Matchstick Puzzles:                  Torsten Sillke, 1999-06-07
                                         last update 2001-03-01

problem A: [Tromholt 1889 2nd, problem 161 p65, solution p107]
           and often reprinted [Brymay, problem 10 p11 solution p22]
           [Janzen 1955, p62][Brooke 1973, p7, 42][Obermair 1975, p48, 130]
           [Para 1978, problem 54, problem 274][Vogt 1985, problem 12]
           [Wells 1992, p157][Post 1940]
 Build 2 squares and 4 right trianlges with 8 matches.

problem B: [Tromholt 1909, problem 174 p85, solution 114 p167]
 Build 2 regular 5-gons and 5 isosceles trianlges with 10 matches.

problem C: [Sperling p61-62, 153][Sillke]
 Build 2 regular 6-gons and 6 trianlges with 12 matches.

problem D: [Sillke]
 Build 2 regular 12-gon and 12 right triangles with 36 matches.




standard solution A:

 A . . . . D     the matches are at
 . . . G . .     AB, BC, CD, AD
 . H . . . .     AE, BF, CG, DH
 . . . . F .
 . . E . . .
 B . . . . C

How large is the inner square, or is this solution valid?
If a triangle is isosceles and right the third side must be 0.
As this problem did not appear in [Tromholt 1909] it seems to me that 
Tromholt noticed that the solution is not good and invented problem B.

So let us relax the constraints and try to build
2 quadrangles and 4 isosceles triangles in this form.
But all angles of the inner quadrangle must be bigger than
a right angle. Therefore this is impossible too unless
all triangles have one side of length 0.
(Let alpha = angle(ADH), beta = angle(DAH) = angle(DHA) then
 alpha + 2*beta = pi. If alpha > 0 then beta = (pi-alpha)/2 < pi/2 and
 therefore angle(GHE) = pi - beta = alpha + beta = (pi+alpha)/2 > pi/2.)
Note that [Michael Schuyt 1989, problem 21.1] asks for isosceles triangles.

solution B:

 Given a regular 5-gon. Draw all diagonals. In the middle there will
 be a smaller regular 5-gon. Place the inner matches along the 
 diagonals you get 5 isosceles triangles with angles 36, 72, 72 degree.
 Note that that problem A seems to be derived from this.
 (The figure given by Tromholt shows a smaller inner pentagon.)

solution C:

 A trick solution:

    A   F      Build 6 equilateral triangles as shown on the left.
               At the center point G you will create a hexagon
  B   G   E    with with the base width of the matches.

    C   D


References:

- Gyles Brandreth;
  Games and Puzzles with Coins and Matches,
  Carousel Books, London 1976, p 53, 101 

- Maxey Brooke;
  Tricks, Games and Puzzles with Matches, New York, 1973

- Brymay (series lable);
  The 'Brymay' Puzzle Book; Fun among the Matches,
  Bryant and May, Limited (undated, but probably early 19th century)

- Martin Gardner;
  Mathematical Circus,
  Alfred A. Knopf (1979) New York
  Chap 2. Matches
   problem 2: 6 matches made 4 regions (in the plane)

- Willi Janzen;
  Denksport: Kniffe und Knobeleien, Listiges und Lustiges,
  Freidrich M. H\"orhold Verlag, Hildesheim, 1952

- Gilbert Obermair; 
  Streichholzspielereien, M\"unchen, 1975

- Karl-Heinz Paraquin;
  Denkspielebuch, Ravensburger 1973

- Para (Karl-Heinz Paraquin);
  Denkspiele, Ravensburger Taschenb\"ucher 466, 1978
  ISBN 3-473-39466-1

- Gr\"une Post (Hrsg.);
  Lustiger Zeitvertreib,
  Deutscher Verlag Berlin, ca. 1940, S. 33, 60
  (Die Gr\"une Post war eine Sonntagszeitung, die 
  von 1927 bis 1944 in Berlin erschien.)

- Walter Sperling 
  Am\"usanter Zeitvertreib: Neue Denksportaufgaben f\"ur jung und alt 
  Albert M\"uller Verlag 
  R\"schlikon-Z\"rich 
  p 61-62, 153. 

- Michael Schuyt;
  Phantastische Z\"undholzspiele,
  DuMont Buchverlag, K\"oln, 1989, ISBN 3-7701-2362-X
  
- Sophus Tromholt;
  Streichholzspiele,
  2nd edition, Otto Spamer Verlag, Leipzig, 1889 (Preface: Dresden, August 1889)
  14th edition, Leipzig, 1909
  (reprint 14th: Zentralantiquariat der DDR, Leipzig, 1986, ISBN 3-7463-0021-5,
            preface by R\"udiger Thiele, spelling adapted)

- Dieter Vogt;
  Streichholzspiele,
  Insel Taschenbuch 797, Frankfurt/M 1985, ISBN 3-458-32497-6

- David Wells; 
  The Penguin Book of Curious and Interesting Puzzles,
  Penguin Books, London 1992, p. 157, 346. 

- Toothpick Would;
  http://www.madras.fife.sch.uk/maths/
  43 match puzzles (does not contain this problem)

Thanks:
 Many thanks for Heinrich Hemme and Jerry Slocum for looking 
 for problem A and B in their puzzle book collections.

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mailto:Torsten.Sillke@uni-bielefeld.de
http://www.mathematik.uni-bielefeld.de/~sillke/