I think Richard Boyce above got it pretty much right.
You need a differential between the distances between the centre magnet and the ring magnets and the external magnet, but the internal magnet must be stronger.
The large magnet in the centre of the ring has a stronger overall field so it attracts more strongly further afield.
However, as the external magnet approaches, the inverse square law kicks in and the repulsion of the outer ring magnets increases more rapidly (they are closer) than the attraction of the inner until the attraction and repulsion balance out.
1) If you try to pull them apart, then the repulsion of the outer drops off faster than the attraction of the inner so the inner resists.
2) If you try to push them together, the outer repulsion increases more rapidly than the attraction of the inner, so the outer resists.
If you superimposed two field curves on the same graph, one stronger one and one less so, but displaced the stronger one horizontally, where they intersected would give the balance point.
Quite a cute little experiment, but I'm at a loss as to why the nice old guy can't figure it out.