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Thinking outside the box

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"Outside the box" redirects here. For other uses, see Outside the Box (disambiguation). Not to be confused with Out-of-the-box functionality.
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Thinking outside the box (also thinking out of the box[1][2] or thinking beyond the box and, especially in Australia, thinking outside the square[3]) is a metaphor that means to think differently, unconventionally, or from a new perspective. The phrase also often refers to novel or creative thinking.

History

The origin of the phrase is unclear. "Think beyond the boundary"-metaphors, that is, metaphors that allude to think differently or with less constraints, seem to have an old history. For example, in 1888, The Annual Register records the phrase think outside the lines.[4][5]

Since at least 1954, the nine dots puzzle (see below) has been used as a metaphor of the type "think beyond the boundary". Early phrasings include go outside the dots (1954),[6][7] breakthrough thinking that gets outside the nine-dot square (1959),[8][9] and what are the actual boundaries of the problem? (1963).[10][7]

In 1969, Norman Vincent Peale writes this in an article for the Chicago Tribune, quote:[11][12]

The "nine dots" puzzle. The goal of the puzzle is to link all 9 dots using four straight lines or less, without lifting the pen. The solution can be found further down in this article.
There is one particular puzzle you may have seen. It's a drawing of a box with some dots in it, and the idea is to connect all the dots by using only four lines. You can work on that puzzle, but the only way to solve it is to draw the lines so they connect outside the box. It's so simple once you realize the principle behind it. But if you keep trying to solve it inside the box, you'll never be able to master that particular puzzle.
That puzzle represents the way a lot of people think. They get caught up inside the box of their own lives. You've got to approach any problem objectively. Stand back and see it for exactly what it is. From a little distance, you can see it a lot more clearly. Try and get a different perspective, a fresh point of view. Step outside the box your problem has created within you and come at it from a different direction.
All of a sudden, just like the puzzle, you'll see how to handle your problem. And just like the four lines that connect all the dots, you'll discover the course of action that's just right in order to set your life straight.

In 1970, the phrase think outside the dots appears without mentioning the nine dots puzzle.[13][7]

Finally, in 1971, the specific phrase think outside the box is attested, again appearing together with the nine dots puzzle.[14][15] In 1976, the phrase is used in England[16] and 1978 in the USA[17], both without mentioning the nine dots puzzle.

Beyond the above attestations, there are several unconfirmed accounts of how the phrase got introduced. According to Martin Kihn, it goes back to management consultants in the 1970s and 1980s challenging their clients to solve the "nine dots" puzzle.[18] According to John Adair, he introduced the nine dots puzzle in 1969, from which the saying comes.[19] According to The Creative Thinking Association of America, Mike Vance popularized the phrase "thinking out of the box".[20] Moreover, it is claimed that the use of the nine-dot puzzle in consultancy circles stems from the corporate culture of the Walt Disney Company, where the puzzle was used in-house.[citation needed]

Nine dots puzzle

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

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

The puzzle has appeared under various other names over the years. Currently, the most popular name appears to be nine dots puzzle.[citation needed]

History

In 1867, in the French chess journal Le Sphinx, an intellectual precursor to the nine dots puzzle appeared credited to Sam Loyd.[21][22]. This 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]

In 1907, the nine dots puzzle appears in an interview with Sam Loyd in The Strand Magazine:[24][22]

The Columbus Egg Puzzle as it appeared in The Strand Magazine in 1907.
“[...] 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.[25][22]

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.[26] From at least 1910, Pearson's "nine dots"-version appeared in puzzle sections.[27][28][29]

Christopher Columbus' Egg Puzzle as it appeared in Sam Loyd's Cyclopedia of Puzzles

In 1914, Sam Loyd's Cyclopedia of Puzzles is published posthumously by his son (also named Sam Loyd).[30] The puzzle is therein explained as follows:[31][22]

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.[32]

In the 1941 compilation The Puzzle-Mine: Puzzles Collected from the Works of the Late Henry Ernest Dudeney, the puzzle is attributed to Dudeney himself and not Loyd.[33][page needed]

Solution

One solution of the nine dots puzzle.

It is possible to mark off the nine dots in four lines.[34] To do so, one goes outside the confines of the square area defined by the nine dots themselves. The phrase "thinking outside the box" 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.[35]

The inherent difficulty of the puzzle has been studied in experimental psychology.[36][37]

Generalization

If, instead of the 3-by-3 square lattice, we consider the n-by-n square lattice, then what is the least amount of lines needed to connect the dots without lifting the pen? Or, stated in mathematical terminology, what is the minimum-segment unicursal polygonal path covering the n × n array of dots?

Various such extensions were stated as puzzles by Dudeney and Loyd with different added constraints.[38]

In 1955, Murray S. Klamkin showed that if n > 2, then 2n - 2 line segments are sufficient and conjectured that it's necessary too.[39][38] In 1956, the conjecture was proven by John Selfridge.[40][38][22]

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.[38] 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.[23]

The single straight line solution

Solution by rolling the paper

Thinking outside of the sheet of paper, we can connect the dots with a single straight line. To do so, cone the paper three-dimensionally, aligning the dots along a spiral, 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.[41]

The Nine Dots Prize

The Nine Dots Prize, named after the puzzle,[42] is a competition-based prize for "creative thinking that tackles contemporary societal issues."[43] 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.[44]

See also

Notes

  1. ^ As it turned out later, the added constraint to only move the pen akin to the queen in chess (i.e., vertically, horizontally, or diagonally) can in fact be dropped. The optimal number of moves remains 14.[23]

References

  1. ^ "box - definition of box in English - Oxford Dictionaries". Retrieved 21 November 2016.
  2. ^ "think outside the box - Definition, meaning & more - Collins Dictionary". Retrieved 21 November 2016.
  3. ^ "Thinking Outside The Square". Retrieved 21 November 2016.
  4. ^ The Annual Register: a review of public events at home and abroad, for the year 1887. London: Rivingtons, Waterloo Place. 1888. p. 168. [Lord Hartington] said that [...] the Liberal party became a one-man party, which scarcely ventured to think outside the lines prescribed by its dictator.
  5. ^ O'Toole, Garson (2010-05-03). "Antedating of "Outside the Box"".
  6. ^ Anderson, John F. (1954-10-30). "Down to Earth". Dallas Morning News. p. 1. An instructor at M.I.T. began his course with a group of graduate students one day by walking to the blackboard and drawing nine dots in this fashion [...] We are not here to go through old routines. Don't let your thinking be contained in a small square of knowledge. Learn to go outside the dots and you may be the one to solve man's most puzzling problems.
  7. ^ a b c Wilton, David (2021-07-19). "think outside the box". Wordorigins.org.
  8. ^ Humphrey, Hal (1959-11-13). "'Breakthrough thinking' gets outside 9-dot square". Detroit Free Press. p. 41. One of our biggest advertising agencies [writes] "Breakthrough thinking is the fresh approach, the new concept, that gets outside the nine-dot square."
  9. ^ Tréguer, Pascal (2021-04-28). "'to think outside the box': meaning and origin". word histories.
  10. ^ Platzer, Norbert A. J. (1963-04-18). "Incentiveness, Motivation, Training Needs of a Scientist". The Springfield Union. p. 52. [...] the next aplitude for creative thinking [is] defining the problem. We must ask ourselves: "What are the actual boundaries of the problem?" Perhaps some of you have seen this little problem before. Here are nine dots [...]. This teaches us, we should avoid imposing limitations that are not in our problem, as I told you before in the cases of Kettering and Reppe.
  11. ^ Peale, Norman Vincent (1969-10-25). "Blackmail Is the Problem". Chicago Tribune. p. 13.
  12. ^ Liberman, Mark (2005-06-02). "Language Log: X-ing outside the Y".
  13. ^ Westell, Anthony (1970-05-23). "Canada Entering Big League In Research". Ottawa Journal. p. 7. The problem, says William Dav[i]d Hopper, is to think "outside the dots" about the questions of how to feed a hungry world. He means that the need is to think imaginatively, creatively, about the development of less-developed countries, and not merely to keep pouring more money and technology into patterns of foreign aid established, not very successfully, over the past 20 years.
  14. ^ Notaro, Michael R. (1971). "Management of Personnel: Organization Patterns and Techniques". Data Management. Vol. 9/#9. Data Processing Management Organization. p. 76.
  15. ^ "box, n.2". OED Online. Oxford University Press.
  16. ^ Hall, David (1976-07-18). "Ex officer's strategy for business success". Sunday Telegraph. p. 24. [...] it is abundantly clear that Service people can turn their hand to many jobs provided they can think "outside the box."
  17. ^ Robert S., Mendelsohn (1978-03-31). "People's Doctor". The Newark Advocate. p. 10. Some of my best teachers have been those who utilize the techniques of shock and surprise to rouse me out of conventional habits of thought, forcing me to question accepted teaching and stimulating me to think "outside the box."
  18. ^ Kihn, Martin. "'Outside the Box': the Inside Story," FastCompany 1995
  19. ^ Adair, John (2007). The art of creative thinking how to be innovative and develop great ideas. London Philadelphia: Kogan Page. p. 127. ISBN 9780749452186.
  20. ^ Biography of Mike Vance at Creative Thinking Association of America.
  21. ^ 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.
  22. ^ 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.
  23. ^ 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.
  24. ^ Bain, George Grantham (1907). "The Prince of Puzzle-Makers. An Interview with Sam Loyd". The Strand Magazine. p. 775.
  25. ^ Pearson, A. Cyril Pearson (1907). The Twentieth Century Standard Puzzle Book. p. 36.
  26. ^ "Advertising for Elgin Creamery Co". Evening star. Washington, D.C. 1908-03-02. p. 6.
  27. ^ "Three Puzzles Are Amusing". The North Platte semi-weekly tribune. North Platte, Nebraska. 1910-05-20. p. 7.
  28. ^ "Three Puzzles are Amusing". Audubon County journal. Exira, Iowa. 1910-07-14. p. 2.
  29. ^ "After Dinner Tricks". The Richmond palladium and sun-telegram. Richmond, Indiana. 1922-06-22. p. 6.
  30. ^ Gardner, Martin (1959). "Chapter 9: Sam Loyd: America's Greatest Puzzlist". Mathematical puzzles & diversions. New York, N.Y.: Simon and Schuster. pp. 84, 89.
  31. ^ Sam Loyd, Cyclopedia of Puzzles. (The Lamb Publishing Company, 1914)
  32. ^ Facsimile from Cyclopedia of Puzzles - Columbus's Egg Puzzle is on right-hand page
  33. ^ J. Travers, The Puzzle-Mine: Puzzles Collected from the Works of the Late Henry Ernest Dudeney. (Thos. Nelson, 1941)
  34. ^ "Sam Loyd's Cyclopedia of 5000 Puzzles, Tricks, and Conundrums With Answers". 1914. p. 380.
  35. ^ Daniel Kies, "English Composition 2: Assumptions: Puzzle of the Nine Dots", retr. Jun. 28, 2009.
  36. ^ 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.
  37. ^ 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.
  38. ^ a b c d Dudeney, Henry; Gardner, Martin (1967). "536 Puzzles And Curious Problems". p. 376.
  39. ^ Klamkin, M. S. (1955-02-01). "Polygonal Path Covering a Square Lattice (E1123)". The American Mathematical Monthly. 62 (2): 124. doi:10.2307/2308156.
  40. ^ Selfridge, John (June 1955). "Polygonal Path Covering a Square Lattice (E1123, Addentum)". The American Mathematical Monthly. 62 (6): 443. doi:10.2307/2307008.
  41. ^ W. Neville Holmes, Fashioning a Foundation for the Computing Profession, July 2000
  42. ^ "The Nine Dots Prize Identity". Rudd Studio. Retrieved 19 November 2018.
  43. ^ "Home". The Nine Dots Prize. Kadas Prize Foundation. Retrieved 19 November 2018.
  44. ^ "Nine Dots Prize". CRASSH. The University of Cambridge. Retrieved 19 November 2018.

Further reading

Look up outside the box in Wiktionary, the free dictionary.
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