In Australian Rules football a score can be worth either 6 points or 1 point. The cheerleaders wave their pompoms each time their team scores, with the number of waves equaling the number of points that their team has scored thus far. So if a score of 1 point was followed by a score of 6 points, there would be one occasion for waving the pompoms once, and a second occasion, on which the pompoms would be waved 7 times (the total score at that point).

In a recent match, the team didn't do so well: the total number of waves that the cheerleaders gave was fewer than 50.

If I told you what that total number of waves was, you'd be able to deduce that the number of occasions on which they waved could be any of three different possibilities. Even if I then told you on how many occasions they waved, you could still find three different orders of 6's and 1's scored that would have led to that total number of waves.

On how many occasions did they wave, and what was the total number of waves?