A horizontal row comprising of $2n + 1$ squares has $n$ red counters placed at one end and $n$ blue counters at the other end, being separated by a single empty square in the center. For example, when $n = 3$.
<imgclass="img-responsive center-block"alt="three squares with red and blue counters placed on opposite ends of the row, separated by one empty square"src="https://cdn.freecodecamp.org/curriculum/project-euler/swapping-counters-1.gif"style="background-color: white; padding: 10px;">
A counter can move from one square to the next (slide) or can jump over another counter (hop) as long as the square next to that counter is unoccupied.
<imgclass="img-responsive center-block"alt="allowed moves of the counter"src="https://cdn.freecodecamp.org/curriculum/project-euler/swapping-counters-2.gif"style="background-color: white; padding: 10px;">
Let $M(n)$ represent the minimum number of moves/actions to completely reverse the positions of the colored counters; that is, move all the red counters to the right and all the blue counters to the left.
It can be verified $M(3) = 15$, which also happens to be a triangle number.
If we create a sequence based on the values of n for which $M(n)$ is a triangle number then the first five terms would be: 1, 3, 10, 22, and 63, and their sum would be 99.