Thinking Outside The Box - Nine Dots Puzzle

The notion of something outside a perceived "box" is related to a traditional topographical puzzle called the nine dots puzzle.

The origins of the phrase "thinking outside the box" are obscure; but it was popularized in part because of a nine-dot puzzle, which John Adair claims to have introduced in 1969. Management consultant Mike Vance has 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.

The nine dots puzzle is much older than the slogan. It appears in Sam Loyd's 1914 Cyclopedia of Puzzles. In the 1951 compilation The Puzzle-Mine: Puzzles Collected from the Works of the Late Henry Ernest Dudeney, the puzzle is attributed to Dudeney himself. Sam Loyd's original formulation of the puzzle entitled it as "Christopher Columbus's egg puzzle." This was an allusion to the story of Egg of Columbus.

The puzzle proposed an intellectual challenge—to connect the dots by drawing four straight, continuous lines that pass through each of the nine dots, and never lifting the pencil from the paper. The conundrum is easily resolved, but only by drawing the lines 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. The puzzle only seems difficult because people commonly imagine a boundary around the edge of the dot array. The heart of the matter is the unspecified barrier which is typically perceived.

Ironically, telling people to "think outside the box" does not help them think outside the box, at least not with the 9-dot problem. This is due to the distinction between procedural knowledge (implicit or tacit knowledge) and declarative knowledge (book knowledge). For example, a non-verbal cue such drawing a square outside the 9 dots does allow people to solve the 9-dot problem better than average. However, a very particular kind of verbalization did indeed allow people to solve the problem better than average. This is to speak in a non-judgmental, free association style. These were the instructions in a study which showed facilitation in solving the 9-dot problem:

While solving the problems you will be encouraged to think aloud. When thinking aloud you should do the following: Say whatever’s on your mind. Don’t hold back hunches, guesses, wild ideas, images, plans or goals. Speak as continuously as possible. Try to say something at least once every five seconds. Speak audibly. Watch for your voice dropping as you become involved. Don’t worry about complete sentences or eloquence. Don’t over explain or justify. Analyze no more than you would normally. Don’t elaborate on past events. Get into the pattern of saying what you’re thinking about now, not of thinking for a while and then describing your thoughts. Though the experimenter is present you are not talking to the experimenter. Instead, you are to perform this task as if you are talking aloud to yourself.

Considering a generalization of the same puzzle to a nxn grid (n>3), we need at least 2n-2 straight lines to fit all the dots without lifting the pencil from the paper.