MathDB

Three grasshoppers

A

B

and

C

are placed on a line. Grasshopper

B

sits at the midpoint between

A

and

C

. Every second, one of the grasshoppers jumps over one of the others to the symmetrical point on the other side (if

X

jumps over

Y

to point

X'

, then

XY

YX'

). After several jumps it so happened that they returned to the three initial points (but maybe in different order). Prove that in this case

B

returns to his initial middle position.
(AK Kovaldzhy)

Problem Statement