Tribute to Invariance

Invariance: Sample Activities

We have already discussed the puzzles about Breaking Chocolate Bar and Squares and Circles. There are quite a few more puzzles of this kind, 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. Such 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 dominos 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, and answer the same question.
   
   
3. Changing Colors #2
   As before, but in each move the 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 there are 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 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 the dates up. Verify that the sum does not depend on your choices.
   
   
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
    The numbers 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, the foundation of the Theory of Impartial Games, is modeled in a variety of games: Nimble, Northcott's game, Plainim, Scoring, Turning Turtles.
    
    

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

Copyright © 1997 Alexander Bogomolny