Engaging math activities for the summer break - Day 6
Much of problem solving in mathematics is about finding a representation in a way that simplifies if not trivializes a given problem. Come to think of it, putting a word problem into algebraic terms - as an equation or a system of equations - is ultimately finding another representation of the problem, a representation more amenable to being solved. On Day 4, there was an example of three different representations of the common TicTacToe game. Now, I'll give three examples of puzzles resolved with similar representations.
In this puzzle, 12 squares are arranged in a chain pattern with four chips of different colors covering the first four squares. You are allowed to drag a chip to a square 5 places from its current location, either clockwise or counterclockwise. The purpose of the puzzle is to have the chips in the originally occupied four squares but in the reverse order, or to determine that this is not possible.
7 chips on an 8-star
The object is to place 7 chips at the vertices of the star. A coin may only be placed by dragging the cursor along an edge from one vertex to another.
Four knights puzzle
The board is a 3×3 part of the chessboard. Two blue and two red knights are placed at the corners, as shown
The only allowed move is to drag a knight to another square following the usual chess rules. Of course, no two knights may occupy the same square. The purpose of the puzzle is to swap red and blue knights.
One helpful suggestion that works for all three puzzles is to enumerate possible figure (chips, knights) locations in the order of the permitted moves. And then redraw the boards in accordance with the new order.
For the looping chips this gives
With the enumeration shown on the right, the beginning configuration looks different:
and the moves reduce to sliding to an adjacent square. On this new board the chips never pass over each other so that their sequence and order always remain the same. Positions of the chips in the original and the new board uniquely determine each other, implying that the puzzle has a negative solution, notwithstanding the appearances.
Four the second puzzle, the renumeration leads to an unfolded octagon
An observation to make is that a chip at a vertex disqualifies both incident edges for subsequent moves. The solution is then to make moves backwards: if your first move is from 1 to 6 then the next one has to be from 4 to 1, and the next from 7 to 4, and so on. This will assure plcing all 7 chips.
For the 4 knights,
As in the first puzzle, the knights now slide to the adjacent squares so that the right board specifies also the sequence of moves. The moves are forced to be executed around the board.
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