Volume 36, 1903
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Art. XLIX.—A Note on Drawing for Competitions.

[Read before the Auckland Institute, 6th July, 1903.]

The question was recently put to me, in connection with a local association of clubs, whether it were possible for seven competing clubs to arrange their fixtures, in which each was to play every other, so that each club should play its matches at home and away alternatively throughout, thus removing the objectionable feature of a succession of matches played either all at home or all away.

The problem was a somewhat interesting one in arrangements, and the result came out in a form that was obviously

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extensible to the case of any odd number of clubs. The arrangement cannot be completely carried out for the case of an even number of clubs, but a very close approximation to it can be made.

I have never heard of any rule or system being used in the arranging of inter-club fixtures. On the other hand, the common circumstances of the home and away fixtures being somewhat irregularly distributed, and of a few straggling fixtures generally remaining at the end of a season, for which apparently no places could be found on the dates when the main body of fixtures were played off, indicate the want of a rule which would enable a better arrangement to be carried out. It has appeared to me, therefore, advisable to give publicity to the result referred to above. It is as follows:—

Let the number of clubs be odd, and represent them by consecutive integers beginning from unity. Write down these integers in succession in a column, beginning with unity, until we have written down the middle integer; then write the next integer on the right of it and begin to ascend, writing down the remaining integers on the right of those previously written down. The last integer will be written down in the second row. the first integer remaining alone in the first row. Then begin a parallel column with the integer 2, and proceed in a similar way, writing down the integers in succession down the column until the last row is reached, and then write the next integer on the right and begin to ascend. The last integer will be written in the third row, but in the vacant place in the second row write unity. Then begin another parallel column with the integer 3, then another with 4, and so on until columns have been written down beginning with each of the integers used to represent the clubs. When the last integer has been used, continue in each case with the consecutive integers, beginning with unity, but never write a second integer in the first row.

The columns now represent the consecutive rounds, the numbers written together representing the competing clubs. Any club will play every other once. The clubs indicated in the first row have byes in the respective rounds. By subscribing “h” and “a,” indicating “at home” and “away” respectively, alternately to the numbers in each column in the order in which they were written down, but beginning with the second in each case, we have the home club indicated for each match, so that no club shall play two matches in succession either at home or away.

Table I., following, indicates the result in the case of seven clubs, and Table II. the general result for (2n + 1) clubs.

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[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

Table I.
Scheme of Matches between Seven Competitors.
First Round. Second Round. Third Round. Fourth Round. Fifth Round. Sixth Round. Seventh Round
1 2 3 4 5 6 7
2h 7a 3h 1a 4h 2a 5h 3a 6h 4a 7h 5a 1h 6a
3a 6h 4a 7h 5a 1h 6a 2h 7a 3h 1a 4h 2a 5h
4h 5a 5h 6a 6h 7a 7h 1a 1h 2a 2h 3a 3h 4a

Thus, e.g., in the first round 1 has a bye, 2 (at home) plays 7 (away), 3 (away) plays 6 (at home), and 4 (at home) plays 5 (away).

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

Table II.
Scheme of Matches between (2n + 1) Competitors.
First Round. Second Round. Third Round. (2n + 1)th Round.
1 2 3 2n + 1
2h (2n+ 1)a 3h 1a 4h 2a 1h 2na
3a 2nh 4a (2n + 1)h 5a 1h 2a (2n-1)h
4h (2n-1)a 5h 2na 6h (2n + 1)a 3h (2n-2)a
5a (2n-2)h 6a (2n-1)h 7a 2nh 4a (2n-3)h
(n + 1)(n + 2) (n + 2)(n + 3) (n + 3)(n + 4) n(n + 1)

If this rule were adopted the only drawing necessary would be for the numbers to represent the several clubs, everything else would then follow.

It has already been mentioned that it is impossible to completely achieve this arrangement in the case of the number of clubs being even. The nearest approach that is possible appears to be as follows:—

Write down columns as in the case above of an odd number of clubs, taking the odd number which is one less than the actual number of clubs. Then take the even number representing the number of clubs and write it at the side of each of the solitary numbers in the first row, but alternately on the right and on the left of them consecutively. The several columns then represent, as before, the competing clubs for the successive rounds, each club playing every other once. There are, of course, no byes.

If, further, we subscribe “h” and “a,” with the same meanings as before, to the numbers in each column alternately, descending with the numbers on the left and rising with those on the right, we get the home club indicated for

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each match in such a way that each club plays either two matches at home or two away in succession, but this occurs not more than once for any club in the whole competition.

Table III., indicating the result for eight clubs, and Table IV., indicating the general result, the number of clubs being 2n, will help to make this clear.

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

Table III.
Scheme of Matches between Eight Competitors.
First Round. Second Round. Third Round. Fourth Round. Fifth Round. Sixth Round. Seventh Round
1h 8a 8h 2a 3h 8a 8h 4a 5h 8a 8h 6a 7h 8a
2a 7h 3a 1h 4a 2h 5a 3h 6a 4h 7a 5h 1a 6h
3h 6a 4h 7a 5h 1a 6h 2a 7h 3a 1h 4a 2h 5a
4a 5h 5a 6h 6a 7h 7a 1h 1a 2h 2a 3h 3a 4h

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

Table IV.
Scheme of Matches between 2n Competitors.
First Round. Second Round. Third Round. Fourth Round. (2n-1)th Round.
1h 2na 2nh 2a 3h 2na 2nh 4a (2n-1)h 2na
2a (2n-1)h 3a 1h 4a 2h 5a 3h 1a (2n-2)h
3h (2n-2)a 4h (2n-1)a 5h 1a 6h 2a 2h (2n-3)a
4a (2n-3)h 5a (2n-2)h 6a (2n-1)h 7a 1h 3a (2n-4)h
n (n+1 (n+1)(n+2) (n+2)(n+3) (n+3)(n+4) (n-1) n