1.
Fractal
–
A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetry or evolving symmetry, if the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge, Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set, Fractals also include the idea of a detailed pattern that repeats itself. Fractals are different from other geometric figures because of the way in which they scale, doubling the edge lengths of a polygon multiplies its area by four, which is two raised to the power of two. Likewise, if the radius of a sphere is doubled, its volume scales by eight, but if a fractals one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the dimension of the fractal. As mathematical equations, fractals are usually nowhere differentiable, the term fractal was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning broken or fractured, there is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as beautiful, damn hard, increasingly useful, Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractal. The word fractal often has different connotations for laypeople than for mathematicians, the mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background. If this is done on fractals, however, no new detail appears, nothing changes, self-similarity itself is not necessarily counter-intuitive. The difference for fractals is that the pattern reproduced must be detailed, a regular line, for instance, is conventionally understood to be 1-dimensional, if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the Koch snowflake and it is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. This also leads to understanding a third feature, that fractals as mathematical equations are nowhere differentiable, in a concrete sense, this means fractals cannot be measured in traditional ways. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, according to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity. In his writings, Leibniz used the term fractional exponents, also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called self-inverse fractals

2.
Rubik's Cube
–
Rubiks Cube is a 3-D combination puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. As of January 2009,350 million cubes had been sold making it the worlds top-selling puzzle game. It is widely considered to be the worlds best-selling toy, in a classic Rubiks Cube, each of the six faces is covered by nine stickers, each of one of six solid colours, white, red, blue, orange, green, and yellow. In currently sold models, white is opposite yellow, blue is green, and orange is opposite red. On early cubes, the position of the colours varied from cube to cube, an internal pivot mechanism enables each face to turn independently, thus mixing up the colours. For the puzzle to be solved, each face must be returned to have one colour. Similar puzzles have now produced with various numbers of sides, dimensions. Although the Rubiks Cube reached its height of popularity in the 1980s, it is still widely known. Many speedcubers continue to practice it and other twisty puzzles and compete for the fastest times in various categories, since 2003, The World Cube Association, the Rubiks Cubes international governing body, has organised competitions worldwide and kept the official world records. In March 1970, Larry D. Nichols invented a 2×2×2 Puzzle with Pieces Rotatable in Groups, Nicholss cube was held together with magnets. Patent 3,655,201 on April 11,1972, on April 9,1970, Frank Fox applied to patent his Spherical 3×3×3. He received his UK patent on January 16,1974, in the mid-1970s, Ernő Rubik worked at the Department of Interior Design at the Academy of Applied Arts and Crafts in Budapest. He did not realise that he had created a puzzle until the first time he scrambled his new Cube, Rubik obtained Hungarian patent HU170062 for his Magic Cube in 1975. Rubiks Cube was first called the Magic Cube in Hungary, Ideal wanted at least a recognisable name to trademark, of course, that arrangement put Rubik in the spotlight because the Magic Cube was renamed after its inventor in 1980. The first test batches of the Magic Cube were produced in late 1977, Magic Cube was held together with interlocking plastic pieces that prevented the puzzle being easily pulled apart, unlike the magnets in Nicholss design. With Ernő Rubiks permission, businessman Tibor Laczi took a Cube to Germanys Nuremberg Toy Fair in February 1979 in an attempt to popularise it. It was noticed by Seven Towns founder Tom Kremer and they signed a deal with Ideal Toys in September 1979 to release the Magic Cube worldwide, the puzzle made its international debut at the toy fairs of London, Paris, Nuremberg and New York in January and February 1980. After its international debut, the progress of the Cube towards the toy shop shelves of the West was briefly halted so that it could be manufactured to Western safety, a lighter Cube was produced, and Ideal decided to rename it

3.
Mechanical puzzle
–
A mechanical puzzle is a puzzle presented as a set of mechanically interlinked pieces. The oldest known mechanical puzzle comes from Greece and appeared in the 3rd century BC, the game consists of a square divided into 14 parts, and the aim was to create different shapes from these pieces. In Iran “puzzle-locks” were made as early as the 17th century AD, the next known occurrence of puzzles is in Japan. In 1742 there is a mention of a game called “Sei Shona-gon Chie No-Ita” in a book, around the year 1800 the Tangram puzzle from China became popular, and 20 years later it had spread through Europe and America. The company Richter from Rudolstadt began producing large amounts of Tangram-like puzzles of different shapes, in 1893, Angelo John Lewis, using the pen name Professor Hoffman, wrote a book called Puzzles, Old and New. It contained, amongst other things, more than 40 descriptions of puzzles with secret opening mechanisms and this book grew into a reference work for puzzle games and modern copies exist for those interested. The beginning of the 20th century was a time in which puzzles were greatly fashionable, the puzzle shown in the picture, made of 12 identical pieces by W. Altekruse in the year 1890, was an example of this. With the invention of modern manufacture of many puzzles became easier and cheaper. In this category, the puzzle is present in component form, the Soma cube made by Piet Hein, the Pentomino by Solomon Golomb and the aforementioned laying puzzles Tangram and “Anker-puzzles” are all examples of this type of puzzle. Furthermore, problems in which a number of pieces have to be arranged so as to fit into a box are also classed in this category, the image shows a variant of Hoffmans packing problem. Modern tools such as laser cutters allow the creation of complex two-dimensional puzzles made of wood or acrylic plastic, in recent times this has become predominant and puzzles of extraordinarily decorative geometry have been designed. This makes use of the multitude of ways of subdividing areas into repeating shapes, computers aid in the design of new puzzles. A computer allows a search for solution – with its help a puzzle may be designed in such a way that it has the fewest possible solutions. The consequence is that solving the puzzle can be very difficult, the use of transparent materials enables the creation of puzzles, in which pieces have to be stacked on top of each other. The aim is to create a pattern, image or colour scheme in the solution. For example, one consists of several discs in which angular sections of varying sizes are differently coloured. The discs have to be stacked so as to create a circle around the discs. The puzzles in this category are usually solved by opening or dividing them into pieces and this includes those puzzles with secret opening mechanisms, which are to be opened by trial and error

4.
Tower of Hanoi
–
The Tower of Hanoi is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod, the puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. The objective of the puzzle is to move the stack to another rod, obeying the following simple rules. Each move consists of taking the upper disk from one of the stacks, No disk may be placed on top of a smaller disk. With three disks, the puzzle can be solved in seven moves, the minimum number of moves required to solve a Tower of Hanoi puzzle is 2n –1, where n is the number of disks. The puzzle was invented by the French mathematician Édouard Lucas in 1883, there is a story about an Indian temple in Kashi Vishwanath which contains a large room with three time-worn posts in it surrounded by 64 golden disks. Brahmin priests, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the rules of the Brahma. The puzzle is also known as the Tower of Brahma puzzle. According to the legend, when the last move of the puzzle is completed and it is not clear whether Lucas invented this legend or was inspired by it. There are many variations on this legend, for instance, in some tellings, the temple is a monastery and the priests are monks. The temple or monastery may be said to be in different parts of the world — including Hanoi, Vietnam, and may be associated with any religion. In some versions, other elements are introduced, such as the fact that the tower was created at the beginning of the world, the puzzle can be played with any number of disks, although many toy versions have around seven to nine of them. The minimum number of required to solve a Tower of Hanoi puzzle is 2n –1. This is precisely the nth Mersenne number, a simple solution for the toy puzzle is to alternate moves between the smallest piece and a non-smallest piece. When moving the smallest piece, always move it to the position in the same direction. If there is no position in the chosen direction, move the piece to the opposite end. For example, if you started with three pieces, you would move the smallest piece to the end, then continue in the left direction after that. When the turn is to move the non-smallest piece, there is one legal move

5.
Peg solitaire
–
Peg solitaire is a board game for one player involving movement of pegs on a board with holes. Some sets use marbles in a board with indentations, the game is known simply as Solitaire in the United Kingdom where the card games are called Patience. It is also referred to as Brainvita, the August 1687 edition of the French literary magazine Mercure galant contains a description of the board, rules and sample problems. This is the first known reference to the game in print, the standard game fills the entire board with pegs except for the central hole. The objective is, making moves, to empty the entire board except for a solitary peg in the central hole. There are two boards, A valid move is to jump a peg orthogonally over an adjacent peg into a hole two positions away and then to remove the jumped peg. In the diagrams which follow, · indicates a peg in a hole, * emboldened indicates the peg to be moved, and o indicates an empty hole. A blue ¤ is the hole the current peg moved from, a red * is the position of that peg. On the English board the equivalent alternative games are to start with a hole, there is no solution to the European board with the initial hole centrally located, if only orthogonal moves are permitted. This is easily seen as follows, by an argument from Hans Zantema, after every move the number of covered A positions increases or decreases by one, and the same for the number of covered B positions and the number of covered C positions. Hence after a number of moves all these three numbers are even, and after an odd number of moves all these three numbers are odd. Hence a final position with one peg can not be reached, then one of these numbers is one. There are, however, several other configurations where a single initial hole can be reduced to a single peg. A tactic that can be used is to divide the board into packages of three and to them entirely using one extra peg, the catalyst, that jumps out. In the example below, the * is the catalyst, other alternate games include starting with two empty holes and finishing with two pegs in those holes. Also starting with one hole here and ending with one peg there, on an English board, the hole can be anywhere and the final peg can only end up where multiples of three permit. Thus a hole at a can only leave a single peg at a, p, O or C, a thorough analysis of the game is known. This analysis introduced a notion called pagoda function which is a tool to show the infeasibility of a given, generalized, peg solitaire

6.
Disentanglement puzzle
–
A disentanglement puzzle is a type of mechanical puzzle that involves disentangling one piece or set of pieces from another piece or set of pieces. The reverse problem of reassembling the puzzle can be as hard as—or even harder than—disentanglement, there are several different kinds of disentanglement puzzles, though a single puzzle may incorporate several of these features. A plate-and-ring puzzle usually consists of three pieces, one plate or similar displaying many holes and/or indentations a closed or nearly closed ring or a similar item, the plate as well as the ring are usually made from metal. The ring has to be disentangled from the plate, wire puzzles consist of two or more entangled pieces of more or less stiff wire. The pieces may or may not be closed loops, the closed pieces might be simple rings or have more complex shapes. Normally the puzzle must be solved by disentangling the two pieces without bending or cutting the wires, early wire puzzles were made from horseshoes and similar material. Wire-and-string puzzles usually consist of, one piece of string, ribbon or similar, one or several pieces of stiff wire sometimes additional pieces like wooden ball through which the string is threaded. One can distinguish three subgroups of wire-and-string puzzles, Closed string subgroup, Here the pieces of string consist of one closed loop, usually the string has to be disentangled from the wire. Unclosed loose string subgroup, Here the pieces of string are not closed, in this case the ends of the string are fitted with a ball, cube or similar which stops the string from slipping out too easily. Usually the string has to be disentangled from the wire, sometimes other tasks have to be completed instead, such as shifting a ring or ball from one end of the string to another end. Unclosed fixed string subgroup, Here the pieces of string are not closed, in these puzzles the string is not to be disentangled from the wire. One possible task may be to shift a ring or ball from one end of the string to another end

7.
Slitherlink
–
Slitherlink is a logic puzzle developed by publisher Nikoli. Slitherlink is played on a lattice of dots. Some of the squares formed by the dots have numbers inside them, the objective is to connect horizontally and vertically adjacent dots so that the lines form a simple loop with no loose ends. In addition, the number inside a square represents how many of its four sides are segments in the loop, other types of planar graphs can be used in lieu of the standard grid, with varying numbers of edges per vertex or vertices per polygon. These patterns include snowflake, Penrose, Laves and Altair tilings and these add complexity by varying the number of possible paths from an intersection, and/or the number of sides to each polygon, but similar rules apply to their solution. Whenever the number of lines around a cell matches the number in the cell and this is usually indicated by marking an X on lines known to be empty. Another useful notation when solving Slitherlink is a ninety degree arc between two adjacent lines, to indicate that one of the two must be filled. A related notation is an arc between adjacent lines, indicating that both or neither of the two must be filled. These notations are not necessary to the solution, but can be helpful in deriving it, many of the methods below can be broken down into two simpler steps by use of arc notation. A key to many deductions in Slitherlink is that point has either exactly two lines connected to it, or no lines. So if a point which is in the centre of the grid and this is because the point cannot have just one line - it has no exit route from that point. Similarly, if a point on the edge of the grid, and if a corner of the grid has one incoming line which is Xd out, the other must also be Xd out. Application of this simple rule leads to increasingly complex deductions, recognition of these simple patterns will help greatly in solving Slitherlink puzzles. If a 1 is in a corner, the actual corners lines may be Xd out and this also applies if two lines leading into the 1-box at the same corner are Xd out. If a 3 is in a corner, the two edges of that box can be filled in because otherwise the rule above would have to be broken. If a 2 is in a corner, two lines must be going away from the 2 at the border, thus if any two inner or outer segments in one 1 are Xd, the respective inner or outer segments of the other 1 must also be Xd. If two 1s are adjacent along the edge of the grid, the line between them can be Xd out, because there would be no direction for it to continue when it reached the edge. This means that the line must continue on one side of the 2 or the other

8.
Sudoku
–
Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, the puzzle setter provides a partially completed grid, which for a well-posed puzzle has a unique solution. Completed games are always a type of Latin square with a constraint on the contents of individual regions. For example, the same single integer may not appear twice in the row, column. French newspapers featured variations of the puzzles in the 19th century, however, the modern Sudoku only started to become mainstream in 1986 by the Japanese puzzle company Nikoli, under the name Sudoku, meaning single number. It first appeared in a US newspaper and then The Times in 2004, from the efforts of Wayne Gould, Number puzzles appeared in newspapers in the late 19th century, when French puzzle setters began experimenting with removing numbers from magic squares. Le Siècle, a Paris daily, published a partially completed 9×9 magic square with 3×3 subsquares on November 19,1892, on July 6,1895, Le Siècles rival, La France, refined the puzzle so that it was almost a modern Sudoku. It simplified the 9×9 magic square puzzle so that each row, column, and broken diagonals contained only the numbers 1–9, although they are unmarked, each 3×3 subsquare does indeed comprise the numbers 1–9 and the additional constraint on the broken diagonals leads to only one solution. These weekly puzzles were a feature of French newspapers such as LEcho de Paris for about a decade, but disappeared about the time of World War I. Garnss name was present on the list of contributors in issues of Dell Pencil Puzzles and Word Games that included Number Place. He died in 1989 before getting a chance to see his creation as a worldwide phenomenon, whether or not Garns was familiar with any of the French newspapers listed above is unclear. At a later date, the name was abbreviated to Sudoku by Maki Kaji, Sudoku is a registered trademark in Japan and the puzzle is generally referred to as Number Place or, more informally, a portmanteau of the two words, Num Pla. In 1986, Nikoli introduced two innovations, the number of givens was restricted to no more than 32, and puzzles became symmetrical and it is now published in mainstream Japanese periodicals, such as the Asahi Shimbun. Gould devised a program to produce unique puzzles rapidly. Although the 9×9 grid with 3×3 regions is by far the most common, the Times offers a 12×12-grid Dodeka Sudoku with 12 regions of 4×3 squares. Dell Magazines regularly publishes 16×16 Number Place Challenger puzzles, Nikoli offers 25×25 Sudoku the Giant behemoths. A 100×100-grid puzzle dubbed Sudoku-zilla was published in 2010, another common variant is to add limits on the placement of numbers beyond the usual row, column, and box requirements. Often, the limit takes the form of an extra dimension, the aforementioned Number Place Challenger puzzles are all of this variant, as are the Sudoku X puzzles in The Daily Mail, which use 6×6 grids

9.
T puzzle
–
The T puzzle is a tiling puzzle consisting of four polygonal shapes which can be put together to form a capital T. The four pieces are usually one isosceles right triangle, two right trapezoids and an irregular shaped pentagon, despite its apparent simplicity, it is a surprisingly hard puzzle of which the crux is the positioning of the irregular shaped piece. The earliest T puzzles date from around 1900 and were distributed as promotional giveaways, from the 1920s wooden specimen were produced and made available commercially. At 2015, most T puzzles come with a leaflet with additional figures to be constructed, which shapes can be formed depends on the relative proportions of the different pieces. The Latin cross puzzle consists of a reassembling a five-piece dissection of the cross with three isosceles triangles, one right trapezoids and an irregular shaped six-sized piece. When the pieces of the puzzle have the right dimensions. From Chinese origin, the oldest examples date from the first half of the nineteenth century, one of the earliest published descriptions of the puzzle appeared in 1826 in the Sequel to the Endless Amusement. Many other references of the puzzle can be found in amusement, puzzle. The T puzzle is based on the puzzle, but without head and has therefore only four pieces. Another difference is that in the dissection of the T, one of the triangles is usually elongated as a right trapezoid and these changes make the puzzle more difficult and clever than the cross puzzle. The T-puzzle became very popular in the beginning of the 20th century as a giveaway item, the pieces were made from paper or cardboard and served as trade cards, with advertisement printed on them. They usually came in an envelop with instructions and an invitation to write to or call at the company or local dealer for its solution, examples include, Lashs Bitters - the original tonic laxative. This is the earliest known version of the T-puzzle, the angles are cut at 35 degrees which makes the puzzle easier and less confusing. White Rose Ceylon tea, Seeman Brothers, New York and this puzzle is often cited as being the oldest version of the T puzzle, but Lashs Bitters puzzle predates it. Armours dry sausage, Armour and Company, Chicago, the text on the envelope reads The Teaser T, Please accept this interesting little puzzle with our compliments. You will find it a real test to fit the four pieces enclosed in this together to form this perfect letter T. If you fail to solve it, ask your dealer for the solution, and to solve the problem of adding delicious meat dishes to your menu Ask your dealer for Armours Dry Sausage. Wateralls T Puzzle Paints & Varnishes distributed by O. J. Miller & Son, Allentown, the envelope mentions that the puzzle is highly entertaining, interesting, perplexing, aggravating and easy

10.
15 puzzle
–
The 15-puzzle is a sliding puzzle that consists of a frame of numbered square tiles in random order with one tile missing. The puzzle also exists in other sizes, particularly the smaller 8-puzzle, the object of the puzzle is to place the tiles in order by making sliding moves that use the empty space. The n-puzzle is a problem for modelling algorithms involving heuristics. Commonly used heuristics for this problem include counting the number of misplaced tiles, note that both are admissible, i. e. they never overestimate the number of moves left, which ensures optimality for certain search algorithms such as A*. Johnson & Story used a parity argument to show half of the starting positions for the n-puzzle are impossible to resolve. The invariant is the parity of the permutation of all 16 squares plus the parity of the distance of the empty square from the lower right corner. This is an invariant because each move changes both the parity of the permutation and the parity of the taxicab distance, in particular if the empty square is in the lower right corner then the puzzle is solvable if and only if the permutation of the remaining pieces is even. Johnson & Story also showed that the converse holds on boards of size m×n provided m and n are both at least 2, all permutations are solvable. This is straightforward but a little messy to prove by induction on m and n starting with m=n=2, archer gave another proof, based on defining equivalence classes via a hamiltonian path. Wilson studied the analogue of the 15 puzzle on arbitrary finite connected, the exceptional graph is a regular hexagon with one diagonal and a vertex at the center added, only 1/6 of its permutations can be obtained. For larger versions of the n-puzzle, finding a solution is easy and it is also NP-hard to approximate the fewest slides within an additive constant, but there is a polynomial-time constant-factor approximation. The multi-tile metric counts subsequent moves of the empty tile in the direction as one. The number of positions of the 24-puzzle is 25. /2 ≈7. 76×1024 which is too many to calculate Gods number. In 2011, a bound of 152 single-tile moves had been established. The symmetries of the fifteen puzzle form a groupoid, this acts on configurations. Now if we look at the state of the puzzle. Hence it is easy to prove by induction that any state of the puzzle for which the sum is odd cannot be solvable. In particular, if the empty square is in the right corner then the puzzle is solvable if

11.
Tangram
–
The tangram is a dissection puzzle consisting of seven flat shapes, called tans, which are put together to form shapes. The objective of the puzzle is to form a shape using all seven pieces. It is reputed to have invented in China during the Song Dynasty. It became very popular in Europe for a then. It is one of the most popular dissection puzzles in the world, a Chinese psychologist has termed the tangram the earliest psychological test in the world, albeit one made for entertainment rather than for analysis. The origin of the word tangram is unclear, the -gram element is apparently from Greek γράμμα. The tan- element is variously conjectured to be Chinese tan to extend or Cantonese tang Chinese, the tangram had already been around in China for a long time when it was first brought to America by Captain M. Donnaldson, on his ship, Trader, in 1815. When it docked in Canton, the captain was given a pair of Sang-Hsia-kois Tangram books from 1815 and they were then brought with the ship to Philadelphia, where it docked in February 1816. The first Tangram book to be published in America was based on the pair brought by Donnaldson. The puzzle was originally popularized by The Eighth Book Of Tan, a history of Tangram. The book included 700 shapes, some of which are possible to solve, the puzzle eventually reached England, where it became very fashionable. The craze quickly spread to other European countries and this was mostly due to a pair of British Tangram books, The Fashionable Chinese Puzzle, and the accompanying solution book, Key. Soon, tangram sets were being exported in great number from China, made of materials, from glass, to wood. Many of these unusual and exquisite tangram sets made their way to Denmark, danish interest in tangrams skyrocketed around 1818, when two books on the puzzle were published, to much enthusiasm. The first of these was Mandarinen and this was written by a student at Copenhagen University, which was a non-fictional work about the history and popularity of tangrams. The second, Det nye chinesiske Gaadespil, consisted of 339 puzzles copied from The Eighth Book of Tan, Tangrams were first introduced to the German public by industrialist Friedrich Adolf Richter around 1891. The sets were out of stone or false earthenware. More internationally, the First World War saw a resurgence of interest in Tangrams, on the homefront

12.
24 Game
–
The 24 Game is an arithmetical card game in which the objective is to find a way to manipulate four integers so that the end result is 24. For example, for the card with the numbers 4,7,8,8, the game has been played in Shanghai since the 1960s, using playing cards. It is similar to the card game Maths24, the original version of 24 is played with an ordinary deck of playing cards with all the face cards removed. The aces are taken to have the value 1 and the game proceeds by having 4 cards dealt. Some advanced players allow exponentiation, roots, logarithms, and other operations, for short games of 24, once a hand is won, the cards go to the player that won. If everyone gives up, the cards are shuffled back into the deck, the game ends when the deck is exhausted, and the player with the most cards wins. Longer games of 24 proceed by first dealing the cards out to the players, a player who solves a set takes its cards and replenishes their pile, after the fashion of War. Players are eliminated when no longer have any cards. A slightly different version includes the face cards, Jack, Queen, and King, giving them the values 11,12, mental arithmetic and fast thinking are necessary skills for competitive play. Pencil and paper will slow down a player, and are not allowed during play anyway. In the original version of the game played with a standard 52-card deck, additional operations, such as square root and factorial, allow more possible solutions to the game. For instance, a set of 1,1,1,1 would be impossible to solve with only the five basic operations, however, with the use of factorials, it is possible to get 24 as. Krypto 4nums. com, online 24 game General information from Pagat List of all possible solvable combinations