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JUMPS & SLIDES

 A very neat gane-like puzzle is introduced in LUCI unit "Perspectives on Algebra" for investigating patterns. The goal is to rearrange the checkers to exchange their places. Rules are very simple- slideing in the empty space and jumping over the opposite color checker if there is a room after that. Click on this picture to get the interactive applet to play the game and enjoy: By clicking on the checkers you will learn the simple instructions GRAPHICAL RECORDING With each slide move horiazontaly 1 step and with each jump move up vertically one step. Colors can be used to show the color of the moved checker The picture in the left shows an example with 5 red and 5 green checkers. The graphical recording of the moves shows at a glance the symmetry and the "parabolic" (quadratic) behavior of each half of the graph (one is flipped). Interestingly, it also resembles the integral sign! The slides in the graph serve as independent variable, changing by one at a time (which seems natural for no two consecutive slides can happen), whereas the jumps serve as dependent variable. UNEQUAL NUMBERS OF MEN (general case) The numbers of checkers of different color (or chips, I call men) don't have to be equal. Say, we have m and n men. Then, the number of slides is (m+n) and the number of Jumps is (mn), so the total number of moves is mn+m+n. This turns, in particular, into n(n+2) when m=n. The number of moves in general case is less than in ordinary case. We know that mn is largest when m=n for given total number of soldiers m+n. Like in a rectangle: for a given value of the semi-perimeter (m+n) the area is the largest for a square, when m=n. So playing more general game is even simpler, and it has more possibilities to investigate. This could be a useful thing to do if there is time, of course. Besides generalizing arithmetic operations and rules algebra provides also qualitative thinking. It is used by physicists in their descriptions of processes. So, algebra allows to think in terms of processes. For instance, the term (mn) is known as interaction term. There are m red men and n green men in the Jumps&Sildes game. They have to face and jump over each other, they do interact. Each green man faces exactly once with each red man and therefore the total number of jumps must be mn. Also as one can easily see from the graph there are m+n slides altogether, which is equal to the total number of men, as if each man slides exactly once. This is not exactly true but only ON AVERAGE.