- Email: [email protected]
Forming all pairs in a minimal number of steps
ARTICLE IN PRESS
Journal of Mathematical Psychology 49 (2005) 155–159 www.elsevier.com/locate/jmp
Forming all pairs in a minimal number of steps Karl E. Scheibea,, W.W. Comfortb a
Department of Psychology, Wesleyan University, Judd Hall, Middletown, CT 06459-0408, USA Department of Mathematics, Wesleyan University, Exley Science Center, Middletown, CT 06459-0408, USA
b
Received 29 December 2003; received in revised form 28 August 2004 Available online 7 January 2005
Abstract How can N individuals in a closed space meet and greet each other most efficiently? This paper presents a general solution to this problem—an algorithm or ‘‘dance’’ that will achieve universal pairing in the least possible number of moves or steps. A proof of the suggested algorithm is included, showing that it guarantees that every two participants will greet each other once and only once, and that no procedure with this property can be accomplished with fewer steps. Slightly different procedures are required for the odd and even cases. The algorithm has been applied in classroom settings, and could be applied in any social setting where the objective is to initiate efficiently a sense of group cohesion and common purpose. r 2004 Elsevier Inc. All rights reserved. Keywords: Forming pairs; Mathematical induction; Social structure; Dyads
1. Introduction This paper presents a solution to a problem that is of both practical and conceptual importance: Given a finite set of individuals, to find an algorithm or ‘‘dance’’ which ensures most efficiently that every two individuals meet. The procedure is to be orderly and well-defined; the requirement that it be most efficient means that it will achieve every pairing in the least possible number of moves or steps.
2. N individuals in a room In the exposition that follows, one individual in the set is its designated leader. This reflects the history of the actual development of the solution to the problem, for a teacher was interested in devising a technique for assuring that everyone in a class would be introduced to everyone else in the class, including the teacher, in an efficient manner. But the solution to the problem is quite Corresponding author. Fax: +1 860 685 2761.
E-mail address: [email protected] (K.E. Scheibe). 0022-2496/$ - see front matter r 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmp.2004.11.005
general and does not require that any individual in the group be distinguished by special status as a leader or director. In fact, the solution applies equally well to inanimate objects or to abstract entities, the objective in either case being to achieve universal pairing in a most efficient manner. In practice, individuals or objects must be arranged in a specific way and must be instructed or caused to move in a particular way by a sequence of signals. The solution is easiest to conceptualize if we think of the exercise being done by sentient individuals under the instruction of a single leader. An initial way of representing several individuals in a closed space is shown in Fig. 1. (For illustrative purposes we choose N ¼ 22; a number consistent with the number of students in the class where this exercise was initially developed and applied.)1 The problem of getting individuals introduced to each other is often accomplished by milling. That is, the people are instructed to mill about, with the objective that every two people meet. The difficulty, of course, is that milling is inefficient: it takes a lot of time, many 1 See Scheibe (2000, Chapter 12), for a description of the class, a seminar called ‘‘The Dramaturgical Approach to Psychology.’’
ARTICLE IN PRESS K.E. Scheibe, W.W. Comfort / Journal of Mathematical Psychology 49 (2005) 155–159
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pairings are repeated, and there is no guarantee within a finite period of time, allocated in advance, that every two participants will have met.
3. Even and odd cases For a more orderly approach, we arrange the individuals in two facing rows. Figs. 2E and O represent typical configurations, one with N Even (we choose N ¼ 22) and one with N Odd (we choose N ¼ 23). The procedures for achieving efficient matching of all pairs differ for the Even case and the Odd case. In the Even case, with the participants arranged in two rows of equal length, each individual is instructed to greet the person in the opposite or facing position. First greetings are illustrated by the connecting lines in Fig. 2E. At the completion of the first greeting, the leader gives a signal to shift, whereupon everyone in the set (with the exception of the leader) moves one position in a clockwise direction. When everyone has moved one position, the leader gives another instruction to greet. After the greeting is completed, the leader again gives a command to shift. This pair of commands, ‘‘Greet’’ and ‘‘Shift’’, is given N 1 times, whereupon the entire set is returned to its initial position. In this Even case, the leader remains in the same position throughout the exercise, greeting each individual as they move into the opposing position.
X
X X
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X X
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X XX
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Fig. 1. N ¼ 22 individuals, casually distributed.
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(O) Fig. 2. (E) Initial greeting pattern when N ¼ 22: The leader, x22 ; is stationary: (O) Initial greeting pattern when N ¼ 23: Here & remains stationary.
The procedure when N is Odd necessarily differs from that just described. In this case, the leader (denoted x1 in Fig. 2O) initially greets no individual. A ‘‘ghost’’ or ‘‘dummy’’ participant, here denoted &; is paired with the leader in Round 1. Other participants ‘‘greet’’ & in subsequent rounds. Unlike the Even case, the leader now also moves one position at each ‘‘Shift’’ command; furthermore, N pairings of the commands ‘‘Greet’’ and ‘‘Shift’’ (rather than just N 1) must be given in order for the group to return to the initial arrangement. (One may think of the Even case as slightly less wasteful of opportunity than the Odd case, for in the Odd case at each step of the exercise necessarily some one person is left without anyone to greet.) Participants in this exercise generally find one of its features to be counterintuitive. When in the initial configuration (Fig. 2E or 2O), if participants are asked to point to the person they will greet after the first shift, they typically will point to the person one position to the left of the person they face. But because both files shift to the left simultaneously, in fact the person two positions to the left of the person initially greeted will be in position for the second greeting.
4. The mathematical argument While it may seem evident that no procedure can achieve the objective of universal greeting more efficiently than the one illustrated and described above, it is appropriate to present a mathematical proof. The following discussion is quite general, not being limited to the cases N ¼ 22 and 23 of Figs. 2E and 2O. Indeed Figs. 4E and 4O show the initial alignment for (arbitrary) even N ¼ 2m and odd N ¼ 2m þ 1; respectively. We adopt the standard (row,column) matrix notation associated with Fig. 3. There are 2 rows and m columns; for 1pkp2 and 1pppm the notation ðk; pÞ designates the position in row k and column p. To be formal: Our assignment is to establish two properties of the greeting procedure described, namely (A) each participant greets other participant exactly once, and (B) the procedure is maximally efficient in the sense that no procedure with property (A) can take fewer steps. Once (A) is known, (B) is established easily, as follows. The number of greetings to be achieved is the (1, 1) (1, 2) · · · (1, p) · · · (1, m) (2, 1) (2, 2) · · · (2, p) · · · (2, m) Fig. 3. A matrix with m columns and 2 rows.
ARTICLE IN PRESS K.E. Scheibe, W.W. Comfort / Journal of Mathematical Psychology 49 (2005) 155–159
in position ð2; pÞ on the first of these occasions. At their second meeting, in some column q, either
number of pairs in an N-set, namely CðN; 2Þ ¼
157
N! NðN 1Þ ¼ : ðN 2Þ!2! 2
When N is even, say N ¼ 2m; one can form (at most) m pairs in a single round, so the number of rounds must be (at least) NðN 1Þ CðN; 2Þ=m ¼ m 2 ¼ 2mð2m 1Þ=2m ¼ 2m 1 ¼ N 1; the number achieved by our procedure. Similarly when N is odd, say N ¼ 2m þ 1; one can again form no more than m pairs in a single round, so the number of rounds required is at least CðN; 2Þ=m ¼ ½ð2m þ 1Þð2m þ 1 1Þ=2=m ¼ 2m þ 1 ¼ N; again the number used in our procedure. It remains, then, only to prove (A). Case: 1 N is even, say N ¼ 2m: If (A) fails, so that some participant xi never meets some participant xk (kai) then, since there are N 1 rounds of greeting, xi must meet some participant xj at least twice. [The mathematical aficionado will recognize this as an elementary instance of the so-called pigeonhole principle: there can be no injection from an ðN 1Þ-set (here, the N 1 rounds of greeting) into a set with fewer than N 1 elements (here, the set of participants greeted by xi :)] Thus it suffices to show that no two participants greet each other twice. It is clear that the (stationary) leader x2m does greet each participant exactly once. Indeed with numbering as in Fig. 4E, x2m greets participant xi in round i; that is, x2m greets x1 in round 1, x2 in round 2, and so on until round 2m 1; the final round, when x2m greets x2m1 : We must show that no two participants xi and xj (with xi ax2m ; xj ax2m ) can meet twice. Greetings occur (only) between participants in the same column. Suppose now that some participants, xi and xj ; meet twice, say with xi in position ð1; pÞ and xj
x 2m x1
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· · · x2m+1−p ·· · xp
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· · · x2m+3−p ·· · xp
· · · xm+2 · · · x m+1
(E)
(O) Fig. 4. (E) Initial position when N ¼ 2m is even: (O) Initial position when N ¼ 2m þ 1 is odd.
(i) xi is in position ð1; qÞ and xj is in position ð2; qÞ; or (ii) xj is in position ð1; qÞ and xi is in position ð2; qÞ: If (i) occurs then paq since no participant other than the stationary leader visits the same position twice in the 2m 1 rounds of greeting, so either 1opoqpm or 1oqoppm: In the first case participant xi moves from ð1; pÞ to ð1; qÞ in q p rounds, while xj moves from ð2; pÞ to ð2; qÞ in ð2m 1Þ þ ðq pÞ rounds, so q p ¼ ð2m 1Þ ðq pÞ and hence 2ðq pÞ ¼ 2m 1; a contradiction since no integer is simultaneously even and odd; in the latter case a similar argument yields ð2m 1Þ ðp qÞ ¼ p q and hence 2m 1 ¼ 2ðp qÞ; again a contradiction. Thus (i) cannot occur. As to (ii), we notice that xi ; beginning at ð1; pÞ; moves in m p rounds to ð1; mÞ; moves in the next round to ð2; mÞ; and arrives at ð2; qÞ m q rounds later; in sum, xi requires ðm pÞ þ 1 þ ðm qÞ stages to move from ð1; pÞ to ð2; qÞ: Similarly, xj requires p 1 moves to arrive at ð2; 1Þ; then q 1 additional moves to arrive at ð1; qÞ: Thus ðm pÞ þ 1 þ ðm qÞ ¼ ðp 1Þ þ ðq 1Þ and the contradiction 2m þ 3 ¼ 2p þ 2q arises. Since neither (i) nor (ii) can arise in the case that N ¼ 2m is even, the proof of (A) (and hence of (B)) is complete in that case. Case 2: N is odd, say N ¼ 2m þ 1: We assign to the array a ‘‘dummy’’ or ‘‘ghost’’ participant, here denoted &: This brings the number of participants to 2m þ 1 þ 1; so Case 1 applies to that even number. The entire dance now requires ð2m þ 2Þ 1 ¼ 2m þ 1 ¼ N steps. (Each of the genuine original participants xi ‘‘greets’’ & exactly once; in practice this corresponds to the round in which xi is ‘‘odd man out’’.) The proof is complete. Remarks. When N is even, the greeting procedure we have developed keeps one participant in fixed position, while the others rotate clockwise. In practice, the leader has assumed this special position. When N is odd and & is adjoined to the array, we have found it convenient to cede this privileged stationary role to &: The genuine original leader now assumes the position of x1 as in Fig. 4O, ‘‘greeting’’ & in round 1. By this device no human being is seen as distinguished by rank or status from others in the group, and the true leader participates in the ‘‘Greet–Shift’’ sequence with no apparent special distinction. The mathematical analysis required above to verify the validity and efficiency of the procedures defined may strike the reader, as it struck us, as unexpectedly or unnecessarily long. It is reassuring to note that an approach based on Mathematical Induction, while blind
ARTICLE IN PRESS 158
K.E. Scheibe, W.W. Comfort / Journal of Mathematical Psychology 49 (2005) 155–159
to the social niceties, effects a proof with greater dispatch. As usual, the set of 2-sets of an N-set is denoted ½N2 : Theorem. For finite NX2 the set ½N2 can be expressed in SN1 2 the form ½N ¼ S k¼1 Ak when N ¼ 2m is even, and in the form ½N2 ¼ N k¼1 Ak when N ¼ 2m þ 1 is odd, with (in each case) each Ak a pairwise disjoint family of cardinality m. Proof. The indicated statement SðNÞ is obvious when N ¼ 2: Fix K and suppose that SðNÞ is true whenever 2pNoK: We are to prove SðKÞ: If K is odd then SðKÞ follows from SðK 1Þ by the argument given earlier, so we may assume that K is even, say K ¼ 2m: We write K ¼ K 1 [ K 2 with jK 0 j ¼ jK 1 j ¼ m; and we define S ¼ ffb; cg : b 2 K 1 ; c 2 K 2 g; so that ½K2 ¼ ½K 1 2 [ ½K 2 2 [ S: If m isSeven we use the inductive Sm1 hypothesis to write 2 ½K 1 2 ¼ m1 B and ½K ¼ 2 k¼1 k k¼1 C k with each Bk and each C k a pairwise disjoint family of cardinality m2 ; and we set Ak ¼ Bk [ C k for 1pkpm 1: Writing K 1 ¼ fb1 ; b2 ; . . . ; bm g and K 2 ¼ fc1 ; c2 ; . . . ; cm g; for mpkp2m 1 ¼ K we set Ak ¼ ffbi ; ckþi g 2 S : 1pipm mod m. Then ½S2 ¼ SK1 SK1 1g with subscripts 2taken 2 k¼m Ak and the relation ½K ¼ k¼1 Ak expresses ½K in the required form. Thus SðKÞ holds when K ¼ 2m with m even. Sm 2 2 SmIf m is odd we again let ½K 1 ¼ k¼1 Bk and ½K 2 ¼ k¼1 C k be as given by the inductive hypothesis, and with K 1 n [ Bk ¼ fbk g and K 2 n [ C k ¼ fck g we define Ak ¼ Bk [ C k [ ffbk ; ck gg for 1pkpm: (Informally, bk and ck are those members of K 1 and K 2 ; respectively who are ‘‘left out’’ at the kth round of greeting. In this plan they greet each other at stage k.) The indexings K 1 ¼ fb1 ; b2 ; . . . ; bm g and K 2 ¼ fc1 ; c2 ; . . . ; cm g are now determined. Now for m þ 1pkp2m 1 ¼ K 1 we set Ak ¼ ffbi ; ciþk g : 1pipmg; again with subscripts taken mod m. (In these rounds of greeting each bi meets each cj with iaj; but for no i does bi greet ci :)Thus S K1 Ak ¼ ½S2 nffbk ; ck g : 1pkpmg and the relation k¼mþ1S 2 2 ½K ¼ K1 k¼1 Ak again expresses ½K in the required form. Thus SðKÞ holds also for K ¼ 2m with m odd. In summary: We have noted that Sð2Þ holds trivially, and we have shown that if K42 and SðNÞ holds whenever 2pNoK then SðNÞ holds for N ¼ K: It follows from the Principle of Induction that SðNÞ holds for all integers NX2; as required.
5. The silent greeting exercise The mathematical and procedural arguments are now complete. Yet something remains to be said about the psychological and social side of putting the exercise into practice. First of all, the simple act of conducting this exercise communicates something about the intended
character of the group processes that might proceed from such an introduction. The objective is to lay a basis for general and egalitarian participation, including the leader as one of the participants, albeit with special status. Also, it seemed from the outset that in order to conduct such an exercise properly and with maximum effect, it should be done in silence. One reason for this is that the commands to ‘‘Greet’’ and ‘‘Shift’’ cannot easily be heard over the din of a score of voices. But a more powerful reason for devising a silent greeting exercise derives from a suggestion from the German novelist, Thomas Mann, that words tend to falsify human intercourse and that the glance and the touch are more likely to be authentic expressions of sentiment and interest.2 The instruction for silent greetings is for pairs to touch their right palms together and to make their eyes meet. They are told to begin the greeting at the ‘‘Greet’’ command, and to hold it until the ‘‘Shift’’ command, with the leader pacing the commands fairly evenly at intervals of about 3–5 s. Thus, an entire group of 20-some participants can achieve universal contact in less than 2 min. When this exercise is performed as a first exercise in a class or group meeting, there is typically some nervousness and perhaps giggling during the procedure. When groups are allowed to discuss their feelings about conducting the exercise, they typically remark on differences they experienced from person to person in the expression of their eyes, the feel of their hands, the facial expressions encountered. When this same silent greeting exercise is repeated at the end of a 13-week semester of working, studying, and learning together as a group, the emotional character of the experience is quite different. There is now no giggling, no nervousness. Instead, eyes are commonly watered with tears, and the hands are likely to clasp together, not merely touching palms. Students commonly report on the intensity of this experience and compare it with the nervousness and uncertainty they felt when the exercise was done at the beginning of the course. Much of this transformation depends, of course, on the length, character, and intensity of the activities taking place between the two silent greeting exercises. The arguments given above show that this technique for achieving universal pairwise contact is most efficient 2 ‘‘[T]here alone are unconditional freedom, secrecy, and profound ruthlessness. Everything by way of human contact and exchange that lies between is lukewarm and insipid; it is determined, conditioned, and limited by manners and social convention. Here the word is master— that cool, prosaic device, that first begetter of tame, mediocre morality . . . [The] silent regions of human intercourse . . . where strangeness and social rootless still maintain a free, primordial condition and glances meet and marry irresponsibly in dreamlike wantonness.’’ (Mann, 1955, p. 79).
ARTICLE IN PRESS K.E. Scheibe, W.W. Comfort / Journal of Mathematical Psychology 49 (2005) 155–159
for every finite set, irrespective of its size. Given a group of 100 individuals arranged on a field, for example, all 4950 possible pairings may be achieved within 7 min. A thousand people could exchange universal greetings, if they could find an hour in common for the exercise. As a macabre oddity, we note that any one individual transmitting a pathogen might communicate that pathogen to everyone else in the group with just one iteration of the greeting cycle. A more positive aspect is this: The experience of greeting a sequence of people in a way that is simple and yet authentic is uniformly experienced as pleasurable. The conceptual value of our algorithm is that it sets forth in elementary and workable language a solution to the problem of achieving maximally efficient pairings of N entities. At first glance, one might think that the model proposed here has something in common with the ‘‘fast and frugal’’ heuristics proposed by Gigerenzer and others as ways of facilitating human judgment tasks or solving reasoning problems (see (Gigerenzer, Todd, & ABC Research group , 1999). Indeed, the algorithm described here is both ‘‘rapid and clean’’ in the sense introduced by those authors. However, it is distinctly not a descriptive heuristic, as are the devices proposed by Gigerenzer et al. It is possible that chance, or some collective intelligence, might guide people into forming lines and exchanging greetings in the manner suggested here, but the likelihood of this happening without explicit instruction is close to zero for all but very small values of N. Our model is normative, not descriptive. Given that one wishes to achieve universal pairing within a set of individuals, this algorithm provides the most efficient means of achieving that result. Why and under what circumstances might one wish to achieve universal pairing? The example from the classroom has already been described. While for many classrooms the relationships among students might be entirely unimportant, in this case the objective was to create as quickly as possible a sense of group cohesion in
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order to facilitate cooperation and collaboration among class members. It has long been observed that forms of patterns of communication within groups determine such outcomes as speed and effectiveness of group function, group morale, speed of group decisionmaking, and the creative capacity of groups (see Leavitt, 1951). The common use of the circle form of seating within the human potentials movement is testimony to the intuitive appeal of achieving universal contact within a group (see Hare, Borgatta, & Bales, 1965). But use of the circle arrangement does not guarantee universal contact of all possible pairs—it merely makes that contact possible. The objective of developing group cohesion and facilitating the formation of common group identity is common to many settings. Examples include academic classes, management training seminars, ensemble performance groups, religious or political groups, therapy groups, military units and athletic teams. In all of these arenas, functioning groups must somehow be forged out of a set of isolated individuals. The device set forth in this paper is easily implemented and has proved to be an effective and even elegant means of thickening the social soup. The mathematical arguments here set forth provide assurance that no more efficient means of achieving universal pairing may be found.
References Gigerenzer, G., Todd, P., & ABC Research group (1999). Simple heuristics that make us smart. New York: Oxford University Press. Hare, A. P., Borgatta, E. F., & Bales, R. F. (1965). Small groups: Studies in social interation. Knopf: New York. Leavitt, H. J. (1951). Some effects of certain communication patterns on group performance. Journal of Abnormal and Social Psychology, 46, 38–50. Mann, T. (1955). Confessions of Felix Krull, confidence man. Random House: New York. Scheibe, K. E. (2000). The drama of everyday life. Harvard University Press: Cambridge.
Journal of Mathematical Psychology 49 (2005) 155–159 www.elsevier.com/locate/jmp
Forming all pairs in a minimal number of steps Karl E. Scheibea,, W.W. Comfortb a
Department of Psychology, Wesleyan University, Judd Hall, Middletown, CT 06459-0408, USA Department of Mathematics, Wesleyan University, Exley Science Center, Middletown, CT 06459-0408, USA
b
Received 29 December 2003; received in revised form 28 August 2004 Available online 7 January 2005
Abstract How can N individuals in a closed space meet and greet each other most efficiently? This paper presents a general solution to this problem—an algorithm or ‘‘dance’’ that will achieve universal pairing in the least possible number of moves or steps. A proof of the suggested algorithm is included, showing that it guarantees that every two participants will greet each other once and only once, and that no procedure with this property can be accomplished with fewer steps. Slightly different procedures are required for the odd and even cases. The algorithm has been applied in classroom settings, and could be applied in any social setting where the objective is to initiate efficiently a sense of group cohesion and common purpose. r 2004 Elsevier Inc. All rights reserved. Keywords: Forming pairs; Mathematical induction; Social structure; Dyads
1. Introduction This paper presents a solution to a problem that is of both practical and conceptual importance: Given a finite set of individuals, to find an algorithm or ‘‘dance’’ which ensures most efficiently that every two individuals meet. The procedure is to be orderly and well-defined; the requirement that it be most efficient means that it will achieve every pairing in the least possible number of moves or steps.
2. N individuals in a room In the exposition that follows, one individual in the set is its designated leader. This reflects the history of the actual development of the solution to the problem, for a teacher was interested in devising a technique for assuring that everyone in a class would be introduced to everyone else in the class, including the teacher, in an efficient manner. But the solution to the problem is quite Corresponding author. Fax: +1 860 685 2761.
E-mail address: [email protected] (K.E. Scheibe). 0022-2496/$ - see front matter r 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmp.2004.11.005
general and does not require that any individual in the group be distinguished by special status as a leader or director. In fact, the solution applies equally well to inanimate objects or to abstract entities, the objective in either case being to achieve universal pairing in a most efficient manner. In practice, individuals or objects must be arranged in a specific way and must be instructed or caused to move in a particular way by a sequence of signals. The solution is easiest to conceptualize if we think of the exercise being done by sentient individuals under the instruction of a single leader. An initial way of representing several individuals in a closed space is shown in Fig. 1. (For illustrative purposes we choose N ¼ 22; a number consistent with the number of students in the class where this exercise was initially developed and applied.)1 The problem of getting individuals introduced to each other is often accomplished by milling. That is, the people are instructed to mill about, with the objective that every two people meet. The difficulty, of course, is that milling is inefficient: it takes a lot of time, many 1 See Scheibe (2000, Chapter 12), for a description of the class, a seminar called ‘‘The Dramaturgical Approach to Psychology.’’
ARTICLE IN PRESS K.E. Scheibe, W.W. Comfort / Journal of Mathematical Psychology 49 (2005) 155–159
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pairings are repeated, and there is no guarantee within a finite period of time, allocated in advance, that every two participants will have met.
3. Even and odd cases For a more orderly approach, we arrange the individuals in two facing rows. Figs. 2E and O represent typical configurations, one with N Even (we choose N ¼ 22) and one with N Odd (we choose N ¼ 23). The procedures for achieving efficient matching of all pairs differ for the Even case and the Odd case. In the Even case, with the participants arranged in two rows of equal length, each individual is instructed to greet the person in the opposite or facing position. First greetings are illustrated by the connecting lines in Fig. 2E. At the completion of the first greeting, the leader gives a signal to shift, whereupon everyone in the set (with the exception of the leader) moves one position in a clockwise direction. When everyone has moved one position, the leader gives another instruction to greet. After the greeting is completed, the leader again gives a command to shift. This pair of commands, ‘‘Greet’’ and ‘‘Shift’’, is given N 1 times, whereupon the entire set is returned to its initial position. In this Even case, the leader remains in the same position throughout the exercise, greeting each individual as they move into the opposing position.
X
X X
X
X
X X
X X
X XX
X X
X X
X
XX
X
X X
Fig. 1. N ¼ 22 individuals, casually distributed.
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(O) Fig. 2. (E) Initial greeting pattern when N ¼ 22: The leader, x22 ; is stationary: (O) Initial greeting pattern when N ¼ 23: Here & remains stationary.
The procedure when N is Odd necessarily differs from that just described. In this case, the leader (denoted x1 in Fig. 2O) initially greets no individual. A ‘‘ghost’’ or ‘‘dummy’’ participant, here denoted &; is paired with the leader in Round 1. Other participants ‘‘greet’’ & in subsequent rounds. Unlike the Even case, the leader now also moves one position at each ‘‘Shift’’ command; furthermore, N pairings of the commands ‘‘Greet’’ and ‘‘Shift’’ (rather than just N 1) must be given in order for the group to return to the initial arrangement. (One may think of the Even case as slightly less wasteful of opportunity than the Odd case, for in the Odd case at each step of the exercise necessarily some one person is left without anyone to greet.) Participants in this exercise generally find one of its features to be counterintuitive. When in the initial configuration (Fig. 2E or 2O), if participants are asked to point to the person they will greet after the first shift, they typically will point to the person one position to the left of the person they face. But because both files shift to the left simultaneously, in fact the person two positions to the left of the person initially greeted will be in position for the second greeting.
4. The mathematical argument While it may seem evident that no procedure can achieve the objective of universal greeting more efficiently than the one illustrated and described above, it is appropriate to present a mathematical proof. The following discussion is quite general, not being limited to the cases N ¼ 22 and 23 of Figs. 2E and 2O. Indeed Figs. 4E and 4O show the initial alignment for (arbitrary) even N ¼ 2m and odd N ¼ 2m þ 1; respectively. We adopt the standard (row,column) matrix notation associated with Fig. 3. There are 2 rows and m columns; for 1pkp2 and 1pppm the notation ðk; pÞ designates the position in row k and column p. To be formal: Our assignment is to establish two properties of the greeting procedure described, namely (A) each participant greets other participant exactly once, and (B) the procedure is maximally efficient in the sense that no procedure with property (A) can take fewer steps. Once (A) is known, (B) is established easily, as follows. The number of greetings to be achieved is the (1, 1) (1, 2) · · · (1, p) · · · (1, m) (2, 1) (2, 2) · · · (2, p) · · · (2, m) Fig. 3. A matrix with m columns and 2 rows.
ARTICLE IN PRESS K.E. Scheibe, W.W. Comfort / Journal of Mathematical Psychology 49 (2005) 155–159
in position ð2; pÞ on the first of these occasions. At their second meeting, in some column q, either
number of pairs in an N-set, namely CðN; 2Þ ¼
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N! NðN 1Þ ¼ : ðN 2Þ!2! 2
When N is even, say N ¼ 2m; one can form (at most) m pairs in a single round, so the number of rounds must be (at least) NðN 1Þ CðN; 2Þ=m ¼ m 2 ¼ 2mð2m 1Þ=2m ¼ 2m 1 ¼ N 1; the number achieved by our procedure. Similarly when N is odd, say N ¼ 2m þ 1; one can again form no more than m pairs in a single round, so the number of rounds required is at least CðN; 2Þ=m ¼ ½ð2m þ 1Þð2m þ 1 1Þ=2=m ¼ 2m þ 1 ¼ N; again the number used in our procedure. It remains, then, only to prove (A). Case: 1 N is even, say N ¼ 2m: If (A) fails, so that some participant xi never meets some participant xk (kai) then, since there are N 1 rounds of greeting, xi must meet some participant xj at least twice. [The mathematical aficionado will recognize this as an elementary instance of the so-called pigeonhole principle: there can be no injection from an ðN 1Þ-set (here, the N 1 rounds of greeting) into a set with fewer than N 1 elements (here, the set of participants greeted by xi :)] Thus it suffices to show that no two participants greet each other twice. It is clear that the (stationary) leader x2m does greet each participant exactly once. Indeed with numbering as in Fig. 4E, x2m greets participant xi in round i; that is, x2m greets x1 in round 1, x2 in round 2, and so on until round 2m 1; the final round, when x2m greets x2m1 : We must show that no two participants xi and xj (with xi ax2m ; xj ax2m ) can meet twice. Greetings occur (only) between participants in the same column. Suppose now that some participants, xi and xj ; meet twice, say with xi in position ð1; pÞ and xj
x 2m x1
x 2m−1 x2
· · · x2m+1−p ·· · xp
· · · xm+1 · · · xm
x1
x 2m+1 x2
· · · x2m+3−p ·· · xp
· · · xm+2 · · · x m+1
(E)
(O) Fig. 4. (E) Initial position when N ¼ 2m is even: (O) Initial position when N ¼ 2m þ 1 is odd.
(i) xi is in position ð1; qÞ and xj is in position ð2; qÞ; or (ii) xj is in position ð1; qÞ and xi is in position ð2; qÞ: If (i) occurs then paq since no participant other than the stationary leader visits the same position twice in the 2m 1 rounds of greeting, so either 1opoqpm or 1oqoppm: In the first case participant xi moves from ð1; pÞ to ð1; qÞ in q p rounds, while xj moves from ð2; pÞ to ð2; qÞ in ð2m 1Þ þ ðq pÞ rounds, so q p ¼ ð2m 1Þ ðq pÞ and hence 2ðq pÞ ¼ 2m 1; a contradiction since no integer is simultaneously even and odd; in the latter case a similar argument yields ð2m 1Þ ðp qÞ ¼ p q and hence 2m 1 ¼ 2ðp qÞ; again a contradiction. Thus (i) cannot occur. As to (ii), we notice that xi ; beginning at ð1; pÞ; moves in m p rounds to ð1; mÞ; moves in the next round to ð2; mÞ; and arrives at ð2; qÞ m q rounds later; in sum, xi requires ðm pÞ þ 1 þ ðm qÞ stages to move from ð1; pÞ to ð2; qÞ: Similarly, xj requires p 1 moves to arrive at ð2; 1Þ; then q 1 additional moves to arrive at ð1; qÞ: Thus ðm pÞ þ 1 þ ðm qÞ ¼ ðp 1Þ þ ðq 1Þ and the contradiction 2m þ 3 ¼ 2p þ 2q arises. Since neither (i) nor (ii) can arise in the case that N ¼ 2m is even, the proof of (A) (and hence of (B)) is complete in that case. Case 2: N is odd, say N ¼ 2m þ 1: We assign to the array a ‘‘dummy’’ or ‘‘ghost’’ participant, here denoted &: This brings the number of participants to 2m þ 1 þ 1; so Case 1 applies to that even number. The entire dance now requires ð2m þ 2Þ 1 ¼ 2m þ 1 ¼ N steps. (Each of the genuine original participants xi ‘‘greets’’ & exactly once; in practice this corresponds to the round in which xi is ‘‘odd man out’’.) The proof is complete. Remarks. When N is even, the greeting procedure we have developed keeps one participant in fixed position, while the others rotate clockwise. In practice, the leader has assumed this special position. When N is odd and & is adjoined to the array, we have found it convenient to cede this privileged stationary role to &: The genuine original leader now assumes the position of x1 as in Fig. 4O, ‘‘greeting’’ & in round 1. By this device no human being is seen as distinguished by rank or status from others in the group, and the true leader participates in the ‘‘Greet–Shift’’ sequence with no apparent special distinction. The mathematical analysis required above to verify the validity and efficiency of the procedures defined may strike the reader, as it struck us, as unexpectedly or unnecessarily long. It is reassuring to note that an approach based on Mathematical Induction, while blind
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to the social niceties, effects a proof with greater dispatch. As usual, the set of 2-sets of an N-set is denoted ½N2 : Theorem. For finite NX2 the set ½N2 can be expressed in SN1 2 the form ½N ¼ S k¼1 Ak when N ¼ 2m is even, and in the form ½N2 ¼ N k¼1 Ak when N ¼ 2m þ 1 is odd, with (in each case) each Ak a pairwise disjoint family of cardinality m. Proof. The indicated statement SðNÞ is obvious when N ¼ 2: Fix K and suppose that SðNÞ is true whenever 2pNoK: We are to prove SðKÞ: If K is odd then SðKÞ follows from SðK 1Þ by the argument given earlier, so we may assume that K is even, say K ¼ 2m: We write K ¼ K 1 [ K 2 with jK 0 j ¼ jK 1 j ¼ m; and we define S ¼ ffb; cg : b 2 K 1 ; c 2 K 2 g; so that ½K2 ¼ ½K 1 2 [ ½K 2 2 [ S: If m isSeven we use the inductive Sm1 hypothesis to write 2 ½K 1 2 ¼ m1 B and ½K ¼ 2 k¼1 k k¼1 C k with each Bk and each C k a pairwise disjoint family of cardinality m2 ; and we set Ak ¼ Bk [ C k for 1pkpm 1: Writing K 1 ¼ fb1 ; b2 ; . . . ; bm g and K 2 ¼ fc1 ; c2 ; . . . ; cm g; for mpkp2m 1 ¼ K we set Ak ¼ ffbi ; ckþi g 2 S : 1pipm mod m. Then ½S2 ¼ SK1 SK1 1g with subscripts 2taken 2 k¼m Ak and the relation ½K ¼ k¼1 Ak expresses ½K in the required form. Thus SðKÞ holds when K ¼ 2m with m even. Sm 2 2 SmIf m is odd we again let ½K 1 ¼ k¼1 Bk and ½K 2 ¼ k¼1 C k be as given by the inductive hypothesis, and with K 1 n [ Bk ¼ fbk g and K 2 n [ C k ¼ fck g we define Ak ¼ Bk [ C k [ ffbk ; ck gg for 1pkpm: (Informally, bk and ck are those members of K 1 and K 2 ; respectively who are ‘‘left out’’ at the kth round of greeting. In this plan they greet each other at stage k.) The indexings K 1 ¼ fb1 ; b2 ; . . . ; bm g and K 2 ¼ fc1 ; c2 ; . . . ; cm g are now determined. Now for m þ 1pkp2m 1 ¼ K 1 we set Ak ¼ ffbi ; ciþk g : 1pipmg; again with subscripts taken mod m. (In these rounds of greeting each bi meets each cj with iaj; but for no i does bi greet ci :)Thus S K1 Ak ¼ ½S2 nffbk ; ck g : 1pkpmg and the relation k¼mþ1S 2 2 ½K ¼ K1 k¼1 Ak again expresses ½K in the required form. Thus SðKÞ holds also for K ¼ 2m with m odd. In summary: We have noted that Sð2Þ holds trivially, and we have shown that if K42 and SðNÞ holds whenever 2pNoK then SðNÞ holds for N ¼ K: It follows from the Principle of Induction that SðNÞ holds for all integers NX2; as required.
5. The silent greeting exercise The mathematical and procedural arguments are now complete. Yet something remains to be said about the psychological and social side of putting the exercise into practice. First of all, the simple act of conducting this exercise communicates something about the intended
character of the group processes that might proceed from such an introduction. The objective is to lay a basis for general and egalitarian participation, including the leader as one of the participants, albeit with special status. Also, it seemed from the outset that in order to conduct such an exercise properly and with maximum effect, it should be done in silence. One reason for this is that the commands to ‘‘Greet’’ and ‘‘Shift’’ cannot easily be heard over the din of a score of voices. But a more powerful reason for devising a silent greeting exercise derives from a suggestion from the German novelist, Thomas Mann, that words tend to falsify human intercourse and that the glance and the touch are more likely to be authentic expressions of sentiment and interest.2 The instruction for silent greetings is for pairs to touch their right palms together and to make their eyes meet. They are told to begin the greeting at the ‘‘Greet’’ command, and to hold it until the ‘‘Shift’’ command, with the leader pacing the commands fairly evenly at intervals of about 3–5 s. Thus, an entire group of 20-some participants can achieve universal contact in less than 2 min. When this exercise is performed as a first exercise in a class or group meeting, there is typically some nervousness and perhaps giggling during the procedure. When groups are allowed to discuss their feelings about conducting the exercise, they typically remark on differences they experienced from person to person in the expression of their eyes, the feel of their hands, the facial expressions encountered. When this same silent greeting exercise is repeated at the end of a 13-week semester of working, studying, and learning together as a group, the emotional character of the experience is quite different. There is now no giggling, no nervousness. Instead, eyes are commonly watered with tears, and the hands are likely to clasp together, not merely touching palms. Students commonly report on the intensity of this experience and compare it with the nervousness and uncertainty they felt when the exercise was done at the beginning of the course. Much of this transformation depends, of course, on the length, character, and intensity of the activities taking place between the two silent greeting exercises. The arguments given above show that this technique for achieving universal pairwise contact is most efficient 2 ‘‘[T]here alone are unconditional freedom, secrecy, and profound ruthlessness. Everything by way of human contact and exchange that lies between is lukewarm and insipid; it is determined, conditioned, and limited by manners and social convention. Here the word is master— that cool, prosaic device, that first begetter of tame, mediocre morality . . . [The] silent regions of human intercourse . . . where strangeness and social rootless still maintain a free, primordial condition and glances meet and marry irresponsibly in dreamlike wantonness.’’ (Mann, 1955, p. 79).
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for every finite set, irrespective of its size. Given a group of 100 individuals arranged on a field, for example, all 4950 possible pairings may be achieved within 7 min. A thousand people could exchange universal greetings, if they could find an hour in common for the exercise. As a macabre oddity, we note that any one individual transmitting a pathogen might communicate that pathogen to everyone else in the group with just one iteration of the greeting cycle. A more positive aspect is this: The experience of greeting a sequence of people in a way that is simple and yet authentic is uniformly experienced as pleasurable. The conceptual value of our algorithm is that it sets forth in elementary and workable language a solution to the problem of achieving maximally efficient pairings of N entities. At first glance, one might think that the model proposed here has something in common with the ‘‘fast and frugal’’ heuristics proposed by Gigerenzer and others as ways of facilitating human judgment tasks or solving reasoning problems (see (Gigerenzer, Todd, & ABC Research group , 1999). Indeed, the algorithm described here is both ‘‘rapid and clean’’ in the sense introduced by those authors. However, it is distinctly not a descriptive heuristic, as are the devices proposed by Gigerenzer et al. It is possible that chance, or some collective intelligence, might guide people into forming lines and exchanging greetings in the manner suggested here, but the likelihood of this happening without explicit instruction is close to zero for all but very small values of N. Our model is normative, not descriptive. Given that one wishes to achieve universal pairing within a set of individuals, this algorithm provides the most efficient means of achieving that result. Why and under what circumstances might one wish to achieve universal pairing? The example from the classroom has already been described. While for many classrooms the relationships among students might be entirely unimportant, in this case the objective was to create as quickly as possible a sense of group cohesion in
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order to facilitate cooperation and collaboration among class members. It has long been observed that forms of patterns of communication within groups determine such outcomes as speed and effectiveness of group function, group morale, speed of group decisionmaking, and the creative capacity of groups (see Leavitt, 1951). The common use of the circle form of seating within the human potentials movement is testimony to the intuitive appeal of achieving universal contact within a group (see Hare, Borgatta, & Bales, 1965). But use of the circle arrangement does not guarantee universal contact of all possible pairs—it merely makes that contact possible. The objective of developing group cohesion and facilitating the formation of common group identity is common to many settings. Examples include academic classes, management training seminars, ensemble performance groups, religious or political groups, therapy groups, military units and athletic teams. In all of these arenas, functioning groups must somehow be forged out of a set of isolated individuals. The device set forth in this paper is easily implemented and has proved to be an effective and even elegant means of thickening the social soup. The mathematical arguments here set forth provide assurance that no more efficient means of achieving universal pairing may be found.
References Gigerenzer, G., Todd, P., & ABC Research group (1999). Simple heuristics that make us smart. New York: Oxford University Press. Hare, A. P., Borgatta, E. F., & Bales, R. F. (1965). Small groups: Studies in social interation. Knopf: New York. Leavitt, H. J. (1951). Some effects of certain communication patterns on group performance. Journal of Abnormal and Social Psychology, 46, 38–50. Mann, T. (1955). Confessions of Felix Krull, confidence man. Random House: New York. Scheibe, K. E. (2000). The drama of everyday life. Harvard University Press: Cambridge.