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.
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?
- Changing Colors
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.
Changing Colors #2
As before, but in each move the squares in a single row or column
- Plus or
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.
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.
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?
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.
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?
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.
- Counting Diagonals in a Convex
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
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
The famous Nim, the foundation of the Theory of Impartial Games, is
modeled in a variety of games:
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