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Instant Insanity puzzle in the "solved" configuration. From top to bottom, the colors on the back of the cubes are white, green, blue, and red (left side), and blue, red, green, and white (right side)
Nets of the Instant Insanity cubes – the line style is for identifying the cubes in the solution

Instant Insanity is the name given by Parker Brothers to their 1967 version of a puzzle which has existed since antiquity, and which has been marketed by many toy and puzzle makers under a variety of names, including: Devil's Dice (Pressman); DamBlocks (Schaper); Logi-Qubes (Schaeffer); Logi Cubes (ThinkinGames); Daffy Dots (Reiss); Those Blocks (Austin); PsykoNosis (A to Z Ideas), and many others.[1]

The puzzle consists of four cubes with faces colored with four colors (commonly red, blue, green, and white). The objective of the puzzle is to stack these cubes in a column so that each side of the stack (front, back, left, and right) shows each of the four colors. The distribution of colors on each cube is unique, and the order in which the four cubes are stacked is irrelevant as long as each side shows every color.

This problem has a graph-theoretic solution in which a graph with four vertices labeled B, G, R, W (for blue, green, red, and white) can be used to represent each cube; there is an edge between two vertices if the two colors are on the opposite sides of the cube, and a loop at a vertex if the opposite sides have the same color. Each individual cube can be placed in one of 24 positions, by placing any one of the six faces upward and then giving the cube up to three quarter-turns. Once the stack is formed, it can be rotated up to three quarter-turns without altering the orientation of any cube relative to the others. Ignoring the order in which the cubes are stacked, the total possible number of arrangements is therefore 3,456 (24 * 24 * 24 * 24 / (4 * 4!)). The puzzle is studied by D. E. Knuth in an article on estimating the running time of exhaustive search procedures with backtracking.[2]

Every position of the puzzle can be solved in eight moves or less.[3]

The first known patented version of the puzzle was created by Frederick Alvin Schossow in 1900, and marketed as the Katzenjammer puzzle.[4] The puzzle was recreated by Franz Owen Armbruster, also known as Frank Armbruster, and independently published by Parker Brothers and Pressman, in 1967. Over 12 million puzzles were sold by Parker Brothers alone. The puzzle is similar or identical to numerous other puzzles[5][6] (e.g., The Great Tantalizer, circa 1940, and the most popular name prior to Instant Insanity).

One version of the puzzle is currently being marketed by Winning Moves Games USA.

Solution

[edit]
A graph of the opposite faces of the cubes, the line styles corresponding to the cubes in the image of their nets above

Given the already colored cubes and the four distinct colors are (Red, Green, Blue, White), we will try to generate a graph which gives a clear picture of all the positions of colors in all the cubes. The resultant graph will contain four vertices one for each color and we will number each edge from one through four (one number for each cube). If an edge connects two vertices (Red and Green) and the number of the edge is three, then it means that the third cube has Red and Green faces opposite to each other.

To find a solution to this problem we need the arrangement of four faces of each of the cubes. To represent the information of two opposite faces of all the four cubes we need a directed subgraph instead of an undirected one because two directions can only represent two opposite faces, but not whether a face should be at the front or at the back.

So if we have two directed subgraphs, we can actually represent all the four faces (which matter) of all the four cubes.

  • First directed graph will represent the front and back faces.
  • Second directed graph will represent the left and right faces.

We cannot randomly select any two subgraphs - so what are the criteria for selecting?

We need to choose graphs such that:

  1. the two subgraphs have no edges in common, because if there is an edge which is common that means at least one cube has the pair of opposite faces of exactly the same color, that is, if a cube has Red and Blue as its front and back faces, then the same is true for its left and right faces.
  2. a subgraph contains only one edge from each cube, because the sub graph has to account for all the cubes and one edge can completely represent a pair of opposite faces.
  3. a subgraph can contain only vertices of degree two, because a degree of two means a color can only be present at faces of two cubes. Easy way to understand is that there are eight faces to be equally divided into four colors. So, two per color.

After understanding these restrictions if we try to derive the two sub graphs, we may end up with one possible set as shown in Image 3. Each edge line style represents a cube.

Mapping the edges of the two directed subgraphs to the left (L) and right (R), and front (F) and back (B) faces solves the puzzle

The upper subgraph lets one derive the left and the right face colors of the corresponding cube. E.g.:

  1. The solid arrow from Red to Green says that the first cube will have Red in the left face and Green at the Right.
  2. The dashed arrow from Blue to Red says that the second cube will have Blue in the left face and Red at the Right.
  3. The dotted arrow from White to Blue says that the third cube will have White in the left face and Blue at the Right.
  4. The dash-dotted arrow from Green to White says that the fourth cube will have Green in the left face and White at the Right.

The lower subgraph lets one derive the front and the back face colors of the corresponding cube. E.g.:

  1. The solid arrow from White to Blue says that the first cube will have White in the front face and Blue at the Back.
  2. The dashed arrow from Green to White says that the second cube will have Green in the front face and White at the Back.
  3. The dotted arrow from Blue to Red says that the third cube will have Blue in the front face and Red at the Back.
  4. The dash-dotted arrow from Red to Green says that the fourth cube will have Red in the front face and Green at the Back.

The third image shows the derived stack of cube which is the solution to the problem.

It is important to note that:

  1. You can arbitrarily label the cubes as one such solution will render 23 more by swapping the positions of the cubes but not changing their configurations.
  2. The two directed subgraphs can represent front-to-back, and left-to-right interchangeably, i.e. one of them can represent front-to-back or left-to-right. This is because one such solution also render 3 more just by rotating. Adding the effect in 1., we generate 95 more solutions by providing only one. To put it into perspective, such four cubes can generate 243 × 3 = 41472 configurations.
  3. It is not important to take notice of the top and the bottom of the stack of cubes.[7]

Generalizations

[edit]
A variant with playing card suits is easier, unless the symbols can be oriented anyhow

Given n cubes, with the faces of each cube coloured with one of n colours, determining if it is possible to stack the cubes so that each colour appears exactly once on each of the 4 sides of the stack is NP-complete.[8][9]

The cube stacking game is a two-player game version of this puzzle. Given an ordered list of cubes, the players take turns adding the next cube to the top of a growing stack of cubes. The loser is the first player to add a cube that causes one of the four sides of the stack to have a color repeated more than once. Robertson and Munro[10] proved that this game is PSPACE-complete, which illustrates the observation that NP-complete puzzles tend to lead to PSPACE-complete games.

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Instant Insanity

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from Grokipedia
Instant Insanity is a classic mechanical stacking puzzle consisting of four cubes, each face colored with one of four distinct colors—typically red, blue, green, and white. The goal is to arrange and stack the cubes into a single tower such that each of the four vertical sides of the tower displays all four colors exactly once, with no color repeating on any side. The puzzle's origins trace back to the early , with the first known patented version invented by Frederick A. Schossow of , , in 1900 under the name "Katzenjammer Puzzle." This original design featured four small wooden cubes and a tray, using a similar arrangement of symbols (equivalent to colors) to challenge players to arrange them in a row such that each of the four sides displays all four symbols without repetition. Schossow's invention laid the foundation for subsequent variations, establishing the core mechanic of aligning multicolored faces to avoid repetitions. In 1967, California computer programmer and puzzle designer Franz O. Armbruster (1929–2013), also known as Frank Armbruster, recreated and popularized the puzzle in its modern plastic form, which was marketed by Parker Brothers as Instant Insanity. Armbruster's version retained the four-cube structure but introduced durable plastic components and the evocative name suggesting quick solvability, though solutions often require systematic trial or mathematical insight. This edition became a commercial success, with Parker Brothers selling over 12 million copies, introducing the puzzle to a wide audience through toy stores and educational settings. Mathematically, Instant Insanity is analyzed using , where each cube is represented by a graph of opposite-face color pairs, and the solution involves finding disjoint paths or cycles that satisfy the side conditions. The puzzle has exactly two solutions when considering rotations of the entire stack, making it deceptively challenging with 4! × 24^4 = 7,962,624 possible configurations, though graph-based methods reduce the search space dramatically. Variations, such as those with more cubes or colors, have been studied for , proving for generalizations beyond four pieces.

History

Origins and Early Versions

The puzzle now known as Instant Insanity traces its origins to the early , with the first documented version invented by Frederick A. Schossow, a resident of , . In , Schossow patented a mechanical puzzle consisting of four small wooden cubes, each face marked with one of four suits—spades, clubs, diamonds, or hearts—arranged in a specific configuration to limit possible arrangements. The objective was to stack the cubes into a linear tower such that each of the four long sides displayed all four suits exactly once, without repetition, mirroring the core challenge of later color-based iterations. Marketed under the name "Katzenjammer Puzzle," this design used suits as proxies for colors, emphasizing visual distinction and the tantalizing difficulty of achieving a unique solution through . Preceding Schossow's invention, similar dice- and block-based stacking puzzles appeared under various names, evoking frustration and intrigue. One early moniker was "The Great Tantalizer," applied to versions around 1900 that involved arranging colored blocks or dice to form coherent patterns on multiple sides, often with stacking or alignment goals that tested spatial reasoning. These precursors typically featured four distinct markers per block while retaining the non-repeating side requirement. Another historical name, "Devil's Dice," referred to comparable puzzles produced by multiple manufacturers. Schossow's Katzenjammer Puzzle established the foundational mechanics of four aligned blocks with opposing faces constrained for solvability, influencing subsequent adaptations that shifted from suits to colors while preserving the stacking rules. This early framework paved the way for broader commercialization in the mid-20th century, culminating in ' 1967 release of Instant Insanity.

Popularization and Commercial Success

The modern version of Instant Insanity was recreated in 1967 by computer programmer Franz O. Armbruster (also known as Frank Armbruster) and commercialized by as a branded iteration of Frederick A. Schossow's 1900 patented puzzle design. The company positioned the puzzle as an engaging suitable for children ages 7 and older, with packaging featuring vibrant illustrations of the colorful cubes and taglines highlighting its challenging yet solvable nature to appeal to family audiences. Promotional campaigns emphasized the puzzle's quick-to-learn mechanics and mental stimulation, contributing to its rapid adoption as a gift item during the late 1960s. The puzzle achieved significant commercial success, with selling over 12 million units worldwide between 1966 and 1967, reportedly outselling even Monopoly during that period and earning a mention in the 1966 Guinness Book of World for its popularity. This surge transformed Instant Insanity from a niche curiosity into a mainstream toy, spawning numerous imitations and establishing it as a staple in the category. In the , the puzzle has seen reprints and renewed availability through Winning Moves Games, which acquired the and reissued versions starting in the , including wooden editions in the to cater to contemporary collectors and puzzle enthusiasts. Its enduring legacy is reflected in cultural recognition, such as inclusion in the National Museum of American History's collection as a representative example of mid-20th-century toys.

Puzzle Components and Objective

The Cubes

The Instant Insanity puzzle consists of four cubes of equal size, approximately 1 inch on each side, with each of the six faces colored in one of four distinct colors: , , and . The cubes are constructed from durable , ensuring the colors remain vibrant and non-fading despite frequent manipulation. Each features a unique distribution of colors, with repeats on some faces to create the challenge, but crucially, no pair of opposite faces on any cube shares the same color. This design forces players to consider rotations and orientations carefully when arranging the cubes. The exact color distributions and opposite face pairs for the standard set of cubes are detailed below.
CubeColor DistributionOpposite Face Pairs
13 red, 1 blue, 1 green, 1 whitered-blue, red-green, red-white
21 red, 2 blue, 2 green, 1 whitered-green, blue-white, blue-green
31 red, 2 blue, 1 green, 2 whiteblue-red, blue-white, green-white
41 red, 2 blue, 2 green, 1 whitered-blue, green-white, green-blue
These configurations ensure that each cube has at least one color appearing more than once, contributing to the puzzle's complexity while maintaining balance across the set.

Goal of the Puzzle

The objective of Instant Insanity is to arrange four s, each with faces colored using one of four colors (, , and ), into a single vertical tower where both the stacking order and the orientation of each cube are crucial to achieving the solution. When the tower is formed, each of its four long sides—front, back, left, and right—must exhibit exactly one instance of each color across the four visible faces, ensuring no color repeats on any individual side. The top and bottom faces of the overall tower do not factor into the win condition and can be any colors. Each admits 24 distinct orientations, determined by selecting one of 6 faces to be on top and then rotating the cube in one of 4 ways around the vertical axis. Accounting for the permutations in stacking order, this yields a total of 4!×244=7,962,6244! \times 24^4 = 7{,}962{,}624 possible arrangements prior to any consideration of symmetries. The puzzle is designed such that a valid solution can be reached from any random starting configuration in eight moves or fewer, where a move typically involves reorienting or repositioning a single .

Solving Strategies

Brute Force and Trial-and-Error

Brute force and trial-and-error methods for solving rely on systematically testing cube orientations and stack orders through manual experimentation, making them accessible for casual solvers without requiring mathematical tools. The puzzle has 41,472 possible arrangements when accounting for reduced symmetries in manual trials, such as fixing one cube's position to limit rotations to three axes, though the full search space reaches 7,962,624 without such pruning (4! cube orders × 24^4 orientations). Only one unique solution exists up to rotation and reflection of the entire tower, as the puzzle is designed with exactly two solutions that are mirrors of each other. A practical starting point is to identify the cube with three faces of the same color—typically three reds—and place it at the base or a fixed position to anchor the stack, as its repeated colors limit viable orientations. With the order of cubes fixed initially, test the 24 possible orientations (6 choices for the bottom face times 4 rotations) for each subsequent cube sequentially, eliminating partial stacks early if any side already shows a repeated color on the visible faces. To streamline this, count total colors across all cubes (e.g., 7 reds, 6 blues, 5 greens, 6 whites in the standard set) and determine pairs to hide on top and bottom faces (e.g., 3 reds, 2 blues, 1 green, 2 whites), ensuring the remaining faces provide one of each color per side. Useful heuristics include prioritizing cubes with unique color pairs on opposite faces for the front-back or left-right views, as these constrain options effectively, and building the stack incrementally from bottom to top while verifying each added cube against the current side colors. Beginners typically take 10-30 minutes to solve via this approach, with an average around 15 minutes for those familiarizing themselves through repeated trials. A common pitfall is fixating on all six faces per cube, overlooking that top and bottom pairs are hidden and do not affect the visible sides, which can lead to unnecessary rotations and frustration. For those seeking greater efficiency beyond trial-and-error, graph-theoretic methods offer a structured alternative by modeling color oppositions.

Graph-Theoretic Solution

The graph-theoretic solution to Instant Insanity transforms the puzzle into a problem of selecting and orienting edges in a to satisfy the color constraints on the tower's sides. This approach, introduced in combinatorial texts, systematically identifies valid configurations by modeling opposite faces as edges and ensuring balanced color appearances through cycle structures. Begin by labeling the four cubes as 1 through 4. For each , determine its three pairs of opposite faces based on the fixed coloring; these pairs may connect different colors or the same color (resulting in a loop). Represent each pair as an undirected edge between the corresponding color vertices, labeling the edge with the cube number. This yields three edges per cube, capturing all possible axis assignments (front-back, left-right, or top-bottom). Construct a multigraph GG with vertex set {R,B,G,W}\{R, B, G, W\} (red, blue, green, white) and incorporate all 12 edges from the four cubes. Multiple edges between the same pair of vertices are possible, as are loops if a cube has identical colors on opposite faces. The standard cubes have the following opposite pairs (derived from their fixed face arrangements):
  • Cube 1: RR-RR, BB-RR, GG-WW
  • Cube 2: RR-GG, WW-BB, RR-WW
  • Cube 3: RR-GG, BB-WW, BB-WW
  • Cube 4: RR-BB, GG-WW, BB-GG
(Note: Cube 1 has three RR faces; Cube 3 has two adjacent BB faces.) To find a solution, select two disjoint s of edges SFBS_{FB} and SLRS_{LR}, each containing exactly one edge from each (thus four edges total, using labels 1-4 once). These subsets represent the front-back pairs and left-right pairs, respectively; the unused pairs per become top-bottom. For each SS, the subgraph formed by its edges must be 2-regular—every vertex has degree exactly 2—ensuring it consists of a single 4-cycle or two 2-cycles covering all vertices. This condition guarantees that colors can be assigned without repetition on opposite sides of the tower. Next, orient the edges in each 2-regular subgraph to form a directed graph where every vertex has in-degree 1 and out-degree 1, equivalent to a disjoint union of directed cycles. For a 4-cycle, traverse the cycle in one direction; for two 2-cycles, orient each as a directed 2-cycle (A → B → A). The orientation assigns directions consistently: for the front-back subgraph, direct each edge from the back-face color (tail) to the front-face color (head). Similarly, for left-right, direct from left-face color (tail) to right-face color (head), maintaining a consistent viewing convention. The directed cycles determine the stacking sequence and orientations. For the front-back cycle, the sequence of labels along the directed path gives the bottom-to-top order in the tower (up to cyclic shift). Each directed edge labeled kk from color A to B orients kk with back A and front B. The front colors (heads) will then be all distinct, and the back colors (tails) will be all distinct, as the cycle structure permutes the colors. Apply the same to the left-right cycle; its sequence must match the front-back sequence (or its reverse, allowing for 180-degree tower ). The orientations ensure no conflicts, as front-back and left-right pairs are disjoint per . In the standard Instant Insanity puzzle, a valid pair of subsets yields the unique solution (up to symmetries). One such configuration assigns to front-back the edges R-G (cube 2), G-W (cube 4), W-B (cube 3, using one B-W), B-R (cube 1). Orienting as the 4-cycle R → G (cube 2), G → W (cube 4), W → B (cube 3), B → R (cube 1) places cubes in order 2 (bottom), 4, 3, 1 (top), with cube 2: back R, front G; cube 4: back G, front W; cube 3: back W, front B; cube 1: back B, front R. The corresponding left-right cycle, using remaining edges like R-R (cube 1, loop for top-bottom possible but adjusted), R-W (cube 2), R-G (cube 3), G-B (cube 4), confirms the same order with appropriate side orientations, resulting in all sides displaying each color once.

Mathematical Aspects

Graph Representation

The graph representation of Instant Insanity models the puzzle as a G=(V,E)G = (V, E), where the vertex set V={R,B,G,W}V = \{R, B, G, W\} consists of the four colors (, , , ), and the edge set EE comprises 12 edges corresponding to the three pairs of opposite faces on each of the four . Each contributes exactly three edges, one for each pair of opposite colors, with edges labeled by their respective to distinguish them in the . A key property of this representation is that the edges capture the opposite face pairs, as any valid orientation of a assigns two of these pairs to the visible sides—specifically, one pair to the front-back faces and one to the left-right faces—while the third pair is oriented top-bottom and thus hidden from view. This structure enforces the puzzle's constraints: for the front-back side, the selected subgraph (one edge per ) must be 2-regular in the undirected sense, meaning each vertex has degree 2, which ensures that each color appears exactly once on the front across all cubes and once on the back. Equivalently, in a directed interpretation where edges are oriented from front to back, this subgraph forms a in which each vertex has in-degree and out-degree 1, forming a cycle cover consisting of one or more disjoint directed cycles, guaranteeing a balanced appearance of colors on opposing faces. The standard Instant Insanity puzzle features no self-loops, as no cube has identical colors on opposite faces, though the permits multiple edges between the same pair of vertices if different cubes share identical opposite color pairs. This directed cycle structure equates to a matching, where the cycle encodes a between the front colors and back colors across the cubes, ensuring all colors are distinctly represented on each visible side.

Complexity and Number of Solutions

The total number of possible configurations for the standard Instant Insanity puzzle, considering all permutations of the four cubes and their orientations, is 4!×244=7,962,6244! \times 24^4 = 7,962,624. for the 192 symmetries consisting of permuting the cube positions (4!) and the 8 tower symmetries (rotations and flips), the number of distinct arrangements up to these symmetries is 41,472. Despite this large configuration space, the standard puzzle has exactly two valid solutions, which are related by a 180° of the tower. These solutions are equivalent under tower symmetries including and reflections, yielding precisely one unique solution up to such transformations. A brute-force approach, after fixing the base to eliminate some symmetries, requires checking approximately 41,472 trials, though optimized can reduce this further by pruning invalid partial stacks early. In the graph-theoretic model, the number of solutions equals the number of pairs of edge-disjoint 2-regular spanning subgraphs of the configuration , where each subgraph selects exactly one edge per and forms a of cycles covering all four colors. This equivalence to counting disjoint cycle covers highlights the puzzle's combinatorial structure. The deliberate choice of cube colorings—selected from 240 distinct cubes (up to )—ensures this scarcity of solutions, in contrast to random colorings that often yield zero or multiple solutions.

Generalizations

Variants of the Puzzle

One notable commercial variant is "Drive Ya Crazy," produced by Meffert's in the 1990s, which expands the puzzle to six cubes using six colors: , orange, , green, blue, and cyan, with each cube featuring all six colors on its faces. The objective remains similar, requiring players to stack the cubes into a taller tower such that each of the four sides displays all six colors exactly once. The Hungarian version, "Buvos Golyok," produced by Politoys, replaces the cubes with four spherical balls housed in , where each ball has six colored spots representing the faces. The goal is to rotate the balls within the frame to achieve a configuration where each vertical column shows all four colors without repetition, mimicking the stacking alignment of the original puzzle. "Mutando," a German variant manufactured by Logika Spiele, uses four cubes with a distinct arrangement of four colors that allows for multiple solutions beyond the standard tower. In addition to stacking for diverse colors on each side, players can arrange the cubes into a 2×2×1 flat block where each of the six exposed faces displays a single uniform color. Other commercial iterations include "Watch It!," which substitutes colors with clock faces on the cubes (showing 2:00, 4:00, 8:00, and 10:00) and requires stacking so that the four times on each side sum to 24 hours, yielding 11 possible solutions. Earlier versions such as "Cube-4" by (1980) and "Trikki 4" from in the replicate the four-cube format with minor color substitutions, like swapping red for and for orange, while retaining the core stacking goal. Another variant is "Dorobo Five Cube Instant Insanity" by Hanayama, featuring five cubes with five colors (such as , orange, , , and ), where the objective is to stack them so each side displays all five colors exactly once. Modern DIY adaptations often employ affordable materials like cardboard boxes or wooden cubes with colored stickers to replicate the puzzle, enabling educators and hobbyists to create custom versions with the standard four colors. Some rule tweaks in these and other variants emphasize visibility of all six faces, such as arranging the cubes in a linear row to expose tops, bottoms, and sides without stacking, or forming a 2×2 base configuration where each side shows all four colors while the top and bottom display distinct patterns.

Theoretical Extensions

Theoretical extensions of Instant Insanity explore generalizations beyond the standard four-cube puzzle, focusing on computational and structural properties. In the n-cube version with n colors, where each cube's faces are colored using the n colors and the goal is to stack them such that each long face displays each color exactly once, deciding solvability is NP-complete. This result holds by reduction from known NP-complete problems, establishing that no efficient algorithm exists for arbitrary n unless P=NP. A two-player variant, where players alternately rotate and position cubes toward a shared stacking goal under normal play convention (last player to move wins), models positions as states in the configuration graph and is proven . This complexity arises from the exponential state space and the need to evaluate winning strategies, highlighting how single-player puzzles often yield harder two-player . The puzzle connects to broader applications, particularly finding edge-disjoint cycle covers in 4-regular , where vertices represent colors and edges (labeled by ) encode opposite-face pairs. Solvability requires decomposing the multigraph into two edge-disjoint 4-cycles (one for front-back faces, one for left-right), each using exactly one edge per cube label; extensions consider higher-dimensional hypercubes or non-cube polyhedra like Platonic solids, where analogous decompositions determine stackings with uniform face exposures. Seminal research includes algorithmic analyses from the 1970s onward, such as early computational solutions via , though no direct 1975 paper by B. D. Beach in Mathematics of Computation was identified; instead, Robertson and Munro's 1978 work formalized the and game complexities. No established connections to Latin squares or orthogonal arrays appear in the literature. Unsolvability in modified colorings can be proven via graph degree constraints: for instance, if a color vertex has degree less than 4 in the (as each of the two cycle covers requires degree 2), or if the incidences violate for bipartite matchings in rotation subgraphs, no valid decomposition exists, rendering the puzzle insoluble.

References

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  19. https://www.researchgate.net/publication/336399065_A_New_Technique_to_Solve_the_Instant_Insanity_Problem
  20. https://www.projectenigma.org/puzzle-manual/instant-insanity
  21. https://www.crcpress.com/Graphs-Digraphs/Chartrand-Lesniak-Zhang/p/book/9781439895184
  22. https://circles.math.ucla.edu/circles/lib/data/Handout-3647-3243.pdf
  23. https://www.pearson.com/us/higher-education/product/West-Introduction-to-Graph-Theory/9780131437371.html
  24. https://apps.dtic.mil/sti/tr/pdf/AD0665834.pdf
  25. https://ptwiddle.github.io/Graph-Theory-Notes/s_intro_instantinsanity.html
  26. https://math.stackexchange.com/questions/183314/instant-insanity-question
  27. https://math.stackexchange.com/questions/866825/how-to-find-out-the-number-of-ways-to-solve-instant-insanity
  28. https://www.cs.brandeis.edu/~storer/JimPuzzles/ZPAGES/zzzInstantInsanity.html
  29. https://mathequalslove.net/diy-instant-insanity-puzzle/
  30. https://www.instructables.com/Instant-Insanity-Puzzle/
  31. http://www.cs.bme.hu/~friedl/alg/games.pdf
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