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6.AO.1. PLACE FOUR POINTS EQUIDISTANTLY = MAKE FOUR

TRIANGLES WITH SIX MATCHSTICKS
I am adding the problem of making three squares with nine matchsticks here a it uses the same thought process -- see Mittenzwey and see the extended discussion at Anon., 1910.
Pacioli. De Viribus. c1500. Ff. 191r - 192r. LXXX. Do(cumento). commo non e possibile piu ch' tre ponti o ver tondi spere tocarse in un piano tutti (how it is not possible for more than three points or discs or spheres to all touch in a plane). = Peirani 252-253. Says you can only get three discs touching in the plane, but you can get a fourth so they are all touching by making a pyramid.

Endless Amusement II. 1826? Prob. 21, p. 200. "To place 4 poles in the ground, precisely at an equal distance from each other." Uses a pyramidal mound of earth.

Young Man's Book. 1839. P. 235. Identical to Endless Amusement II.

Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 6, p. 178 (1868: 189). Plant four trees at equal distances from each other.

Frank Bellew. The Art of Amusing. 1866. Op. cit. in 5.E. 1866: pp. 97-98 & 105-106; 1870: pp. 93 94 & 101 102.

Mittenzwey. 1880.

Prob. 161, pp. 32 & 84; 1895?: 184, pp. 37 & 86; 1917: 184, pp. 34 & 83. Use six sticks to make four congruent triangles. Solution is a rectangle (should be a square) with its diagonals, but then two of the sticks have to be longer than the others.

Prob. 163, pp. 32 & 84; 1895?: 186 & 194, pp. 37 & 86-87; 1917: 186 & 194, pp. 34 & 83-84. Use six equally long sticks to make four congruent triangles -- solution is a tetrahedron. The two problems in the 1895? are differently phrased, but identical in content, while the first solution is a picture and the second is a description.

Prob. 171, pp. 33 & 85; 1895?: 195, pp. 38 & 87; 1917: 195, pp. 34 & 84. Use nine equal sticks to make three squares. Solution is three faces of a cube.

F. Chasemore. Loc. cit. in 6.W.5. 1891. Item 3: The triangle puzzle, p. 572.

Hoffmann. 1893.

Chap. VII, no. 15, pp. 290 & 298 = Hoffmann-Hordern, pp. 195. Four matches.

Chap. X, no. 19: The four wine glasses, pp. 344 & 381 = Hoffmann-Hordern, pp. 238 239, with photo on p. 239 of a version by Jaques & Son, 1870-1900. I usually solve the second version by setting one glass on top of the other three, but here he wants the centre of the feet of the glasses to be equally spaced and he turns one glass over and places it in the centre of the other three, appropriately spaced.

Loyd. Problem 34: War ships at anchor. Tit Bits 32 (22 May & 12 Jun 1897) 135 & 193. Place four warships equidistantly so that if one is attacked, the others can come to assist it. Solution is a tetrahedron of points on the earth's oceans.

Parlour Games for Everybody. John Leng, Dundee & London, nd [1903 -- BLC], p. 30. "With 6 matches form 4 triangles of equal size."

Pearson. 1907. Part III, no. 77: Three squares, p. 77. Make three squares with nine matches. Solution is a triangular prism!

Anon. Prob. 66. Hobbies 31 (No. 781) (1 Oct 1910) 2 & (No. 784) (22 Oct 1910) 68. Use nine matches to make three squares. "... the only possible solution" is to make two adjacent squares with seven matches, then bisect each square to produce a third square which overlaps the other two.

I re-invented this problem in Apr 1999 and posted it on NOBNET on 19 Apr 1999. Solution (1) is the idea I had when I made up the puzzle, but various friends gave more examples and then I found solution (3).

(1). Arrange the nine matches to form the following.

_

|_| |_|

| |
Then 4 is a square, 9 is a square and 49 is a square.

(2). Use the matches to form a triangular prism. One may object that this also makes two triangles.

(3). Make three squares forming three faces of a cube, all meeting at one corner. Cf Mittenzwey 171.

(4). Make two adjacent squares with seven of the matches. Now bisect each of the squares with a match parallel to the common edge of the squares. This produces a row of four adjacent half-squares as below. The middle two form a new square. Here one may object that the squares are overlapping.

─── ───

│ │ │ │ │

─── ───

(5). Use the matches to make the figures 0, 1 and 4.

One can use the matches to make squares whose edge is half the match length, but one only needs eight matches to make three squares.

There are other solutions which use the fact that matches have squared off ends and have square cross-section, but these properties do not hold for paper matches torn from a matchbook or for other equivalent objects like toothpicks and hence I don't consider them quite reasonable.

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!!

Williams. Home Entertainments. 1914. Tricks with matches: To form four triangles with six matches, p. 106.

Blyth. Match-Stick Magic. 1921. Four triangle puzzle, p. 23. Make four triangles with six matchsticks.

King. Best 100. 1927. No. 59, pp. 24 & 53. = Foulsham's no. 20, pp. 8 & 12. Use six matches to make four triangles.