How many wards do I have to use to make sightstones efficient?

Solution 1:

A sightstone costs 950g. It gives you 180 health, the same as the ruby crystal that goes into creating it. Assuming the ruby crystal is of value to you, that means you have 950g-475g=475g invested in the warding portion of the sightstone alone. If that health isn't of value to you, or you would not have gotten a ruby crystal, then you should really not discount (all of) the ruby crystal's price.

This means you have somewhere between 475g (best case) and 950g (worst case, because you don't value HP at all) invested in the sight stone.

Green wards cost 75g. 475/75 = 6.33 wards. 950/75 = 12.67 wards.

In general, if you use it 7 or more times (meaning going back once after you buy), you've probably made back your cost. If you don't value health much, think of it more like 10 uses to break even (go back twice). If you don't value health at all, 13 uses (go back three times).

As a further consideration, remember that you can sell it back later in the game, if this turns out to be useful for you. This could reduce the number of wards you consider your "break even" point, especially if you don't value the health from the item. If you had bought an equivalent number of wards, you would not be able to recoup the sell price of the sightstone later in the game (380g)

EDIT: With regards to upgrading, the sightstone only upgrades into the ruby sightstone. This adds the effect of an additional ruby crystal for MORE than the price of a ruby crystal (price of crystal + cost to combine). While this is a useful thing if you're out of item slots and have gold to upgrade, it doesn't really apply to situations where you'd be worried about efficiency. It does allow you to have more charges (5 instead of 4) so you wouldn't have to back as often, and as pointed out in the comments allows 3 simultaneous ward placements.

Along the lines of upgrading, when/if you get to the point of the game where inventory slots are at a premium, keep in mind that this allows you to have wards+health in one slot, instead of having a slot reserved just for wards.

Solution 2:

If a ward placed prevents or nets your team even 1 death/kill, the sightstone has paid for itself, imo. Keep this in mind as League is not really a game about min/max NECESSARILY, so much as "winning continually." If you win early game, it's very easy to win mid game, etc. If wards let you do this, it doesn't matter if they're mathematically cost-effective. I know you're looking for the math, but I feel like this point HAS to be made.