Tribute to Invariance

Invariance: Sample Activities

We have already discussed the Breaking Chocolate Bar and Squares and Circles puzzles. There are quite a few more, some of which are very well known. Most of the puzzles may be played in numerous settings. In Squares and Circles, it's of course possible to use the same basic shape painted in 2 (or 3) colors. In puzzles where a move consists ostensibly in changing colors, one may turn coins instead. Puzzles are so numerous that just counting them is in itself a good exercise.

  1. Chessboard
      Consider a chess board with two of the diagonally opposite corners removed. Is it possible to cover the board with pieces of domino whose size is exactly two board squares? The size and the shape of the board may vary. What about removing other pairs of squares?

  2. Changing Colors #1
      In a 4x4 chessboard, all squares are colored white or black. Given an initial coloring of the board, we are allowed to recolor it by changing the color of all squares in any 3x3 or 2x2 subboard. Is it possible to get every possible recoloring of the board from the "all-white" coloring by applying some number of these "subboard recolorings"? In addition to square introduce "diagonal" moves. The same question.

  3. Changing Colors #2
      As before, but in one move squares in a single row or column change colors.

  4. Plus or Minus
      Write a series of integers. Take turns putting the plus or minus signs between two consecutive numbers. When all the spaces are filled with signs, evaluate the result. If the result is odd, the first player wins. Otherwise, the second is the winner.

  5. Solitaire on a Circle
      Chips of two colors are arranged on a circle. When one is removed, its immediate neighbors (if any) change colors. The goal is to leave a single chip of a given color.

  6. Splitting Piles
      One is presented with several piles of objects. A move consists in splitting an arbitrary pile into two. The objects in the two piles are counted, the two numbers are multiplied and the result is stored somewhere. When no further pile splitting is possible, the stored numbers are retrieved and summed up. What is the result?

  7. Squares, Circles, and Triangles
      A puzzle mentioned in the text. In a group of objects of three kinds, select two of different kinds and replace them with either one or two objects of the third kind. Great exercise in the arithmetic modulo 2 and 3.

  8. Sums and Products
      Several numbers are written down. On a single move two of the present numbers are selected and replaced with a single number according to the rule: A and B are replaced with (AB+A+B). Can you predict the result?

  9. Calendar Magic
      Take a wall or desk calendar. Outline a square array of dates. Inside each such square select dates so as to have one in each row and each column. Add dates up. Verify that the sum does not depend on the process of selection.

  10. Counting Diagonals in a Convex Polygon
      Draw a convex polygon. Start connecting its vertices with non-intersecting diagonals. When you can't draw any more without intersecting existing diagonals, count their total number. See that it does not depend on the manner in which diagonals have been drawn.

  11. Fif
      Number from 1 through 9 are written in a row. Players take turns selecting numbers. The goal is to get three numbers that add up to 15. This is a model for the ubiquitous Tic-Tac-Toe.

  12. Nim
      The famous Nim, being the foundation of the Theory of Impartial Games, is modeled in a variety of games: Nimble, Northcott's game, Plainim, Scoring, Turning Turtles.

  1. Sam Loyd's fifteen
      Every one knows this one. Solution, even if not elementary, can be explained to a preschooler. Offers a great counting exercise.

  2. Ford's touching circles
      Construction of mediant fractions preserves a fundamental arithmetic identity insuring that new fractions are in lowest terms if their predecessors were in lowest terms.

  3. Euclid's Game
      For a given set of numbers you are allowed to add a difference of any two numbers present unless, of course, it's already there. The number of steps it takes to write down all possible numbers only depends on the starting pair.

  4. Peg Solitaire
      In the well known central solitaire, the task is to remove all pegs but one. One gets a score of genius if the last peg is left at the center of the puzzle. If a single peg is left elsewhere, the player is dubbed outstanding. Invariance helps us realize that an oustanding player must be a complete idiot not to become a genius.

  5. Swapping Rows and Columns
      In a square array of numbers one is permitted to either swap any two columns or any two rows. The question is when is it possible to change one array into another. The problem is solved with the realization that, while the row and column operation may change the order of the elements in the array, they do not change the contents of either columns or row looked at as the sets of their elements.

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