Michael T. Marran
University of Wisconsin- Porkside Kenosho, 53140
Assigning Laboratory Partners Different partners every week
At the beginning of each semester I am faced with the problem of drawing up a schedule for Physical Chemistry Laboratory which assigns student pairs to particular experiments for each week. In our laboratory there is only one experimental apparatus available for each experiment and the number of experimental setups is equal to or only slightly greater than the number of experiments required of each student. The task of constructing a schedule is complicated by attempting to make assignments i n such a way as to eliminate instances of two individuals working together more than once. If the number of student pairs and the numher of experiments required approaches the numher of weeks in a semester, i t becomes extremely difficult to assign every student a different experiment and a different partner each week. One apparent solution to this problem is to allow students to sign up for experiments. This procedure has two shortcomings. Better students in a class often sign up to work togetKer leaving poorer students consistenth paired together. Also if the same pair of students work together on several experiments the part-nership sometimes develops into one in which one person does all the thinking and the other does the lab work. A second shortcoming of a signup procedure is that it is hard to avoid irreconkahle schedule conflicts in the final weeks of the semester. If I find it difficult t o make conflict-free assignments by an organized scheme, why should the shotgun signup approach he any better? In fact, it is worse. The only other ohvious solution, that of setting up additional experiments, is not generally acceptable because i t requires grading a wider variety of reports (in the case of additional new experiments) and the purchase of more equipment. The Problem
Let there he N experiments to he performed by 2P students. To simplify discussion, P, the number of student pairs, will initially be set equal to N. A schedule may he constructed in tabular form by allowing row position to indicate date(e.g., week 1, week 2, etc.). Column position will then indicate experiment. A schedule is obtained by assigning each student a numher and entering two numbers a t each position in that table indexed by row and column. The requirement that every student perform each experiment once translates into the requirement (for P = N) that every numher must appear in each column once and only once. Because a student may work on only one experiment a t a time, the complete requirement is that (1) each number may appear in a column or row once and only once. The resulting schedule is an N X N array having pairs of student numbers as entries. Our objective is to obtain such a schedule in which (2) no two students are paired more than once. This is possible for all cases exc e p t N = 2 a n d N = 6. The Solulion
The problem described above is a classical problem in combinatorial mathematics: the construction of orthogonal Latin squares.l The desired schedule is ohtained as the superposition of two orthogonal Latin squares. Latin squares have property (1) above and orthogonal Latin 336 / Journal of Chemical Education
squares ensure property (2). When two orthogonal Latin squares are superposed t o form a single square, the result is referred to as a Graeco-Latin square. This name arises because many authors distinguish between entries by forming one of the Latin squares using Latin letters and the other using Greek letters. Several methods for constructing orthogonal Latin squares are discussed in the mathematics literature; however, none of them are particularly suited for our pumose. They are concerned with constructing complete sets of orthogonal squares and are thus unduly complicated. Only a single pairof orthogonal squares is required in order to form a Graeco-Latin square. In the discussion below, schedules are constructed for a class of size 2N. Without loss of generality the class may be divided into halves, where the first half is numbered 1 through N and the second half is numbered N + 1 through 2N. Graeco-Latin Squares for N Odd When N is odd a simple prescription exists for constructing a Graeco-Latin square. This may be illustrated by the schedule for N = 5
The first sequence (1, 2, 3, 4, 5) is entered in each row, changing each entry by performing a cyclic permutation on the sequence. The second sequence (6, 7, 8, 9, 10) is entered in the same fashion except that the direction of permutation is reversed. Graeco-Latin Squares for N Even I have yet to discover a simple prescription for constructing N-even squares. In lieu of a general prescription, examples may be given for the cases N = 4,8,10,12
'For example, see Hall, Jr., M., "Combinatorial Theory," BlaisdellPublishingCo., Waltham, MA, 1967, Chapter 13. R. and yates, F., -statistical T~~~~~~~~ ~ i ~ ~ ~ ~ ~ ~and ~i ~ ~d~ i ~~~ ~l~sixth l ~~ d ~,~~~f~~~ , ~ ~ publish. h ~ , l, ~ i n g ~ o . ~ nN~~ c . , yo&, 1963, pp.24and88. 3Tarry, G., Comptes Rendus de 1'Assoe. Froncaise pour L'nunncernentdes Sciences, 1,122 (1900): 2.170 (1901).
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Applications
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These examples are taken from the b w k by Fisher and Yates2 wherein a method is described for constructing higher-order squares from lower-order squares. Their method does not apply, however, to the cases N = 4s + 2, s = 0 , 1 , 2 , 3 ....
The Euler Conjecture and the Case N = 6 In 1782 Euler posed the problem of arranging 36 military officers of six different ranks and from six different regiments in a square formation. Each row and each column of this formation was to contain one and only one officer of each rank and one and only one officer from each regiment. Euler's problem can be reduced to construction of a Graeco-Latin square of order N = 6. As a result of his analysis, he conjectured that Graeco-Latin squares do not exist for orden N = 4s 2, s = 0, 1, 2, 3 . . . . The conjecture is obviously true for N = 2. In 1900 Tarry verified the conjecture for N = 6 by systematic enumeration of all possible arrangemenh3 In 1960 the conjecture was shown to be false for the higher order cases,4 N = 10, 14, 18 . . . . The examples above include one of these cases, viz., N = 10. I have not constructed, nor have I seen reported, Graeco-Latin squares of orders N = 14, 18 . . . . Techniques for doing so are reported by Bose et aL4 Because the N = 6 case may he a fairly common one for lahoratory assignments, a schedule is given below which approximates a Graeco-Latin square. A single repeated partnership is assigned for every student in the last two rows
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Uniqueness of Solutions The solutions given are not unique. Row and column designations can be exchanged and rows (columns) may be interchanged to alter the structure of a square. On occasion I have found it useful to alter the N = 6 table to obtain assignments which do not duplicate partnerships for everyone two weeks in a row. An alternate example is given below which has higher symmetry and distributes duplicate partnerships throughout the table
The schedules described above apply to many situations other than the rare situation where the number of student ~ -a~i -r -is r -s - the same as the number of exneriments reauired. to these I have not yet figured Out a way to the eeneral case where dunlicate annaratus is available for .. certain experiments. ~ h e ' t h i r d treatment below is sometimes applicable in this situation, depending on the num~~~~
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The number of students is not an even number. This situation is treated by adding a dummy student to the class list. Students paired with the dummy must perform that exneriment alone. If the exneriment reauires occasional 2 d from a second person thk instructor can provide it. If the exneriment can onlv be performed bv two persons, the assignment table st&ctur& may be altered b; interchanging rows (columns). The number fP)of pairs is greater than the number (N) of experiments required: f P > N). This is the situation I encounter most frequently. This situation is treated by introducing P - N phony experiments (I call them R & R) and then constructing a P x P table. Students adsigned to phony experiments are excused from lab that week. Note that it requires P weeks for all students to complete N experiments. Thus, in a 15 week semester, a maximum of thirty students may enroll in a laboratory course, even if only ten experiments are required (as long as there are only ten experimental setups available). The number (El of experimental setups available is greater than the number (N) of experiments required and greater than or equal to the number (Pi of pairs: fE > N; E 2 P). This situation can he treated by constructing an E x E table and truncating the table after P rows (weeks). The number (N) of experiments required is greater than the number (P) of pairs: IN > P). This situation can often he treated without resorting to one of the schemes desribed here. Since N > P implies that some experimental apparatus is unused each week, conflicts are not usually a problem. While it may not be the most efficient schedule, bne can be made u p by dividing the experiments into P X P blocks, adding phony experiments where they are needed to fill out a block. One advantaee of this method is that everyone finishes the first I' experiments at the same time and before bezinninz the second P experimenti. T h ~ s may be convenientin terms of lab setups-or report grading. Experiments are classified into subgroups. This situation might arise if a subgroup of experiments are interrelated. For example, five of the ten experiments required might all be devoted to the study of the physical properties of a particular binary liquid system. This situation may he treated by dividing the class in half and viewing the class as two separate classes. A schedule may then be constructed hy one of the techniques described above. Acknowledgment
The author acknowledges the helpful comments of Dr. Richard Orr. 'Base, R. C., Parker, E. T., and Shrikhande, S., Can. J. Math., 12, 189 (1960).
Volume 52, Number 5, May 1975
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