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A FIBONACCI PROPERTY OF WYTHOFF PAIRS ROBERT SILBER North Carolia State Uiversity, Raleigh, North Carolia 27607 I this paper we poit out aother of those fasciatig "coicideces" which are so characteristically associated with the Fiboacci umbers. It occurs i relatio to the so-called safe pairs (a, b ) for Wythoffs Nim [1, 2, 3]. These pairs have bee extesively aalyzed by Carlitz, Scoville ad Hoggatt i their researches o Fiboacci represetatios [4, 5, 6, 7], a cotext urelated to the game of im. The latter have carefully established the basic properties of the a ad b, so that eve though that which we are about to report is ot described i their ivestigatios, it is a ready cosequece of them. For coveiece ad for reasos of precedece, we refer to the pairs (a, b ) as Wyth off pairs. The first forty Wythoff pairs are listed i Table 1 for referece. We recall that the pairs are defied iductively as follows: (a-j, b-i) = (1, 2), ad, havig defied (aj, bj), (a2, b2),, (a, b ) f a +i is defied as the smallest positive iteger ot amog ai, bj, a2, b2,, a, b ad the b +i is defied asa / + ( + 1). Each positive iteger occurs exactly oce as a member of some Wyth off pair, ad the sequeces la ) ad {b } are (strictly) icreasig. t _ Wythoff [1] showed that a = faj ad b = fa 2 J, a beig the golde ratio (1 + V5)/2; a more elegat proof of this appears i [3]. This coectio of the Wythoff pairs with the golde ratio suggests to ay "Fiboaccist" that the Fiboacci umbers are ot very far out of the picture. The work of Carlitz et.'al. that we have metioed shows that the a ad b play a fudametal part i the aalysis of the Fiboacci umber system. We ow look at aother coectio with Fiboacci umbers. Table 1 The First Forty Wythoff Pairs 1 2 3 4 5 6 7 8 9 10 3/7 1 3 4 6 8 9 11 12 14 16 b 2 5 7 10 13 15 18 20 23 26 12 13 14 15 16 17 18 19 20 3 17 19 21 22 24 25 27 29 30 32 b 28 31 34 36 39 41 44 47 49 52 ~W 22 23 24 25 26 27 28 29 30 a I F 35 37 38 40 42 43 45 46 48 b 54 57 60 62 65 68 70 73 75 78 31 32 33 34 35 36 37 38 39 40 3 50 51 53 55 56 58 59 61 63 64 ~~b~\ 81 83 86 89 91 94 96 99 102 104 I examiig Table 1, it is iterestig to observe that the first few Fiboacci umbers occur paired with other Fiboacci umbers: (a l,bj = (1,2), (a 2,b 2 ) = (3,5), (a 5, b s ) = (8,13), (a l3, b l3 ) = (21,34), (a 34, b da ) = (55, 89). It is ot difficult to establish that this patter cotiues throughout the sequece of Wythoff pairs, usig the fact that lim (b /a ) = a ad also lim (F + j/f ) = a. However, a almost immediate proof ca be had, based o Eq. (3.5) of [4], which states that (1) a +b = a b 380

NOV. 1976 A FIBONACCI PROPERTY OF WYTHOFF PAIRS 381 for each positive iteger. We defer the proof mometarily, it beig our itetio to state ad prove a geeralizatio of the foregoig. Clearly there are may Wythoff pairs whose members are ot Fiboacci umbers; the first such is (a 3, b 3 ) = (4,7). The pair (4,7) ca be used to geerate a Fiboacci sequece i the same way X\\dX(a u bj = (1,2) ca be cosidered to determie the usual Fiboacci umbers; we take G-j = 4, 62 = 7, G +2= G +f + G. The first few terms of the resultig Fiboacci sequece are It is (perhaps) a bit startlig to observe that 4, 7, 1 1, 1 8, 2 9, 4 7, -. ' (a 3,b 3 ) = (4,7), (a,b ) = (11, 18), (a lb,bj = (29,47). The pair (a 4, b 4 )= (6, 10) similarly geerates a Fiboacci sequece 6, 10, 16, 26, 42,68, - ad sure eough (a 4/ bj = (6, 10), (a 10, b j = (16, 26), (a 26, b j = (42, 68). It is time for our first theorem. Theorem. Let G u G 2, G 3, - be the Fiboacci sequece geerated by a Wythoff pair (a, b ). The every pairft? 1# G 2 ), (G 3, GJ, (G 5, GJ, - i s agai a Wythoff pair. Proof. By costructio, every Wythoff pair satisfies (2) a k + k = b k. Cosider the first four terms of the geerated Fiboacci sequece: Accordig to Eq. (1) so that Equatio (2) with k = b gives a,b,a +b,a +2b. so that the four terms uder cosideratio are i fact a +b = at, a +2b = a b + b. a b + b = b bp, a, b, a b, b b. Thus we have prove that i geeral (G 3, 4 ) is a Wythoff pair whe (G u G 2 ) is. But (G 5, G 6 ) ca be cosidered as cosistig of the third ad fourth terms of the Fiboacci sequece geerated by (G 3, G A ), ad the latter is already kow to be a Wythoff pair. I this way, the theorem follows by iductio. Thus we see that each Wythoff pair geerates a sequece of Wythoff pairs; the pairs followig the first pair of the sequece will be said to be geerated by the first pair. We defie a Wythoff pair to be primitive if o other Wythoff pair geerates it. It is clear that if (a m, b m ) geerates (a, b ), the m <. For this reaso, oe ca determie the first few primitive pairs by the followig algorithm, aalogous to Eratosthees' sieve. The first pair (1,2) is clearly primitive. All those geerated by (1,2) are elimiated (up to some specified poit i the table). The first pair remaiig must agai be primitive, ad all pairs geerated by that primitive are elimiated. The process is repeated. The first few primitive pairs so determied are pair umbers 1,3,4,6,8,9, 11, 12, 14, 1 6, - which we recogize at oce to be the sequece This occasios our ext theorem. a x, a 2, a 3,.

382 A FIBONACCI PROPERTY OF WYTHOFF PAIRS [NOV. Theorem. A Wythoff pair (a, h ) is primitive if ad oly if = a^ for some positive iteger k. Proof. We have see that the terms of the Fiboacci sequece geerated by ay Wythoff pair (a, b )are of the form *.b.*b.b b.a bb.b bb.~. From this it is obvious that ay o-primitive pair (a, b ) must have = b k for some positive iteger k, which makes every pair (a, b ) with = a^ a primitive pair. O the other had, the sequece a, b, a bp, b b, - geerated by (a,b ) shows clearly that each pair (a bk, b bf< ) is geerated by (a^, b^); thus every primitive pair (a, b ) must have/7 = a^. This theorem shows that the umber of primitive pairs is ifiite, ad has the followig corollary. Corollary. There exists a sequece of Fiboacci sequeces which simply covers the set of positive itegers. A iterestig property of the primitive pairs turs up whe we calculate successively the determiats I 3/7 b 31 b 1 J ' restrictig our attetio to those (a, b ) which are primitive. We fid that 4 7 9 15 = 1, = 3, 6 10 12 20 ad so o. This suggests that the value of the determiat applied to the k primitive is k - 1. By the foregoig theorem, we kow that the k th primitive is i fact (a a., b a.), so the suggested idetity becomes 2 3a k -b ak = k- 1, which follows readily from Eq. (3.2) of [4]. We coclude by iterpretig our results i terms of the fidigs i [4] ad [6]. Accordig to the latter, the Wy'thoff pairs (a, b ) are those pairs of positive itegers with the followig two properties: first, the caoical Fiboacci represetatio of b is exactly the left shift of the caoical Fiboacci represetatio of a, ad secod, the right-most 1 appearig i the represetatio of a occurs i a eve umbered positio. (I base 2 this would be aalogous to sayig that b = 2a ad that the largest power of 2 which divides * is odd). No two 1's appear i successio i the represetatios of a ad/?^. If we adda^ ad/?,,, each 1 i the represetatio of b will combie with its shift i the represetatio of a to yield a 1 i the positio immediately to the left of the added pair, sice F + F i = F +i. This meas that b +a has a represetatio which is exactly the left shift of b. By exactly the same reasoig, (b + a ) + b has a represetatio which is exactly the left shift of b + a, ad so forth. Hece, the Fiboacci sequece geerated by ay Wythoff pair, whe expressed caoically i the Fiboacci umber system, cosists of cosecutive left shifts of the first term of the sequece. I the simplest case, the pair (1,2) geerates the usual Fiboacci sequece, which i the Fiboacci umber system would be expressed ad the geerated pairs would be 10, 100, 1000, 10000, 100000, - (10, 100), (1000, 10000),

1976] A FIBONACCI PROPERTY OF WYTHOFF PAIRS 383 which have the requisite properties that each b is the left shift of a ad that each a has its right-most 1 i a eve-umbered positio. The ext case correspods to the sequece geerated by (4, 7); 4, 7, 11, 18, -. I Fiboacci, this appears as 1010, 10100, 101000, 1010000,-. This procedure ca be traced back a additioal step to the idex of the pair (a, b ). Doig so provides i additio a simple iterpretatio of the primitive pairs i terms of Fiboacci represetatios. There is a prescriptio i [6] for geeratig a Wythoff pair (a, b ) from its idex, but it ecessitates the so-called secod caoical Fiboacci represetatio. For preset purposes it suffices to remark that the secod caoical represetatio of ay ca be obtaied by addig 1 to the usual caoical represetatio of - 1. (For example,the caoical represetatio of 7 is 10100, so the secod caoical represetatio of 8 is 10101). The umbers a ad b are the obtaied from successive left shifts of the secod caoical represetatio of. Thus, i the example of (4,7), we obtai the secod caoical represetatio of 4 as 1000 + 1 = 1001 ad geerate a b a +b = a b, etc. 10001:10010 100100 1001000 We have see that the primitive pairs correspod to the case = a^. It is readily established o the basis of the results i [4] that the umbers a^ are precisely those umbers whose secod caoical Fiboacci represetatios cotai a 1 i the first positio (as follows: first, a umber is a b/< if ad oly if its caoical represetatio cotais its right-most 1 i a odd positio-which is ever the first-ad, secod, a secod caoical represetatio fails to be caoical if ad oly if it cotais a 1 i the first positio). It follows that the primitive pairs (a, b ) are precisely those for which, the secod caoical represetatio of havig a 1 i the first positio, the caoical represetatio of a eds i 10 ad that of h eds i 1QO. Other terms of the geerated Fiboacci sequece have additioal zeroes, the locatio of ay umber i the sequece beig exactly depedet o the umber of termial zeroes i its caoical represetatio. This eables oe to determie for ay positive iteger/? exactly which primitive Wythoff pair geerates the Fiboacci sequece i which that appears, as well as the locatio of i that sequece. First determie the caoical Fiboacci represetatio of. The portio of the represetatio betwee the first ad last 1's, iclusive, is the secod caoical represetatio of the umber k of the primitive Wythoff pair which geerates the Fiboacci sequece cotaiig. Oe left shift produces a^; aother produces b^. Coutig a^ as the first term, b/< as the secod, ad so forth, the i th term will equal, where / is the umber of zeroes prior to the first 1 i the Fiboacci represetatio of A7. For example, let = 52. I Fiboacci, 52 is represeted 101010000. Sice 10101 represets 8 ad the represetatio termiates i four zeroes, 52 must be the fourth term of the Fiboacci sequece geerated by the primitive pair (a 8, b B ). As cofirmatio we ote that this particular sequece is 12,20,32,52,84,-. AFOOTWOTE Because of the coectio of the Wythoff pairs with Wythoff's Mim, the precedig prescriptio for geeratig. Wythoff pairs is clearly also a prescriptio for playig Wythoff's Nim usig the Fiboacci umber system. This gives the Fiboacci umber system a role i this game quite aalogous to the role of the biary umber system i Bouto's Nim [8]. The aalysis of Wythoff's Nim usig Fiboacci represetatios ca be made self-cotaied ad elemetary, certaily ot requirig the level of mathematical sophisticatio required to follow the ivestigatios i [4, 5, 6, 7]. For the beefit of those iterested i mathematical recreatios, we provide this aalysis i a compaio paper [9]. It is iterestig to ote that the role of the Fiboacci umber system i im games was already aticipated by Whiiha [10] i 1963.

384 A FIBONACCI PROPERTY OF WYTHOFF PAIRS NOV. 1976 REFERENCES 1. W. A. Wythoff, '"A Modificatio of the Game of Nim," NieuwArchiefvoorWiskude, 2d Series, Vol. 7, 1907, pp. 199-202. 2. A. P. Domoryad, Mathematical Games ad Pastimes, N. Y., Macmilla Co., 1964, pp. 62-65. 3. H. S. M. Coxeter, "The Golde Sectio, Phyllotaxis, ad Wythoff's Game,"Scripta Mathematica, Vol. 19, 1953, pp. 135-143. 4. L Carlitz, R. Scoville ad V. E. Hoggatt, Jr., "Fiboacci Represetatios," The Fiboacci Quarterly, Vol. 10, 1972, pp. 1-28. 5. L Carlitz, et. al., "Lucas Represetatios," The Fiboacci Quarterly, Vol. 10, 1972, pp. 29-42. 6. L Carlitz, et. al., "Addedum to the Paper 'Fiboacci Represetatios'," The Fiboacci Quarterly, Vol. 10, 1972, pp. 527-530. 7. L Carlitz, R. Scoville ad T. Vaugha, "Some Arithmetic Fuctios Related to Fiboacci Numbers," The Fiboacci Quarterly, Vol. 11, 1973, pp. 337-386. 8. C.L. Bouto, "Nim, A Game with a Complete Mathematical Theory," Aals of Mathematics, 1901-1902, pp. 35-39. 9. R. Silber, "Wythoff's Nim ad Fiboacci Represetatios," to appear, The Fiboacci Quarterly. 10. M. J. Whiiha, "Fiboacci Nim," The Fiboacci Quarterly, Vol. 1, No. 4, 1963, pp. 9-13. *******