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5.X. COUNTING FIGURES IN A PATTERN
New section -- there must be older examples. There are two forms of such problems depending on whether one must use the lattice lines or just the lattice points.

For counting several shapes, see: Young World (c1960); Gooding (1994) in 5.X.1.

5.X.1. COUNTING TRIANGLES
Counting triangles in a pattern is always fraught with difficulties, so I have written a program to do this, but I haven't checked all the examples here.
Pearson. 1907. Part II.

No. 74: A triangle of triangles, p. 74. Triangular array with four on a side, but with all the altitudes also drawn. Gets 653 triangles of various shapes.

No. 75: Pharaoh's seal, pp. 75 & 174. Isosceles right triangles in a square pattern with some diagonals.

Anon. Prob. 76. Hobbies 31 (No. 791) (10 Dec 1910) 256 & (No. 794) (31 Dec 1910) 318. Make as many triangles as possible with six matches. From the solution, it seems that the tetrahedron was expected with four triangles, but many submitted the figure of a triangle with its altitudes drawn, but only one solver noted that this figure contains 16 triangles! However, if the altitudes are displaced to give an interior triangle, I find 17 triangles!!

Loyd. Cyclopedia. 1914. King Solomon's seal, pp. 284 & 378. = MPSL2, No. 142, pp. 100 & 165 c= SLAHP: Various triangles, pp. 25 & 91. How many triangles in the triangular pattern with 4 on a side? Loyd Sr. has this embedded in a larger triangle.

Collins. Book of Puzzles. 1927. The swarm of triangles, pp. 97-98. Same as Pearson No. 74. He says there are 653 triangles and that starting with 5 on a side gives 1196 and 10,000 on a side gives 6,992,965,420,382. When I gave August's problem in the Weekend Telegraph, F. R. Gill wrote that this puzzle with 5 on a side was given out as a competition problem by a furniture shop in north Lancashire in the late 1930s, with a three piece suite as a prize for the first correct solution.

Evelyn August. The Black-Out Book. Harrap, London, 1939. The eternal triangle, pp. 64 & 213. Take a triangle, ABC, with midpoints a, b, c, opposite A, B, C. Take a point d between a and B. Draw Aa, ab, bc, ca, bd, cd. How many triangles? Answer is given as 24, but I (and my program) find 27 and others have confirmed this.

Anon. Test your eyes. Mathematical Pie 7 (Oct 1952) 51. Reproduced in: Bernard Atkin, ed.; Slices of Mathematical Pie; Math. Assoc., Leicester, 1991, pp. 15 & 71 (not paginated - I count the TP as p. 1). Triangular pattern with 2 triangles on a side, with the three altitudes drawn. Answer is 47 'obtained by systematic counting'. This is correct. Cf Hancox, 1978.

W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. How many triangles, pp. 43 & 130. Take a pentagon and draw the pentagram inside it. In the interior pentagon, draw another pentagram. How many triangles are there? Answer is 85.

Young World. c1960. P. 57: One for Pythagoras. Consider a L-tromino. Draw all the midlines to form 12 unit squares. Or take a 4 x 4 square array and remove a 2 x 2 array from a corner. Now draw the two main diagonals of the 4 x 4 square - except half of one diagonal would be outside our figure. How many triangles and how many squares are present? Gives correct answers of 26 & 17.

J. Halsall. An interesting series. MG 46 (No. 355) (Feb 1962) 55 56. Larsen (below) says he seems to be the first to count the triangles in the triangular pattern with n on a side, but he does not give any proof.

Although there are few references before this point, the puzzle idea was pretty well known and occurs regularly. E.g. in the children's puzzle books of Norman Pulsford which start c1965, he gives various irregular patterns and asks for the number of triangles or squares.

J. E. Brider. A mathematical adventure. MTg (1966) 17 21. Correct derivation for the number of triangles in a triangle. This seems to be the first paper after Halsall but is not in Larsen.

G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Prob. 2/12, pp. 23 & 75. Consider an isosceles right triangle with legs along the axes from (0,0) to (4,0) and (0,4). Draw the horizontals and verticals through the integer lattice points, except that the lines through (1,1) only go from the legs to this point and stop. Draw the diagonals through even-integral lattice points, e.g. from (2,0) to (0,2). How many triangles. Says he found 27, but his secretary then found 29. I find 29.

Ripley's Puzzles and Games. 1966. Pp. 72-73 have several problems of counting triangles.

Item 3. Consider a Star of David with the diameters of its inner hexagon drawn. How many triangles are in it? Answer: 20, which I agree with.

Item 4. Consider a 3 x 3 array of squares with their diagonals drawn. How many triangles are there? Answer: 150, however, there are only 124.

Item 5. Consider five squares, with their midlines and diagonals drawn, formed into a Greek cross. How many triangles are there? Answer: 104, but there are 120.

Doubleday - 2. 1971. Count down, pp. 127-128. How many triangles in the pentagram (i.e. a pentagon with all its diagonals)? He says 35.

Gyles Brandreth. Brandreth's Bedroom Book. Eyre Methuen, London, 1973. Triangular, pp. 27 & 63. Count triangles in an irregular pattern.

[Henry] Joseph & Lenore Scott. Master Mind Brain Teasers. 1973. Op. cit. in 5.E. An unusual star, pp. 49-50. Consider a pentagram and draw lines from each star point through the centre to the opposite crossing point. How many triangles? They say 110.

[Henry] Joseph and Lenore Scott. Master Mind Pencil Puzzles. 1973. Op. cit. in 5.R.4.

Diamonds are forever, pp. 35-36. Hexagon with Star of David inside and another Star of David in the centre of that one. How many triangles? Answer is 76.

Count the triangles, pp. 55-56. Ordinary Greek cross of five squares, with all the diagonals and midlines of the five squares drawn. How many triangles> Answer is 104.

C. P. Chalmers. Note 3353: More triangles. MG 58 (No. 403) (Mar 1974) 52 54. How many triangles are determined by N points lying on M lines? (Not in Larsen.)

Nicola Davies. The 2nd Target Book of Fun and Games. Target (Universal-Tandem), London, 1974. Squares and triangles, pp. 18 & 119. Consider a chessboard of 4 x 4 cells. Draw all the diagonals, except the two main ones. How many squares and how many triangles?

Shakuntala Devi. Puzzles to Puzzle You. Op. cit. in 5.D.1. 1976. Prob. 136: The triangles, pp. 85 & 133. How many triangles in a Star of David made of 12 equilateral triangles?

Michael Holt. Figure It Out -- Book Two. Granada, London, 1978. Prob. 67, unpaginated. How many triangles in a Star of David made of 12 equilateral triangles?

Putnam. Puzzle Fun. 1978. No. 91: Counting triangles, pp. 12 & 37. Same as Doubleday - 2.

D. J. Hancox, D. J. Number Puzzles For all The Family. Stanley Thornes, London, 1978.

Puzzle 8, pp. 2 & 47. Draw a line with five points on it, say A, B, C, D, E, making four segments. Connect all these points to a point F on one side of the line and to a point G on the other side of the line, with FCG collinear. How many triangles are there? Answer is 24, which is correct.

Puzzle 53, pp. 24 & 54. Same as Anon.; Test Your Eyes, 1952. Answer is 36, but there are 47.

The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex, 1984.

Problem 40, with Solution at the back of the book. Same as Doubleday - 2.

Problem 116, with Solution at the back of the book. Count the triangles in a 'butterfly' pattern.

Sue Macy. Mad Math. The Best of DynaMath Puzzles. Scholastic, 1987. (Taken from Scholastic's DynaMath magazine.) Shape Up, pp. 5 & 56.

Take a triangle, trisect one edge and join the points of trisection to the opposite vertex. How many triangles? [More generally, if one has n points on a line and joins them all to a vertex, there are 1 + 2 + ... + n-1 = n(n-1)/2 triangles.]

Take a triangle, join up the midpoints of the edges, giving four smaller triangles, and draw one altitude of the original triangle. How many triangles?

1980 Celebration of Chinese New Year Contest Problem No. 5; solution by Leroy F. Meyers. CM 17 (1991) 2 & 18 (1992) 272-273. n x n array of squares with all diagonals drawn. Find the number of isosceles right triangles. [Has this also been done in half the diagram? That is, how many isosceles right triangles are in the isosceles right triangle with legs going from (0,0) to (n,0) and (0,n) with all verticals, horizontals and diagonals through integral points drawn?]

Mogens Esrom Larsen. The eternal triangle -- a history of a counting problem. Preprint, 1988. Surveys the history from Halsall on. The problem was proposed at least five times from 1962 and solved at least ten times. I have sent him the earlier references.

Marjorie Newman. The Christmas Puzzle Book. Hippo (Scholastic Publications), London, 1990. Star time, pp. 26 & 117. Consider a Star of David formed from 12 triangles, but each of the six inner triangles is subdivided into 4 triangles. How many triangles in this pattern? Answer is 'at least 50'. I find 58.

Erick Gooding. Polygon counting. Mathematical Pie No. 131 (Spring 1994) 1038 & Notes, pp. 1-2. Consider the pentagram, i.e. the pentagon with its diagonals drawn. How many triangles, quadrilaterals and pentagons are there? Gets 35, 25, 92, with some uncertainty whether the last number is correct.

When F. R. Gill (See Pearson and Collins above) mentioned the problem of counting the triangles in the figure with all the altitudes drawn, I decided to try to count them myself for the figure with N intervals on each side. The theoretical counting soon gets really messy and I adapted my program for counting triangles in a figure (developed to verify the number found for August's problem). However, the number of points involved soon got larger than my simple Basic could handle and I rewrote the program for this special case, getting the answers of 653 and 1196 and continuing to N = 22. I expected the answers to be like those for the simpler triangle counting problem so that there would be separate polynomials for the odd and even cases, or perhaps for different cases (mod 3 or 4 or 6 or 12 or ??). However, no such pattern appeared for moduli 2, 3, 4 and I did not get enough data to check modulus 6 or higher. I communicated this to Torsten Sillke and Mogens Esrom Larsen. Sillke has replied with a detailed answer showing that the relevant modulus is 60! I haven't checked through his work yet to see if this is an empirical result or he has done the theoretical counting.

Heather Dickson, Heather, ed. Mind-Bending Challenging Optical Puzzles. Lagoon Books, London, 1999, pp. 40 & 91. Gives the version m = n = 4 of the following. I have seen other versions of this elsewhere, but I found the general solution on 4 Jul 2001 and am submitting it as a problem to AMM.

Consider a triangle ABC. Subdivide the side AB into m parts by inserting m 1 additional points. Connect these points to C. Subdivide the side AC into n parts by inserting n-1 additional points and connect them to B. How many triangles are in this pattern? The number is [m²n + mn²]/2. When m = n, we get n³, but I cannot see any simple geometric interpretation for this.