including this section, to make the entire paper rea sonablyself-contained,though we arenot actually advocating the neglect of our earlierpaperson this subject . Section 2 openswith the definition of a symbol bal a 2 k1k 2 ar k r which may be regarded as encoding the instructions for folding a strinof tape to form a star {á} -qon, with ai ,b odd, and i a l < 2 .  The "code" is described in a typical case in Section 1 and, in general, in Appendix 1 (Section 4) . However, this symbol also constitutes an interesting algorithm for determining the quadti-ohdelc of 2 mod b, that is, the smallestpositiveinte ger  4  such that  2 2 = tl mod b .  Indeed,  if  a l is prime r to b, then the quasi-order is k = 1 k i andthe parityof k  i=1 r determines whether 2 = 1 or 2k - -1 . Of course, the quasi-order, reinforced with the information provided by the parity of r, providesmuchmore information than the order of 2 mod b . Examples are given in Appendix 2 (Section 5) to show how to apply the algorithm to obtain the symbol (0 .1) and then how, in a given case, to obtain, from the symbol, the factor complementary to b in 2 k ± 1 . In Section 2 we describe the symbols, prove some basic Droperties, and enunciate the Quasi-Order Theorem . The theorem is proved in Section 3, where we also obtain some refinements of the theorem of further number-theoretical interest . We remarkthatan independent proof of the Quasi-Order Theorem was shown to us by Gerald Preston . This proof was based on the notion of Hasse functions (see, for example, [41) ; however, the direction of proofdoes not take us through Theorem 2 .5, which has an immediate application to paper-folding . The papercloseswith thetwo appendices alreadyreferred to ; in the firstwe go back to the geometrical significance of the symbols,and, in the second, we discuss, as examples, Fermat and Mersenne non-primes . A feature of the earlier papers [2] and [31 missing from the present paper was the aeneralizationfrom 'base 2' -- the onlybase of geometrical interest, since we modestly con fine ourselves to  b .ihect :ng  angles -- to 'base t' ,  where  t is an arbitrary positiveinteger * 1 . It appearsthat this generalization leads to interesting difficultieswhenwe try to introduce the analogs of our symbols in base t, since, in this general context, they may fail to exist for a given b . We propose to devote a sequel [61 to the study of generalized symbols and the (generalized) quasi-order problem . 1 . How to fold regular star polycTons First we supposethat appropriate bold, of cAe"e, lines have beenmade on our straight stripof paper and we describe the actual construction nrocess for foldinga {a}-gon l , where a and b are mutuallyprimeintegerswith a<b . SuoDose, as illustrated in Figure 1, that we have a straight strip of paper that has creasesalongstraight lines emanating frommarked vertices Ai,i=0,1, . . ., at the topand bottom edges, and that, for a fixed k, those at the particular vertices A nk, n=0,1,2, . . . . b,  which are on the top edge,  formidentical angles br . Suppose further that thesevertices are equally spaced (we describebelow howyou might obtainsuch a strip) . Figure 1 (a) shows the beginning of the strip . If we fold this stripon AnkAnk+2 (as shown in Figure 1(b)) and then on AnkAnk+l  (as shown in Figure 1(c)),  the direction of the top edge of the tapewillbe rotated through an angleof 2(b ir) and the tane will be oriented the same way, with respect to the center of the polygon being delineated by its top edge . We call these two folds through A nk, in that order, a 2(b r)-tiuíAt at A nkl and observe that, if a 2(b r)-twist is performed at A nk for n = 0, 1, 2, . . ., b-1, thetop edgeof the tape will have turned through an angle of 2aw andthe point A bk will then be coincident with A o . Thus thetop edge of the tape will have visited every a th vertex of a bounding regular convex b-gon, and hence determines a regularstar la}-gon . 1 A closedsequenceof b edgesthat visit, in order, every a th vertex (mod b) of a boundingregular convex b-gon . We include the regular convex  b-gon as the special case  a =1 . 36 Fígure i A ktl~ a, A ik+) . We now explain how we obtained the desired crease lines in the strip of tape in the firstplace . Recall thatwe are seeking to constructa star {e}-gonwhere a, b are mutually primepositive integers with  a < 2 .  We assume first that a, b are odd . Thuswe wish to have a strip of paper on which the angle b u appears at regular intervals along the top edge . We designate the direction from left to right as the 4oAmAd direction on the tape . We beginby markinga point A o on the top of the tape and making an .ínítíal crease line going in the downward forward direction from A o to' A 1 at the bottom of tape, and abz(une that the angle it makes with the top edge is a  we call this the  putatíve angle . The we continue to b 3 7 form new creaselines according to the following four rules : (1) The first new crease lineemanatesfrom the vertex A 1 . (2) Each new crease line goes in the forward direction along the strip of paper . (3) Each new crease line always bí6ect6 the angle between the last crease line and the edge of the tape from which it emanates . (4) The bisection of angles at any vertex continues until a crease line produces a putativeangle of the form b r  where a' is an odd number ; then the folding stops at that vertex and commences at the intersection point of that last crease linewith the other side of the tape . Let us consider the exampleb = 11, a = 3 . Then we cansee that if we begin with an angle of 11 r at Ao (as shown in Figure 2(a)) and adhere to the aboveruleswe will obtain a stripof tapewith the angles and creases (dotted lines) indicated in Figure 2(b) . Adhering to the notation for the primary foldingprocedures in [11, [21 and [31, we could write this more generalized folding procedure as As before, this notation means that if we beginfolding on the strip of paperat the placewhere there is one crease line sloping upwaAdb then the first d l refers to the one bisection (producing a line in a downward direction) at A l0n (for an = 0,1,2, . . .) on thetop of the tape ; the u 3 refers to 3 8 {d 1 u 3 d 1u1 d 3 u 1 }  .  (1 .1) the 3 bisections (producing creases in an upward direction) made at the bottom of the tape through AlOn+l ; etc . However, the foldingprocess is duplícated halfway through, so it suffices to write just the firstthree exponents in (1 .1) . In fact, we can denote (1 .1) evenmore simply as {1,3,1}  (1 .2) with the understandina that we fold dk luk2 d k3 u k4 . . . with the k i , k 2 , k 3 , . . . cycling, in order, repeatedly through the values 1, 3, 1, . . . We call (1 .1) or (1 .2) a oA pevú .od 3 . Note that, in this procedureswe have hitherto considered of period 1 ({d n u n }) _a b primary foldingprocedure terminology, the primaryfolding in 11, 2, 3] were all or period 2  (Id m u n }, m * n) . It is easy to see that, starting with any putative angle a < z),  we will alwaysobtain (a, b odd, mutually prime, by our rules a primary foldina 'produces' this putative angle tative angle 11 angle 11 n at indeed, our crease lines could havebeen used to fold a star 11 { 3} -gon, theycould also have been used to fold a 11-gon and a star  { 5}-L}This feature of our with its creaselines obviously applies in general : other b-gons will be available to us from the tape yielding the procedure k 1 ,k 2t . . .,k r which angle . We alsonote that, startingwith the 11 n at thetop of the tape, we produced a puir at the boton of the tape, then a putative the top of the tape, and so on . Thus if, convex tape furnished star star a < 2,  there is always a completely determined unique symbol like the one above (we do not need a,b relatively prime) . Appropriately interpreted, we can use this symbol to read off the foldingprocedure that produces the angle of  a along b thetop edaeof the tape, so that a symbol such (1 .3) encodes a folding procedure for producinga star {á}-gon, and also tells us whatother star polygons we can obtain from the same tape (of course, for each symbol a diagram similarto Figure 6 can be drawn to illustrate the relative positions of the angles an ) b Beforewe closethissectionwe would like to point out that the foldingprocess described above is the mobt e6bící .ent one possible . That is, there could not be any folding procedure ia^ star oY th .i .s type that wouiü procure t .11G rcÑ,t+~- , 'red a u u  ro iy 7 on s wi_th - . . . . . . fewer folds . It is also optimal from the point of view of "difficultyof execution", for it keeAs the number of bisectionsat each vertex to a minimum . These last comments are explained as follows . If the folding procedure {kl,k2# . . .0kr} produces the angle  r, then (see (2 .3) and (2 .4) bl 2 k ±1, where b r k = E k . If we adopt the procedures described in this seci=1 i tion we willhave a procedure  {£l, 92 ,.... Qs}  such that s R =  E  Q .  is the  smaUut  number  m  suchthat  bl 2 m ±l, j=1 that is, the quo-6í-ondeA of 2 mod b . Moreover, r willbe a multipleo .f s and, suitably cycling the Qj , each k i is a multiple of 2 1 . 46 All thesefacts are contained in thenumber-theoretical results of the next two sections . 2 . S - ~rn bols a nd the quasi-order of 2 mod b By the symbol b a l a 2 . . . ar k l k 2 . . . kr b =  a l +  2 k ia i+l ,  i =  1, 2,  . we understand that b is an odd positive integer, that a l is an odd positive integer < 2, i = 1,2, . . ., r,  and that k l ,k21 . . .k r are positive integers such that ' r,  a r+1  =  a l .  (2 .2) Let us aaree where convenient,to define al for all integers i by making a l periodic in i, with period r, and similarly for k i . We note that, given odd positiveintegers a, b with  a < 2,  there is always a symbol (2 .1) with  a l = a,  and that the symbol is uniqueup to .ít~on ; herewe say that (2 .1) arises by iteration if there exists sir suchthat a l+s = a i' ki+s =k i ' for all i . A proper iteration, that is, one in which s ~ r,  is called a  hepetctdon . Given b,kl, . . .,kr, the equations (2 .2) haveuniquesolutions, in the "unknowns" a ., namely i Ba i = bA i , i = 1, 2, . . ., r,  (2 .3) r where  B  =  2 k -  (-1) r ,  k  =  E  ki ,  (2 .4) i=1 and  A .=2 1 ' -k i-1 -2k-ki-l-kl-2+ . . .+(-1)r2ki-( - 1) r i  i=1,2, . . .,r . (2 .5) We note, for future use, that A i ¿s índependent oj ki _ l . We also remark that the solutions (2 .3) of the equations (2 .2) alwaysexist, but that (for a given odd positive integer b) the numbers al given by (2 .3) may fail to be integers . However, we have immediately Proposition 2 .1  (i)  The 6olutc :ou ob (2 .2) avce natíonal numbM  al satís1yíng  0 < a l<2 ; (ii) íñ any a l .í .6 an ¡rntegeh, then aie al ah .e odd .ínte .geA6 . Proof (i) It is clear form (2 .4) and (2 .5) that B, A i are odd positiveintegers . Thus from (2 .3), each a l is a positive rationalnumber . Now 2k¡ai+l = b - a l < b,  since a l > 0 . Since  al+1  is positive and  k i > 1,  we infer that  a l+1 < 2' ki-1 To prove (ii), observe that a l-l = b - 2  a . . Thus if a l is an integer,  ai_1  is an odd integer, andthe resultfollows by finite induction . 48 As an application, consider B, A i , givenby (2 .4), (2 .5) . As alreadyobserved, B and A i are odd positive integers forall i . Moreover, it follows immediately from ki (2 .3) that the solution of the equationsB = x i+2 x i+l , i = 1,2, . . . . r,xi+l = x l ,  is  x i = A i ,  so that k . B = A i + 2 1Ai+ .l .  (2 .6) is a svmbol . Thus, by Proposition 2 .1, A l A 2 . . . Ar k 1 k 2 . . . kr B (2 .7) we will also need the following elementary propositions ; the first is proved in [21 . Proposition 2 .2  In .the bymbal  (2 .1) , gcd (b,a i )  -í .6 Lndependent Ul 1 . Proposition 2 .3  tiñ, ín .the bymbal  (2 .1) , ki > n, .then al+1 < ñ . 2 Proof This is obvious from (2 .2) . Proposition 2.4  (Periodicity lemma)  11, .i .n (2 .1), theh .e exis .ts an  s  euch that  s i r  and  k i+s - ki joh  aP,C  i,  then  al+s = al dan aCQ .  i . Proof It is clearfrom (2 .5) thatif ki+s =ki for all i, then Ai+s -_ Ai for all i . The result now followsfrom (2 .3) . The periodicity lemmaassertsthat if the sequence k1,k2, . . .,kr is a repeating sequence, then the symbol (2 .1) is obtainéd by the same repetition . If there is no properrepeti tion, we say that the symbol (2 .1) is neduced añd write b a l a 2 k 1 k 2 ar k r (2 .8) Then a general symbol (2 .1) is obtainedby nepeatíng  a unique reduced symbol ; and a reduced symbol (2 .8) is obtained by compkUsb .íng a general symbol . Given positive odd integers a with a < 2,  there is a uniquereduced symbol (2 .8) = a . and b with We come now to our main preliminary result . Theorem 2 .5  Let  k l,k 2 ,. . . . k r  be poeítíve .íníegeu a .~íth E  k i = k > 2 .  Then, bon a g .íven odd .íntegeA  al < .2  ,  we have i=1 k  ala2 . . .a r  al a 2 . . .a r-1 ar 2 -1  .í6 and onty -í~  2 k+l -1 k 1k2 . . . .kr  k1 k 2 . . .k r-1 kr +l in eítheA ccue,  r la even . Proof Assume the left-hand symbol . Then, by (2 .3), If r were odd, we wouldhave 2 k -lla i , an evident contradiction . Thus r is even and al = A i , for all i . So (2 k -  ( - 1) r )a i =  (2 k -  1)A i . Loe now solve the equations 2k+1 - 1 = X i + 2ki Xi+l' r where  k' i=k i , 1 <-i -<r-1, kr = k r + 1,  so that  E ki =k+1 =k', i=1 sav, to obtain (compare (2 .6)) x i = A!, with (compare (2 .5)) A1-2k'-k!_  ~ .1 - 2k'-k1!_1-k1!_2 + . . .+ . (-1) r2kl  -  (-1)r  (2 .9) Thus we obtain the symbol However, we see from (2 .9), recalling that Al is independent of kr, that Al = A 1 = al , establishing the existence of the right-hand symbol of the theorem . The converse is provedsimilarly . There is acompaniontheorem as follows ; we need notgive an exnlicit nroof . Theorem 2 .5 *  Let  kl,k2, . . . ,kr  be pos .ítive íntegeu wíth r Ek i = k ? 1 .  Then, Son a gíven odd íntegeA  al < 2k-1,  we have i=1 A ' A' . . . A'rA' 1 2  1 r k l k 2  ...  k r-1  kr +l 2 k +1 a l a2 k l k 2 . . . . . . a r kr tis and ovney íS al a2 . . ar-1 ar 2 k+1 +1 kl k 2 . . k r-1 k r+1 In "eA case,  r .írs odd . Quasi-OrderTheorem  Le-t  b  be an odd pos .ítc :ve íwtegen, and .let  a . i  be an odd pos .í tíve .íntegen wíth  a l< 2  and a  p~u :me xo b . Then í6  b T4e prove this theorem in the next section but we may imme diatelyanounce the following corollary, relating to the ohden of 2 mod b . Corollary 2 .6  eU .ith che dame hupo .thehes as ín .the 9 .ua~sí-Ondet Theorem, «úe have (i)  .í~  r  íz even,  then the anden o 6  2 mod b  .í s  k  and, even í6  k .ír5 even,  2 k/2 P--1 mod b ; (ii)  í6  r  .í6 odd, then the oAden o6  2 mod b  í s  2k,  and 2 k -1 mod b . 3 . Proof of the Main Theorem prove We are now ready to state our main theorem . a l a 2 . . . ar k l k 2 . . . kr r wíth  E k i = k, we have i=1 (i)  k  .í,b -the m .Lní .mal Q sueh that  bi~~±1, (ii)  b 12 k -1 ,¿~  r  .írs even,  bl2 k+1  í5  r  .irs odd . We firststudy a special case of the main theorem and Theorem 3.1  Let  Q > 2 .  Then í~  ~ -1 r we have  E  Qi i=1 Proof We argue by inductivn on Q , the case Q = 2 being tr_i l vialsince  3 Cl~ .  Thus we assume the theorem for  Q > 2  and prove it for Q +1 .  Let hypothesis, we have 2' -" -1 If  r=1  and  2 1 =l,  the conclusion is trivially true . If not, it followsfrom the periodicity lemma that, for some  i, Qi > 2 . Without real loss of generalitywe may assume that Qr > 2 so that, by Proposition 2 .3, al < 2 Q-1 . Thus, by our inductive A al a2 ... as 2 2 - 1 (3 .2) k1 k 2 ... ks s with E k i IQ . By repetition, if necessary, we find the i=1 symbol a l a2 ... at 29 - 1(3 .3) k 1 k2 . . . kt t with Ek i = Q . By Theorem 2 .5 we deduce the symbol i=1 t Plrite ki = k i , 1 -< i -< t-1, kt =kt +1 .  Then 1 E 1 ki = Q+1 . Compressing, if necessary, we obtain u with  E k' I (Q+1) .  By the uniqueness of the reduced symbol, as i=1 1 a function of b and a o , we inferthat (3 .5) is identical with (3 .1), so that the inductive step is achieved and the theo rem is proved . There is, of course, a companion theorem, with almost ¡den tical proof, namely,  . Theorem 3 .1 * Let  Q > 1 .  Then r we have  E  Q¡  I  Q . i=1 11  11  11 a l a 2 ... at-1  at k 1 k 2 ...  k t-1  kt+l a la2 . . . ar 2 1 a a" . . . all 12  u k ' k' ... k' 1 2  u 2 2 Q r J Proof of the Quasi-Order Theorem First let (3 .4) (3 .5) Thus, by Theorem 3 .1 or 3 .1*, klk 0 . with no restrictionon gcd(a l,b) . r Let  E k . = k and let k  be the minimal Q such that 1=1 1  k  0 bl2 Q ± 1 .  If 2 0 ± 1 = bq,  then, obviously, k  a l q a 2 q . . . arq 2 0 ±1 Now supposethat a l is primeto b . Then, by (2 .3) and (2 .4), (2k -  (_,)r )a ¡ = bAl . Since b is prime to ai , we have bl2 k - (-1) r . Since klk 0 , the minimality of k 0 implies that k = k 0 . Moreover it is plain that bl2 k -1 if r is even and bl2 k +1 if r is odd . Remarks . (i) Note that we haveproved that, if we removefrom the hypothesesof the Quasi-Order Theorem the condition that al be prime to b, and if k is dejíned as the minimal Q Q  r such that bl2 ± 1, then  E k i ¡k . If we write quo(b) for i=1 the cguasi-order of ,? . mod b, then this says that if a la2 . . . a r  r b  I , then E kil áuo(b) . Moreover, the k l k 2 . . . kr i=1 .immedc :ateey translatable into fold-theoretic language! For it tells us that, if we know how to fold our stripof paper to pro k  k+1 duce a star  { 2 a l }-gon,then, to produce a star  {2  a -1}-gon, we introduce one morefold line precisely at thosevertices on thetop edgeof the tape which are destinedto becomevertices of our polygon . 5 . Appendix 2 : á few we ll-chosen examples where, by (2 .5) We note that, if I al a 2 ... ar b with a l = 1, then, by (2 .3), 2 k - ( -1) r = bAl, A  =  20 r-1  -  Z ar-2  +  . 1 J r Ek . =k, i=1 1 (5 .2) with  a . =  E k . .  (5 .3) i=1 1 Moreover, by our main theorem, k = quo (b) . Let us apply this to case b = 641 . We obtain, by our algorithm, 641 [15 159 241 25 77 141 125 129 72 1 4 3 2 2 2 9 Thus we infer, since k = 32, r = 9, that and, from (5 .2) auo(641) = 32 and, indeed, that  232 + 1 =- - 0 mod641 . Moreover, we knowfrom (5 .1) 2 32 + 1 = 641Al, (5 .4) A 1= 2 23 - 2 21 + 219 - 217 + 214 - 2 10 +29 - 27 +1 = 6700417 . This is, of course, Euler's famous factorization showing 5 that 22 + 1 is not a (Fermat) prime . 4 Only the paper-folding fanatic would take the view that the principalinterest of (5 .4) is that it shows how to fold the regularconvex 641-gon andcer tain star 641-gons . As a second example, consider the symbol 23 Here k = 11, r = 6, so that 1 11 3 5 9 7 1 2 21 1 4 4 See, for example, the frontcover of [5] . quo(23) = 11, 2 11 - 1  0 mod 23, and, againby (5 .2), the complementary factor is References Rebut el 16 d'octubne de¡ 1984 Department of Mathematics University of SantaClara SantaClara California 95053 U .S .A . A1 = 27 - 2 6 +25-23 + 2 - 1 = 89 Thus 2 11 - 1 = 23 " 89 and is not a (Mersenne) Prime . [1] Peter Hilton andand JeanPedersen,"Approximating any regu lar polygonby folding paper : An interplay of geometry, ana lysis_andnumber theory", Mathematics Magazine, Vol . 56 . Nó 3, 1983 (141 - 155) . [2] -------------------------, "Regular polygons, star polygons and number theory", Coxeter Festschrift , Math . Sem . Giessen 164, 1984, (217 - 244) . [3] -------------------------, "Folding regular star polygons anrl ni  ymber theor%T" The Mathematical Intell i a en c er . Vol . 7 (1), 1985 (15 - 26) . [4] K .R . Matthews and A .M . Watts, "A generalization of Hasse's generalization of the Syracuse algorithm", Acta Arithmetica XLIII, 1983 (75 - 83) . [5] Mathema t ical Intelligencer , Vol . 6 . Ns 3, 1984, frontcover . [6] Peter Hilton and Jean Pedersen, "On generalized symbols, o_r ders and quasi-orders" (to appear) .