Sources page biographical material

Nineteen trees in nine straight rows.

Five trees in each row I must place,

Or I shall never see her face."

Rudin. 1936. Nos. 105-108, pp. 39 & 99-100.

No. 105: (9, 10, 3).

No. 106: (10, 5, 4) -- two solutions.

No. 107: (12, 6, 4) -- two solutions.

No. 108: (19, 9, 5).

Depew. Cokesbury Game Book. 1939. The orange grower, p. 221. (21, 9, 5).

The Home Book of Quizzes, Games and Jokes. Op. cit. in 4.B.1, 1941. P. 147, prob. 1 & 2. Place six coins in an L or a cross and make two rows of four, i.e. (6, 2, 4), which is done by the simple trick of putting a coin on the intersection.

R. H. Macmillan. Letter: An old problem. MG 30 (No. 289) (May 1946) 109. Says he believes Newton and Sylvester studied this. Says he has examples of (11, 16, 3), (12, 19, 3), (18, 18, 4), (24, 28, 4), (25, 30, 4), (36, 55, 4), (22, 15, 5), (26, 21, 5), (30, 26, 5).

G. C. Shephard. A problem in orchards. Eureka 9 (Apr 1947) 11-14. Given k points in n dimensions, the general problem is to draw N(k, n) hyperplanes to produce k regions, each containing one point. The most common example is k = 7, n = 2, N = 3. [See Section 5.Q for determining k as a function of n and N.] The author investigates the question of determining the possible locations of the seventh point given six points. He gives a construction of a set T such that being in T is necessary and sufficient for three such lines to exist.

J. Bridges. Potter's orchard. Eureka 11 (Jan 1949) 30 & 12 (Oct 1949) 17. Start with an orchard (9, 9, 3). Add 16 trees to make (25, 18, 5). The nine trees are three points in a triangle, with the three midpoints of the sides and the three points halfway between these. Six of the new trees are one third of the way along the sides of the original triangle; another six are one third of the way along the lines joining the midpoints of the original triangle; one point is the centre of the original triangle and the last three are easily seen.

W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. Thirteen rows of three, pp. 45 & 132. (11, 13, 3).

Young World. c1960. Pp. 10-11.

Three coin lines. (9, 10, 3).

Five coin lines. (10, 5, 4).

Eleven coin trick. (11, 12, 3).

Maxey Brooke. Dots and lines. RMM 6 (Dec 1961) 51 55. Cites Jackson and Dudeney. Says Sylvester showed that n points can be arranged in at least (n 1)(n 2)/6 rows of three. Shows (9, 10, 3) and (16, 15, 4).

R. L. Hutchings & J. D. Blake. Problems drive 1962. Eureka 25 (Oct 1962) 20-21 & 34-35. Prob. F. (10, 5, 4) with points in the centres of cells of a chess board. Actually only needs a 7 x 7 board.

Ripley's Puzzles and Games. 1966. Pp. 18-19, item 4. (17, 7, 5).

Doubleday - 3. 1972. Count down, pp. 125-126. Start with a 4 x 4 array of coins. Add four coins so that each row, column and diagonal has the same number. Solution doubles the coins in the 1, 3, 4, 2 positions in the rows.

S. A. Burr, B. Grünbaum & N. J. A. Sloane. The orchard problem. Geometria Dedicata 2 (1974) 397 424. Establishes good examples of (a, b, 3) slightly improving on Sylvester, and establishes some special better examples. Gives upper bounds for b in (a, b, 3). Sketches history and tabulates best values and upper bounds for b in (a, b, 3), for a = 1 (1) 32.

The following have the maximal possible value of b for given a and c.

(3, 1, 3); (4, 1, 3); (5, 2, 3); (6, 4, 3); (7, 6, 3); (8, 7, 3); (9, 10, 3); (10, 12, 3); (11, 16, 3); (12, 19, 3); (16, 37, 3).

The following have the largest known value of b for the given a and c.

(13, 22, 3); (14, 26, 3); (15, 31, 3); (17, 40, 3); (18, 46, 3); (19, 52, 3); (20, 57, 3); (21, 64, 3); (22, 70, 3); (23, 77, 3); (24, 85, 3); (25, 92, 3); (26, 100, 3); (27, 109, 3); (28, 117, 3); (29, 126, 3); (30, 136, 3); (31, 145, 3); (32, 155, 3).

M. Gardner. SA (Aug 1976). Surveys these problems, based on Burr, Grünbaum & Sloane. He gives results for c = 4.

The following have the maximal possible value of b for the given a and c.

(4, 1, 4); (5, 1, 4); (6, 1, 4); (7, 2, 4); (8, 2, 4); (9, 3, 4); (10, 5, 4); (11, 6, 4); (12, 7, 4).

The following have the largest known value of b for the given a and c.

(13, 9, 4); (14, 10, 4); (15, 12, 4); (16, 15, 4); (17, 15, 4); (18, 18, 4); (19, 19, 4); (20, 20, 4).

Putnam. Puzzle Fun. 1978. Nos. 17-23: Bingo arrangements, pp. 6 & 29-30. (21, 11, 5), (16, 15, 4), (19, 9, 5), (9, 10, 3), (12, 7, 4), (22, 21, 4), (10, 5, 4).

S. A. Burr. Planting trees. In: The Mathematical Gardner; ed. by David Klarner; Prindle, Weber & Schmidt/Wadsworth, 1981. Pp. 90 99. Pleasant survey of the 1974 paper by Burr, et al.

Michel Criton. Des points et des Lignes. Jouer Jeux Mathématiques 3 (Jul/Sep 1991) 6-9. Survey, with a graph showing c at (a, b). Observes that some solutions have points which are not at intersections of lines and proposes a more restrictive kind of arrangement of b lines whose intersections give a points with c points on each line. He denotes these with square brackets which I write as [a, b, c]. Pictures of (7, 6, 3), [9, 8, 3], (9, 9, 3), (12, 6, 4), [13, 9, 4], (13, 12, 3), (13, 22, 3), (16, 12, 4), (19, 19, 4), (19, 19, 5), (20, 21, 4), [21, 12, 5], (25, 12, 5), (30, 12, 7), (30, 22, 5), (49, 16, 7) and mentions of (9, 10, 3), (16, 15, 4),