The antique optometer

The power range of the device is ±10.00 DS. Each lens holder has lenses of equal power but opposite sign:

Holder 1: +0.25 DS and -0.25 DS
Holder 2: +0.75 DS and -0.75 DS
Holder 3: +2.25 DS and -2.25 DS
Holder 4: +6.75 DS and -6.75 DS

This particular selection of lenses produces a unique power for every unique permutation of the lens holders, thereby ensuring the maximum possible range.
With this combination any sphere power from -10.00 up to +10.00 can be created. For example +8.25 can be created by combining -0.75, +2.25 and +6.75.
The easiest way to solve this is to it tackle in stages: what would be the maximum range achievable using only one lens holder? The lenses would have to be +0.25 and -0.25 to achieve the stated requirements. A trivial situation.
The next step is to consider what single plus lens added to the range already available (-0.25 to +0.25) extends the range without duplicating any powers already available. What plus lens power added to the most minus power already available (-0.25) gives the next required power in the plus range (+0.50)? The answer is +0.75. The same argument applied to extending the minus range deduces that -0.75 should be the companion lens. Thus, the lenses required in the second lens holder are +0.75 and -0.75. The maximum range available with just two lens holders is -1.00 to +1.00.
The same process applied sequentially to the third and fourth lens holders reveals these should hold the lens pairs ±2.25 and ±6.75.
A practical upshot of this puzzle has a bearing on the choice of plus-minus twirls, flippers or confirmation tests used in practice. Most practitioners have a ±0.25 set and a ±0.50 set, and most practitioners will have had occasion to combine both together in contact to use as a ±0.75 set. It's better to have a ±0.25 set and a ±0.75 set, then, you have access to ±0.25, ±0.50, ±0.75 and ±1.00!