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
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
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.