The nine dots puzzle is a

mathematical puzzle Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that sati ...

whose task is to connect nine squarely arranged points with a pen by four (or fewer) straight lines without lifting the pen. The puzzle has appeared under various other names over the years.

History

In 1867, in the French

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to disti ...

journal ''Le Sphinx'', an intellectual precursor to the nine dots puzzle appeared credited to

Sam Loyd Samuel Loyd (January 30, 1841 – April 10, 1911), was an American chess player, chess composer, puzzle author, and recreational mathematics, recreational mathematician. Loyd was born in Philadelphia but raised in New York City. As a chess com ...

. Said chess puzzle corresponds to a "64 dots puzzle", i.e., marking all dots of an 8-by-8

square lattice In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by their ...

, with an added constraint. TheStrandMagazine1907bColombusEgg

In 1907, the nine dots puzzle appears in an interview with

The Strand Magazine ''The Strand Magazine'' was a monthly British magazine founded by George Newnes, composed of short fiction and general interest articles. It was published in the United Kingdom from January 1891 to March 1950, running to 711 issues, though the ...

: : " ..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. Both versions of the puzzle thereafter appeared in newspapers. From at least 1908, Loyd's egg-version ran as advertising for ''Elgin

Creamery A creamery is a place where milk and cream are processed and where butter and cheese is produced. Cream is separated from whole milk; pasteurization is done to the skimmed milk and cream separately. Whole milk for sale has had some cream re ...

Co'' in Washington, DC., renamed to ''The Elgin Creamery Egg Puzzle''. From at least 1910, Pearson's "nine dots"-version appeared in puzzle sections. Eggpuzzle

In 1914,

's ''Cyclopedia of Puzzles'' is published posthumously by his son (also named Sam Loyd). The puzzle is therein explained as follows: : 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 An egg of Columbus or Columbus' egg ( it, uovo di Colombo ) refers to a brilliant idea or discovery that seems simple or easy after the fact. The expression refers to an apocryphal story, dating from at least the 16th century, in which it is sai ...

.Facsimile from ''Cyclopedia of Puzzles'' - Columbus's Egg Puzzle is on right-hand page
/ref> In the 1941 compilation ''The Puzzle-Mine: Puzzles Collected from the Works of the Late

Henry Ernest Dudeney Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games. He is known as one of the country's foremost creators of mathematical puzzles. Early life ...

'', the puzzle is attributed to Dudeney himself and not Loyd.

Solution

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

thinking outside the box Thinking outside the box (also thinking out of the box or thinking beyond the box and, especially in Australia, thinking outside the square) is a metaphor that means to think differently, unconventionally, or from a new perspective. The phrase als ...

, 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. The inherent difficulty of the puzzle has been studied in

experimental psychology Experimental psychology refers to work done by those who apply experimental methods to psychological study and the underlying processes. Experimental psychologists employ human participants and animal subjects to study a great many topics, in ...

Changing the rules

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. Another way to use only a single line involves rolling the paper into a three-dimensional

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

, so that the dots align along a single

helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...

(which, as a

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

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. 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.

Generalization

If, instead of the 3-by-3

, we consider the -by- 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 Unicursal may refer to: * Eulerian path, a sequential set of edges within a graph that reach all nodes * Labyrinth, a unicursal maze * Unicursal curve, a curve which is birationally equivalent to a line * Unicursal hexagram The unicursal hexa ...

polygonal path In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...

covering the array of dots? Various such extensions were stated as puzzles by

Dudeney Dudeney is a surname. Notable people with the surname include: * Alice Dudeney (1866–1945), English writer, wife of Henry *Henry Dudeney Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who speci ...

and

Loyd Loyd may refer to: Places United States * Loyd, Colorado * Loyd, Illinois * Loyd, Louisiana * Loyd, Mississippi * Loyd, Wisconsin, unincorporated community People Given name * Loyd Auerbach, professor of parapsychology * Loyd Blankenship (bo ...

with different added constraints. In 1955, Murray S. Klamkin showed that if , then line segments are

sufficient In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

and conjectured that it is necessary too. In 1956, the conjecture was proven by

John Selfridge John Lewis Selfridge (February 17, 1927 – October 31, 2010), was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics. Education Selfridge received his Ph.D. in 195 ...

. In 1970, Solomon W. Golomb and

showed that the unicursal polygonal path of segments exists on the array for all with the further constraint that the path be ''closed'', i.e., it starts and ends at the same point. Moreover, the further constraint that the closed path remain within the

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

of the array of dots can be satisfied for all . Finally, various results for the array of dots are proven.

The Nine Dots Prize

The Nine Dots Prize, named after the puzzle, is a competition-based prize for "creative thinking that tackles contemporary societal issues." It is sponsored by the Kadas Prize Foundation and supported by the

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

and the

Centre for Research in the Arts, Social Sciences and Humanities The Centre for Research in the Arts, Social Sciences and Humanities (CRASSH) is an interdisciplinary research centre within the University of Cambridge. Founded in 2001, CRASSH came into being as a way to create interdisciplinary dialogue acros ...

at the

University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...

Notes

References

{{reflist Mathematical puzzles