Wikipedia

Nine dots puzzle

The nine dots puzzle is a mathematical puzzle whose task is to connect nine squarely arranged points with a pen by four (or fewer) straight lines without lifting the pen.

The "nine dots" puzzle. The puzzle asks to link all nine dots using four straight lines or fewer, without lifting the pen.

The puzzle has appeared under various other names over the years.

History edit

In 1867, in the French chess journal Le Sphinx, an intellectual precursor to the nine dots puzzle appeared credited to Sam Loyd.[1][2] Said chess puzzle corresponds to a "64 dots puzzle", i.e., marking all dots of an 8-by-8 square lattice, with an added constraint.[a]

 
The Columbus Egg Puzzle from The Strand Magazine, 1907

In 1907, the nine dots puzzle appears in an interview with Sam Loyd in The Strand Magazine:[4][2]

"[...] Suddenly a puzzle came into my mind and I sketched it for him. Here it is. [...] The problem is to draw straight lines to connect these eggs in the smallest possible number of strokes. The lines may pass through one egg twice and may cross. I called it the Columbus Egg Puzzle."

In the same year, the puzzle also appeared in A. Cyril Pearson's puzzle book. It was there named a charming puzzle and involved nine dots.[5][2]

Both versions of the puzzle thereafter appeared in newspapers. From at least 1908, Loyd's egg-version ran as advertising for Elgin Creamery Co in Washington, DC., renamed to The Elgin Creamery Egg Puzzle.[6] From at least 1910, Pearson's "nine dots"-version appeared in puzzle sections.[7][8][9]

 
Christopher Columbus' Egg Puzzle in Sam Loyd's Cyclopedia of Puzzles, 1914

In 1914, Sam Loyd's Cyclopedia of Puzzles is published posthumously by his son (also named Sam Loyd).[10] The puzzle is therein explained as follows:[11][2]

The funny old King is now trying to work out a second puzzle, which is to draw a continuous line through the center of all of the eggs so as to mark them off in the fewest number of strokes. King Puzzlepate performs the feat in six strokes, but from Tommy's expression we take it to be a very stupid answer, so we expect our clever puzzlists to do better; [...]

Sam Loyd's naming of the puzzle is an allusion to the story of Egg of Columbus.[12]

Solution edit

 
One solution of the nine dots puzzle.

It is possible to mark off the nine dots in four lines.[13] To do so, one goes outside the confines of the square area defined by the nine dots themselves. The phrase thinking outside the box, used by management consultants in the 1970s and 1980s, is a restatement of the solution strategy. According to Daniel Kies, the puzzle seems hard because we commonly imagine a boundary around the edge of the dot array.[14]

The inherent difficulty of the puzzle has been studied in experimental psychology.[15][16]

Changing the rules edit

Various published solutions break the implicit rules of the puzzle in order to achieve a solution with even fewer than four lines. For instance, if the dots are assumed to have some finite size, rather than to be infinitesimally-small mathematical grid points, then it is possible to connect them with only three slightly-slanted lines. Or, if the line is allowed to be arbitrarily thick, then one line can cover all of the points.[17]

Another way to use only a single line involves rolling the paper into a three-dimensional cylinder, so that the dots align along a single helix (which, as a geodesic of the cylinder, could be considered to be in some sense a straight line). Thus a single line can be drawn connecting all nine dots—which would appear as three lines in parallel on the paper, when flattened out.[18] It is also possible to fold the paper flat, or to cut the paper into pieces and rearrange it, in such a way that the nine dots lie on a single line in the plane.[17]

 
Three lines connect thick dots
 
One-line rolled cylinder

Planar generalization edit

 
Cyclic solutions for the 4-by-4 version [19]

Instead of the 3-by-3 square lattice, generalizations have been proposed in the form of the least amount of lines needed on an n-by-n square lattice. Or, in mathematical terminology, the minimum-segment unicursal polygonal path covering an n × n array of dots.

Various such extensions were stated as puzzles by Dudeney and Loyd with different added constraints.[20]

In 1955, Murray S. Klamkin showed that if n > 2, then 2n − 2 line segments are sufficient and conjectured that it is necessary too.[21][20] In 1956, the conjecture was proven by John Selfridge.[22][20][2]

In 1970, Solomon W. Golomb and John Selfridge showed that the unicursal polygonal path of 2n − 2 segments exists on the n × n array for all n > 3 with the further constraint that the path be closed, i.e., it starts and ends at the same point.[20] Moreover, the further constraint that the closed path remain within the convex hull of the array of dots can be satisfied for all n > 5. Finally, various results for the a × b array of dots are proven.[3]

Generalization to higher dimensions edit

The nine dots puzzle can be generalized to higher dimensions, by simply defining the grid   and asking to find (one of) the shortest polygonal chain(s) (i.e., the polygonal chain with the fewest edges) that visits all those   points.

In 2020, it has been proven[23] that the optimal solution for the above-mentioned problem has exactly   links, for every positive integer  . The given constructive solution is a straight generalization of the well-known covering trail that solves the   case.

Moreover, it is even possible to further extend the given optimization problem to any arbitrary grid  , but only the easiest cases have been solved at present (e.g., in 2022 Rinaldi and Ripà showed that the solution for the   case is a trail/cycle of length   for every  , while it is known that the optimal solution for the   case has  , or  , or   links).

The Nine Dots Prize edit

The Nine Dots Prize, named after the puzzle,[24] is a competition-based prize for "creative thinking that tackles contemporary societal issues."[25] It is sponsored by the Kadas Prize Foundation and supported by the Cambridge University Press and the Centre for Research in the Arts, Social Sciences and Humanities at the University of Cambridge.[26]

See also edit

Notes edit

  1. ^ As it turned out later, the added constraint can be dropped, that is, moving the pen only akin to the queen in chess (i.e., vertically, horizontally, or diagonally) and only staying within the square lattice. Even without this constraint, the optimal solution is still 14 moves.[3]

References edit

  1. ^ Journoud, Paul (1867). "Questions Du Sphinx". Le Sphinx: Journal des échecs (in French). 2 (14): 216. Placer la Dame ot l'on voudra, lui faire parcourir par des marches suivies et régulières toutes les cases de I'échiquier, et la ramener au quatorzième coup à son point de départ. Place the queen wherever you want, make her go through all the squares of the chessboard by regular steps, and bring her back to her starting point at the fourteenth move.
  2. ^ a b c d e Singmaster, David (2004-03-19). "Sources In Recreational Mathematics, An Annotated Bibliography (8th preliminary edition): 6.AK. Polygonal Path Covering N X N Lattice Of Points, Queen's Tours, etc". www.puzzlemuseum.com.
  3. ^ a b Golomb, Solomon W.; Selfridge, John L. (1970). "Unicursal Polygonal Paths And Other Graphs On Point Lattices". Pi Mu Epsilon Journal. 5 (3): 107–117. ISSN 0031-952X. JSTOR 24344915.
  4. ^ Bain, George Grantham (1907). "The Prince of Puzzle-Makers. An Interview with Sam Loyd". The Strand Magazine. p. 775.
  5. ^ Pearson, A. Cyril Pearson (1907). The Twentieth Century Standard Puzzle Book. p. 36.
  6. ^ "Advertising for Elgin Creamery Co". Evening star. Washington, D.C. 1908-03-02. p. 6.
  7. ^ "Three Puzzles Are Amusing". The North Platte semi-weekly tribune. North Platte, Nebraska. 1910-05-20. p. 7.
  8. ^ "Three Puzzles are Amusing". Audubon County journal. Exira, Iowa. 1910-07-14. p. 2.
  9. ^ "After Dinner Tricks". The Richmond palladium and sun-telegram. Richmond, Indiana. 1922-06-22. p. 6.
  10. ^ Gardner, Martin (1959). "Chapter 9: Sam Loyd: America's Greatest Puzzlist". Mathematical puzzles & diversions. New York, N.Y.: Simon and Schuster. pp. 84, 89.
  11. ^ Sam Loyd, Cyclopedia of Puzzles. (The Lamb Publishing Company, 1914)
  12. ^ Facsimile from Cyclopedia of Puzzles - Columbus's Egg Puzzle is on right-hand page
  13. ^ "Sam Loyd's Cyclopedia of 5000 Puzzles, Tricks, and Conundrums With Answers". 1914. p. 380.
  14. ^ Daniel Kies, "English Composition 2: Assumptions: Puzzle of the Nine Dots", retr. Jun. 28, 2009.
  15. ^ Maier, Norman R. F.; Casselman, Gertrude G. (1 February 1970). "Locating the Difficulty in Insight Problems: Individual and Sex Differences". Psychological Reports. 26 (1): 103–117. doi:10.2466/pr0.1970.26.1.103. PMID 5452584. S2CID 43334975.
  16. ^ Lung, Ching-tung; Dominowski, Roger L. (1 January 1985). "Effects of strategy instructions and practice on nine-dot problem solving". Journal of Experimental Psychology: Learning, Memory, and Cognition. 11 (4): 804–811. doi:10.1037/0278-7393.11.1-4.804.
  17. ^ a b Tybout, Alice M. (1995). "Presidential Address: The Value of Theory in Consumer Research". In Kardes, Frank R.; Sujan, Mita (eds.). Advances in Consumer Research, Volume 22. Provo, Utah: Association for Consumer Research. pp. 1–8.
  18. ^ W. Neville Holmes, Fashioning a Foundation for the Computing Profession, July 2000
  19. ^ Rob Eastaway, Thinking outside outside the box, Chalkdust Magazine, 2018-03-12
  20. ^ a b c d Dudeney, Henry; Gardner, Martin (1967). "536 Puzzles And Curious Problems". p. 376.
  21. ^ Klamkin, M. S. (1955-02-01). "Polygonal Path Covering a Square Lattice (E1123)". The American Mathematical Monthly. 62 (2): 124. doi:10.2307/2308156. JSTOR 2308156.
  22. ^ Selfridge, John (June 1955). "Polygonal Path Covering a Square Lattice (E1123, Addentum)". The American Mathematical Monthly. 62 (6): 443. doi:10.2307/2307008. JSTOR 2307008.
  23. ^ Ripà, Marco (2020). "Solving the 106 years old 3^k Points Problem with the clockwise-algorithm". Journal of Fundamental Mathematics and Applications. 3 (2): 84–97. arXiv:2401.10163. doi:10.14710/jfma.v3i2.8551.
  24. ^ "The Nine Dots Prize Identity". Rudd Studio. Retrieved 19 November 2018.
  25. ^ "Home". The Nine Dots Prize. Kadas Prize Foundation. Retrieved 19 November 2018.
  26. ^ "Nine Dots Prize". CRASSH. The University of Cambridge. Retrieved 19 November 2018.
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