Ths artcle s wrtten by students. It may nclude omssons and mperfectons, whch were dentfed and reported as mnutely as possble by our revewers n the edtoral notes. The path of ants 07-08 Students names and grades : Dragos Crsan, Andre Petrdean, th grade Insttuton: Colegul Natonal "Eml Racovta", Cluj-Napoca Teacher: Arana Vacaretu Researcher : Lorand Parajd, "Babes-Bolya" Unversty Research topc The problem we were presented wth s: The frst ant makes 0 random moves, each havng a length of one unt and, at each move, she leaves a mark. After the tenth move, she reaches food and sends a rado sgnal back to the anthll. A second ant goes along the frst half of the frst segment, then, from the mddle of t to the mddle of the second and so on, untl she reaches the mddle of the last segment, from where she goes to the food along the second half of the last segment. A thrd ant follows the same strategy, followng the second ant s path. a Is the second path shorter than the frst? b Wll the ants tend to fnd the shortest tnerary between the two ponts? What the second queston actually asks for s to descrbe how a gven polygonal chan behaves, asymptotcally, when the above descrbed algorthm s nfntely repeated. Our research. Results The path wll tend towards a straght lne between the two endponts, n terms of both shape and length.[] Throughout the proof, we have used remarkable results, such as the trangle s nequalty, Pascal s trangle and Walls approxmaton. MATh.en.JEANS 07-08 [Colegul Natonal "Eml Racovta", Cluj-Napoca] Page
. Proof of a We prove the slghtly more general statement: "An ant starts movng at a pont A 0 and makes n moves along the segments A 0 A, A A,, A n A n. If a second ant moves along the segments A 0 M, M M, M M 3,, M n M n, M n A n, where M, M, M n are the mdponts of A 0 A, A A,, A n A n respectvely, then, the second path wll be shorter than the frst one.": A 0 M = A 0A M n A n = A na n M k M k+ = A ka k+ A ka k + A k A k+ for every k from to n. Addng these up, we get to the concluson.3 Proof of b We prove a weaker result: The path tends towards a lne n terms of shape. That s, we consder the xoy Carthesan system, wth the orgn n A 0 and wth Ox = A 0 A n and we consder the coordnates of our ponts. We show that the ordnates tend to 0, and the abscssae tend to be n ascendng order, nsde the ntal segment. We frstly show that the asserton holds for ordnates. For ths, consder the followng table, where every lne represents an antnote that, from now on, unless stated otherwse, the MATh.en.JEANS 07-08 [Colegul Natonal "Eml Racovta", Cluj-Napoca] Page
numberng of lnes and columns starts from 0: lne 0 y y y 3 y n lne y y + y y + y 3 y n + y n lne y y + y y + y + y 3 y n + y n 4 4 4 4 y n y n 4 To be able to determne a closed-form of the general term, we consder the contrbuton of every y separately. Take the followng trangle, constructed by dvdng every lne of Pascal s trangle by, where every term s the arthmetc mean of the two above, just as n the frst table: lne 0 lne lne Now, addng up the coeffcents, we obtan the formula: y j = nα= y α j α+ Consderng M = max{ y, y, y n } and applyng a basc nequalty, we get: y j Mn But, by Walls approxmaton: lm n n n nπ = 4 n lm = 0 Now, we move on to the abscssae. We consder the same type of table as before, the only excepton beng that every lne s nfnte, the last term repeatng ndefntely []. The trangles of coeffcents are the same as above for x, x,, x n, so, by Walls approxmaton, they tend towards 0. The coeffcents of x n are gven by: MATh.en.JEANS 07-08 [Colegul Natonal "Eml Racovta", Cluj-Napoca] Page 3
lne 0 lne lne 3 4 4 Ths "trapezod" T j s just an nfnte sum of the above trangles, so ts terms are of j α=0 the form α, whch are clearly n ascendng order on each lne. That s, the statement s completely proven..4 Proof of b Now, we prove the strong result, that s, the length of the path tends towards the dstance between the ntal endponts. We use both the notatons and the results from the prevous subsecton. If S n denotes the length of the n-th path where the numberng starts from 0, we get: S = +n +n +n = x j + x j + y j + y j x j + x j + y j + y j +n x j + x j + j =0 y j + y j where everythng undefned wth negatve ndces or too large ndces s 0. We show that the frst sum P n converges to x n and the second Q n to 0. As before, we start wth the ordnates: +n Q = y j + y j = = j =0 +n +n M +n n α= y α n α= y α α= j α+ j α+ nα= y α j α+ j α+ n j α + j α + MATh.en.JEANS 07-08 [Colegul Natonal "Eml Racovta", Cluj-Napoca] Page 4
Now, we evaluate the nteror sums, as many terms cancel out: [3] n j + j n+, j α= j α + j α + = j + j n+ j + n j n+ j + + n j We proceed to evaluate the last sum, where other terms cancel out: Q M + j + j n + +n j = +n j = +n j + j n + Q n nm j n + By Walls approxmaton, the last term tends to 0, therefore, so does Q n Q n s nonnegatve. We move on to the abscssae and we let x = x n. Recall the constructon of the abscssae s table and let x j = x j + x j, where x j s a lnear expresson n x, x,, x n and x s of the j form x j = T j n+x, T j beng the "trapezod" mentoned n the prevous secton. So: +n P = x j + x j +n x +n j + x j + j + + x j + x j As above, the frst sum converges to 0, so we compute the second sum: P +n = x j + x j = x j = x j =0 That s, P s smaller than somethng that tends to x and, together wth the result about Q, we obtan S smaller than somethng that tends to x. But, as S xthe shortest path between two ponts s the straght lne, we get that S tends to x and the concluson follows. MATh.en.JEANS 07-08 [Colegul Natonal "Eml Racovta", Cluj-Napoca] Page 5
3 Conclusons and generalsatons The problem, as stated, s completely solved. Although the frst varant of b s weaker, that was the frst we came up wth and only after beng shown why t does not mply the stronger one, dd we manage to prove the latter. In the proof of b, the ndex s tactly assumed even. Although that s not always the case, the computatons for the other case are smlar but, even more, gven the result from a, the proof that the even subsequence converges s enough. An mmedate generalsaton s to extend the eucldean plane to R n, where the proof s dentcal. The generalsaton to a sphere, a torus or any other such surface s not always true, as counter-examples are easly constructed for the frst two. A far more nterestng generalsaton s to consder a metrc on R or R n dfferent from the usual eucldean dstance, and ask under what condtons we can reach the same conclusons, or f there exsts a metrc n whch they fal. These are very general questons that we were not able to tackle yet. Edton Notes [] Here, somethng should be sad about the two notons and about the relatonshp between them. [] Ths sentence s not so clear. Some more explanaton would be useful. [3] These calculatons are rather complex. They should be explaned and developed n more detal. MATh.en.JEANS 07-08 [Colegul Natonal "Eml Racovta", Cluj-Napoca] Page 6