Puzzles and mathematics

SUMMARY: Because problem solving is a core activity of mathematics, it lends itself well to puzzles.

A puzzle is a question, problem, or contrivance designed to challenge and expand the mind and perhaps test ingenuity. Puzzles have been found in virtually all cultures and historical periods, even in mythology. According to legend, the Sphinx prevented anyone from entering Thebes who failed to find the correct answer to the question: What is it that has four feet in the morning, two at noon, and three at twilight?

Mathematicians have long created puzzles and explored their solutions for research and applications. They have also created puzzles for purely recreational purposes. Teachers in many subjects within and outside mathematics use puzzles in the classroom.

There are a number of ways in which words and arrangements of letters or objects are used to create puzzles. Some problems in the Rhind Mathematical Papyrus (1650 BCE) are seen as puzzles. One example is a rhyme that also appears in Leonardo Pisano Fibonacci’s 1202 work Liber Abaci and is still popular today. Here is a modern version:

As I was going to St. Ives,

I met a man with seven wives.

Each wife had seven sacks,

Each sack had seven cats,

Each cat had seven kits.

Kits, cats, sacks, wives,

How many were going to St. Ives?

One may only assume that the narrator was going to St. Ives, not necessarily the other travelers. Mathematically, logic, branching diagrams, multiplication, and addition can be used to determine the final solution.

The Crossing Problem is traditional in several cultures, namely in Africa. The following is a version from Alcuin of York (735–804):

A man wishes to ferry a wolf, a goat, and a cabbage across a river in a boat that can carry only the man and one of the others at a time. He cannot leave the goat alone with the wolf nor leave the goat alone with the cabbage on either bank. How will he safely manage to carry all of them across the river?

To solve this problem, one must recognize that the man may carry an item back and forth across the river as many times as needed and ultimately find appropriate combinations and sequencing. Dynamic versions of this game appear online, adding visual and tactile components to the solving process. Extensions of this problem include adding more items to the list, increasing the size of the boat to carry more items, and adding an island in the middle of the river where objects may be placed. Mathematicians such as Luca Pacioli (1447–1517), Niccolo Tartaglia (1499–1557), Claude-Gaspar Bachet (1581–1638), and Edouard Lucas (1842–1891) investigated this problem. A well-known medieval task consisted of arranging men in a circle so that when every k-th man is removed, the remainder shall be a certain specified man. Several authors commented on this, from Girolamo Cardano (1501–1576) in the sixteenth century to Donald Coxeter (1907–2003) in the twentieth century.

Word Puzzles

Anagrams have a long and mysterious history, being seen as a source of ludic pleasure, but are also believed by some to possess mystic powers. Inside a word or phrase, another one is hiding that one can get by permuting the letters in a different order. For instance, the letters in the word “schoolmaster” may be rearranged to form the related phrase “the classroom.”

Lewis Carroll (1832–1898) invented a forerunner of the crossword: the “doublet.” There are two words presented to the solver, who is required to change one word to the other by replacing only one letter at a time, forming a legitimate word with each transformation. One of his examples is to change “HEAD” into “TAIL,” which can be done via the following sequence: “HEAL,” “TEAL,” “TELL,” “TALL,” and “TAIL.”

Visual Puzzles

Visual puzzles, such as optical illusions, are also popular, and they have long been investigated by mathematicians. Some of these address mathematical questions in disciplines like geometry and visualization, including figures that appear to be impossible.

Samuel Loyd (1841–1911) is referred to by some as “America’s greatest puzzlist.” He reputedly created thousands of puzzles. Some of his inventions were very original, like the Get Off the Earth puzzle. There are 13 men in the figure on the left. Rotating the puzzle, as shown in the figure on the right, produces a drawing that has 12 men. What happened to the 13th man?

Arithmetic Puzzles

Numerical relations and arithmetical principles are often found in puzzles. “Magic squares,” which are square arrays of consecutive numbers with constant sum in columns, rows, and diagonals, illustrate this clearly. One of the oldest, the Chinese lo-shu, dates back thousands of years. Leonhard Euler’s (1707–1783) work on Latin Squares, which are arrays of symbols with no repetitions in rows or columns, is one of the foundations of Sudoku puzzles, which appeared in a U.S. magazine in the 1970s but became famous first in Japan and then in the world. Tartaglia (1500–1557) presented the following numerical problem: A dying man leaves 17 horses to be divided among his three sons in the proportion 1/2 : 1/3 : 1/9. Can the brothers carry out their father’s will? Since 17 is not a multiple of 2, 3, or 9, there is no solution that would give all of the sons a whole number of horses.

Some authors shared problems, even if they lived in different centuries. Fibonacci (1170–1250), Tartaglia, and Bachet (1581–1638) all investigated the question:

If you have a balance, what is the least number of weights necessary to weigh any integer number of pounds from 1 to 40? (Assume you can put weights in either side of the balance.)

“Cryptarithms,” created for training the calculating mind in 1913, were very popular in the twentieth century. In a cryptarithm, one is asked to find the digits erased from a valid calculation. Later, prolific English puzzle inventor Henry Dudeney (1857–1930) substituted letters for the unknown numbers to create another layer of meaning. His first example of an “alphametic” is the equation SEND + MORE = MONEY, where each letter represents a different digit, and the addition is correct.

Rearrangement Puzzles

Some dissection and rearrangement puzzles are based on mathematical principles. Archimedes of Syracuse (287–212 BCE) may have created a 14-piece puzzle, the “Stomachion,” as part of his research. It resembles a version of a “Tangram,” a Chinese puzzle that became very popular in the nineteenth century in the West and is often used in mathematics classrooms in the twenty-first century to investigate dissections and concepts like the Pythagorean Theorem, named for Pythagoras of Samos. The Fibonacci sequence relation

(Fn)² = Fn−1Fn+1 + (-1)ⁿ⁻¹

with n=6 can be used to create a dissection puzzle. Larger values of n generate similar, more impressive puzzles, where the difference of area between a large square and a large rectangle is always included. Some dissection puzzles may lead to optical illusions when the pieces do not fit exactly together, leading to two figures composed of the same pieces that have different areas.

Topological Puzzles

Ring and string puzzles as well as knotted puzzles are examples of topological puzzles, where no discontinuous deformations like cutting the string are allowed. In his De Viribus Quantitatis (c. 1500), cited as the oldest book in recreational mathematics, Luca Pacioli (1445–1517) describes the Chinese Rings, a topological puzzle still popular in the twenty-first century.

Euler’s name is linked to several puzzles. He solved the Bridges of Konigsberg Problem, and this work is usually seen as the starting point of topology and graph theory.

The concept of the Eulerian graph is rooted in Euler’s resolution of the Bridges of Koenigsberg problem.

Movement Puzzles

Numerous puzzles involve patterned movement within some type of framework, and solutions sometimes involve mathematical techniques like numbering, recursion, group theory, and determinants. Peg Solitaire traces its origins to seventeenth-century France. It is a game where a board has all its holes occupied with pegs except for the central one. The objective is to make valid moves (small jump capture) to empty the entire board but for a solitary peg in the central hole.

The Towers of Hanoi is a puzzle invented in 1883 by N. Claus, a pseudonym of the mathematician Edouard Lucas (1842–1891). A pile of discs of decreasing radius lays on one of three poles. Moving one disc at a time without letting a bigger disc rest on a smaller one, the solver is asked to change the pile from one pole to another. The recursive character of the solution to this puzzle makes it somewhat similar to the Chinese Rings.

Other Puzzles

The chessboard is a rich source of puzzles that attracted many mathematicians. In the Knight Tour problem, a knight must visit all the squares of the board just once. Euler is one mathematician who published a solution. Mathematician Johann Carl Friedrich Gauss (1777–1855) was attracted by the 8-Queen Problem, in which eight queens must be placed on a chessboard so they cannot capture any other queen. Some mathematicians have used determinants to solve this problem.

The nineteenth century produced a popular puzzle named “15.” It consists of a sliding device, a 4-by-4 array with the numbers one through 15, and an empty cell. The puzzle was scrambled, and the solver was required to transform the scrambled order back to the natural order with the empty cell in the last position. Sam Loyd offered $1,000 to whoever could reorder a scrambled 14 and 15 in an otherwise solved puzzle. The prize was never claimed. The impossibility of this challenge can be understood when phrased in the language of group theory.

Another puzzle based on mathematics that captivated the world was the Rubik’s Cube, created by Hungarian architect Ernő Rubik (b. 1944) in the 1970s, which became the best-selling puzzle in history. A 3-by-3-by-3 cube with differently colored faces moves by slices, getting scrambled with just a few moves. To find the way back to the starting position is an incredible challenge. This toy puzzle is used to illustrate many group theory concepts. On the other hand, knowledge of group theory facilitates the understanding of the puzzle itself.

Since ancient times, descriptions of “mazes” that must be traversed in a particular pattern of moves have abounded in legend and literature. The Minotaur–Theseus tale is one such example. Stone and hedge labyrinths may still be found in places like Europe, and many puzzle books contain paper mazes. Some mazes can be understood using what is known as “level sequences.”

The “jigsaw puzzle” was invented in England in the mid-1870s by British cartographer and engraver John Spilsbury (1739–1769) as a pedagogical device. Children were asked to rebuild maps. In the twentieth and twenty-first centuries, jigsaw puzzles expanded to include three-dimensional jigsaw puzzles, including spherical three-dimensional puzzles and two-dimensional jigsaw puzzles that are all one color and have all the pieces cut to the same shape. This last style of puzzle is related to tiling. Another mathematical question is how to optimally and efficiently design and cut out puzzle pieces according to certain specifications.

Puzzle designer Scott Kim is considered by some to be a master of symmetry. He has diverse interests in many fields, including mathematics, computer science, puzzles, and education. When discussing these interests, he emphasizes the ties between them rather than their differences. One of his creations is an ambigram in honor of the great Martin Gardner (1914–2010), who invented many puzzles and is known for his recreational mathematics works. An ambigram is a figure that appears the same when rotated 180 degrees or viewed upside down.

In the late twentieth century, puzzle video games like Tetris and Lemmings gained popularity. As technology became widespread in the early twenty-first century, computer puzzles like Minesweeper, Candy Crush Saga, Bejeweled, and many more became ubiquitous on smartphones around the world. Many of these games are addictive because of the complex, extensively planned actions behind each event in the game.

Bibliography

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———. The Tangram Book: The Story of the Chinese Puzzle With Over 2,000 Puzzles to Solve. Sterling Publishing, 2003.

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Spencer, Gwen. “A Conversation with Scott Kim.” Math Horizons, vol. 12, Nov. 2004.

Walsh, Toby. "Candy Crush’s Puzzling Mathematics." American Scientist, Nov.–Dec. 2014, www.americanscientist.org/article/candy-crushs-puzzling-mathematics. Accessed 7 Oct. 2024.