Lesson Three grasshoppers play leapfrog along a line
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<H2>Three grasshoppers play leapfrog along a line</H2> This lesson is advanced. It is intended only for those to whom it is interesting. <H3>Problem 1</H3>Three grasshoppers play leapfrog along a line. At each turn, one grasshopper leaps over another, but not over two others. Can the grasshoppers return to their initial positions after 1991 leaps? <B>Solution</B> <pre> The initial configuration is like this: -----0--------A------B-------------C---------- Three grasshoppers, named A, B and C, were sitting on the straight line (number line !). Let's consider the function f = {{{((X[B]-X[A])*(X[C]-X[B])*(X[C]-X[A]))/(abs(X[B]-X[A])*abs(X[C]-X[B])*abs(X[C]-X[A]))}}} where {{{X[A]}}}, {{{X[A]}}}, {{{X[C]}}} are coordinates of the grasshoppers A, B and C at each current turn of the game. The function f takes the values +1 or -1 depending on the position of A, B and C at each current moment. In the initial position the value of the function f is +1. Notice that f changes the sign to the opposite at each and every turn of the play. Then after 1991-th turn f will have the opposite (negative) sign to that "+1" it had initially. It means that the grasshoppers can not return to (can not be at) their initial positions after 1991-th turn. </pre> <B>Discussion</B> The function f is a kind of a counter on the space of configurations of three grasshoppers on a straight line. The same counter can also be interpreted as a sign of a permutation (in abstract algebra) of three objects A, B and C. The same counter can also be constructed as the determinant of the matrix M with columns A = {{{(matrix(3,1, 1,0,0))}}}, B = {{{(matrix(3,1, 0,1,0))}}}, C = {{{(matrix(3,1, 0,0,1))}}}. In the initial state the columns form the identity matrix M = I = {{{(matrix(3,3, 1,0,0, 0,1,0, 0,0,1))}}} with the determinant 1. Then the columns are subject of permutations repeating the leaps of the grasshoppers A, B, and C, and the determinant of the matrix M in each turn is nothing else as the function f introduced above. Having this counter on the configuration space, it helps us to solve the problem. 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