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Puzzle Typology

Sep 25, 2024, 11 min read

  • The following presents a list of puzzles alongside the general goal of the puzzle and any additional rules present.
  • Although some degree of Mathematics is present for all of these. We exclude mathematical problems.

Dissection Puzzles

  • Cut a shape into smaller shapes and re-arrange them to produce a particular pattern or geometric shape

    • Extension: Frame the problem as going from one shape to another shape while using:
      • The fewest / given number of cuts.
      • The fewest / given number of shapes
    • Extension: Go from one shape to many smaller target shapes. Adding additional constraints (i.e., each smaller shape is the same size)
    • Extension: Go from many given shapes to one target shape.

  • Disappearance Puzzles - arrange the geometric shapes in such a way that in the resulting shape, there is an apparent addition / subtraction of units.

    • Alternatively, explain where the sleight of hand is in the disappearance

Tangram-Like Puzzles

  • Tangram - starting with a set of pieces of different shapes, assemble various figures based on their silhouettes.

    • The STANDARD set is 5 triangles of varying sides + a square and a rhomboid.
      • Extension: Circular Tangram - generalizes Tangram to use pieces that can have curved sides.
      • Extension: Use a different set of given shapes.
      • Usually, though, the pieces can be assembled to form a basic shape (a square, circle, heart, egg, etc…)
    • The player is given a silhouette.
      • We can make it harder by obstructing the silhouette non-intrusively (adding stripes, for example).
      • Silhouettes can be abstract to representational, pertaining to people or objects. Sometimes even these objects may be given in place of the silhouette.
    • The player MUST use all pieces to assemble the figure.
    • In assembling, the pieces MUST touch without overlapping.

  • Polyforms - Similar to Tangram except with the additional constraint that all pieces are identical or similar in some way. The goal is still to produce a similar shape.

    • Polyomino problem - a variant using a polyomino (usually a pentomino). Given a set of polyominos, cover the silhouette of the shape without the pieces overlapping.
    • Extension: Add a constraint where certain patterns may not be produced (for example, four polyominos may not touch at a corner).
    • Extension: Not just 2D shapes, but use 3D shapes too with varying thicknesses.
      • Soma Cube Variants - given a set of 3D polycubes (cubes assembled in a particular 3D shape), make a bigger cube (or a bigger shape) using all the pieces.

  • Matchstick Puzzles - Similar to Tangram except with the additional constraint of all pieces corresponding to sticks (aka edges). The goal is still to produce some shape.

    • Extension: We may only operate on a limited number of moves.
      • Extension: Move matchsticks to produce another configuration.
      • Extension: Remove matchsticks to produce another configuration.
      • Extension: Allow / Forbid matchsticks to be broken
    • Extension: The matchsticks are of different lengths.
    • Extension: Allow / Forbid matchsticks to cross.
    • Extension: The matchsticks must make a 3D structure that satisfies additional constraints.

Miscellaneous Types

  • Domino Puzzles - puzzles that involve the use of dominoes---defined as a piece consisting of two half-tiles. Each half tile has a number of pips on them.
    • Variant: Make a pattern given a set of dominoes such that half-tiles that touched each other had the same number of pips.
    • Variant: Make a pattern such that the “rows”, “columns” or both satisfied an arithmetical constraint (i.e., same sums)
    • Extension: Whether or not to allow dominoes with blank half-tiles and how to handle them exactly.

Logic Puzzles

  • Any form of mathematical puzzle counts here.

Grid Puzzles

Positioning Puzzles

  • Chessboard / Checker Puzzles
    • Involve the use of a chessboard, usually exploiting the property of having white and black cells.
    • Alternatively, it may involve the use of circular checker pieces that slide around each other in a certain way. The goal is to arrange them in a new way.
    • Constraints on movement may also take inspiration from chess moves or checker moves.
  • Connection Problem
    • Form a Graph that connects a set of nodes together, satisfying some sort of property.
  • Moving Peg Puzzles
    • An array of holes is drilled in a board. White pegs are placed in the upper half. Black pegs are placed in the lower half. The center is left vacant.
    • The goal is to transfer the white pegs to the lower half and the black pegs to the upper half.
    • A peg may move either
      • to an adjacent vacant square or
      • by jumping over one neighboring peg to a vacant square
    • Moves must only be forward or backward, parallel to the sides of the square.
  • Peg Solitaire
    • We have a board full of pegs arranged in some way.
    • Pegs can only be moved by jumping over a neighboring peg to a vacant space directly on the other side. Following this jump, the peg that was jumped over is removed.
    • Jumps are only made along lattice lines.
    • The goal is to remove all pegs on the board.
  • Transportation Puzzle
    • We have a set of objects and two (or more) islands with a river separating them.
    • The goal is to move all the objects to the other side of the river.
    • Certain objects cannot be together.
    • Example: Tower of Hanoi
    • Example: Farmer, Fox, Rabbit, Cabbage Problem
  • Shunting Problem
    • Given trains on a set of railroad tracks, find a way to get the trains to where they are supposed to go without them colliding.
    • Constraints on the number of moves or the time taken may be given.
    • Variant: The length of the track may influence the number of rolling vehicles which can be placed on them.
    • Variant: Certain trains may not go on certain tracks.
    • Variant: Build up a train in a specific order, picking up rolling stock from various locations.
    • Variant: Decompose the train, placing rolling stock items in specified locations.
    • Variant: Devise the optimum sequence of rolling stock items in a train, so that it can be efficiently assembled at its starting point and decomposed at its destination(s).
  • Sliding Block Puzzle
    • Consists of a set of blocks arranged on a board. The blocks can slide around.
    • The goal is to arrange the blocks in a certain way.
    • Pieces CANNOT be lifted off the board.
    • Example: The 15 Puzzle
    • Example: Sokoban

Mechanical Puzzles

  • Packing Puzzle - the goal is to fit two items together, but the way they fit together is not as straightforward. The solver must determine how to fit them together.
    • Example: The Melting Block
    • Example: Conway’s Packing Problem
    • Example: The Chinese Cube
    • Example: The Vanishing Sphere

Disassembly Puzzles

  • The challenge for these puzzles is not only to solve it, but to undo the solution (i.e., to reassemble the problem)

  • Burr Puzzle - also called Chinese crosses or Notched stick puzzles. Consists of structures that interlock with each other. The goal is to disassemble the structures.

    • Example: Van der Poel’s puzzle,
    • Example: The Japanese Crystal

  • Magic Boxes - the goal is to unlock the box. The lock has complex mechanisms that the solver must determine and unlock.

    • Example: Haselgrove Box

  • Magic Disk - consists of interlocking disks that must be separated from each other.

  • Rattle Puzzle - consists of a container with an item inside that would rattle when the container was shaken. The goal is to remove the item(s) inside.

    • Example: The Ball in the Cage

Ring String Ball Problems

  • Features any two of a ring, string (solid rod or cord), and a ball. The goal is:
    • Variant: Remove one element from the puzzle.
    • Variant: Move one element with respect to the main component.
    • Variant: Find the sequence of moves such that one part is removed or repositioned.
    • Variant: Solve the puzzle with additional constraints (i.e., the cord and the ring must stay together).
  • Generally when a string is used, the key is generally to be found in a loop drawn over a knot or ball.
  • Example puzzles are found in Van Delft and Botermans.
    • Maleda - the goal is to remove the staple by raising the rings and rods in a certain order.

String Puzzles

  • Whai / Cat’s Cradles - the goal is to form intricate patterns using only string held taut by one’s fingers.
  • Generally feature strings where the objective is:
    • Variant: Rejoin two ends of the string.
    • Variant: Perform a cut on the string that obeys certain constraints.
    • Variant: Pull at the string in a certain way so that knots vanish.
    • Variant: Unknot the cord (can be extended further to escape from bonds.)
    • Variant: Perform an intricate knot subject to constraints.

Wire Puzzles

  • Wire Puzzles - a similar variant to the Burr Puzzle except involving metal pieces that have been bent and interlocked with each other in a convoluted manner.
  • Wire and String Puzzle - consists of a wire and a string (with possibly additional components like balls or knots).
    • Closed String Puzzles - the string consists of one closed loop. The goal is usually to disentangle the string from the wire.
    • Unclosed Loose String - the pieces of the string are not closed and not attached to the wire. Usually the ends are fitted with an object to keep the string from slipping out.
    • Unclosed Fixed String - the pieces of the string are not closed, but somewhere on its length is attached to the wire. The goal here is usually similar to a string puzzle.

Tour Puzzles

  • The puzzle involves the player traveling around a board, possibly requiring the player visit certain points on the board.

Mazes

  • Patterns of intricately winding paths have been cut on the landscape. The goal is to navigate from point to point , traversing through the maze.
  • A classic technique to solving this (which doesn’t always work) is to keep your hand on one side of the maze at all times.
    • This fails when there is more than one entrance and there is a route connecting them that doesn’t pass through the goal, or if there are loops
  • Another technique by Tremeaux (which is just DFS in disguise)
    • Mark one side of the corridor (say the right) as you pass through the maze.
    • Whenever you reach a dead end, retrace your steps, still marking the right hand wall.
    • Whenever you reach a previously visited junction along a path that you are following for the first time, turn around and retrace your steps, still marking the right hand wall.
    • When you reach a previously visited junction along a path that you have used before, take a new path. If there is any available. Otherwise, take any previously used path.
    • Never enter a path that has been marked on both sides.
  • Other methods make use of Graph Algorithms.
  • When making a maze, it is usually best to start with the solution path first.
    • Variant: 3D Mazes
    • Variant: More than one goal (that may need to be reached in a specific order)
    • Variant: More than one entrance.
    • Variant: Entrance and or exit is inside the maze rather than the typical case where they are outside.

Optimization Puzzles

  • Applies to any of the puzzles above (or to new ones).

  • Instead of being given the number of moves, find the optimal number of moves that still solve the problem.

  • This also extends to tasks that are adjacent to Automata or Assembly wherein we want to minimize:

    • The number of instructions
    • The “space” occupied by the solution (analogues to memory or size for example).
    • The time it takes for some program to finish.

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