The jumping frogs puzzle is a puzzle in which stacks of counters on a finite row of equally spaced positions are combined by jumps. A move
carries an entire stack of 
counters exactly 
positions to the left or right, landing on another occupied
position in the row and combining the two stacks. Empty positions remain in the row
and count toward the jump distance. The goal is to combine all counters into a single
stack.
The basic problem starts with a single counter at each of 
positions, where 
is an integer. All counters
can be combined at either endpoint in exactly 
moves (Woll 2017, p. 2).
This follows by mathematical induction. The case 
requires no moves. Suppose that the result holds for 
positions. For a row of 
positions, first combine the 
rightmost counters at the right endpoint by the induction
hypothesis. That stack contains 
counters and is exactly 
positions from the remaining counter at the left endpoint.
Jumping the stack left combines all 
counters. Interchanging left and right gives the other endpoint.
Finally, every move reduces the number of occupied positions by exactly one, which
forces the total to be 
.
