#1 is batch sparging. #2 & #3 are no sparging and will give you lower efficiency than #1.

How is adding 1/3 water at the end and then draining it not sparging?

Oh, man, this is a topic that been debated for years! I think Dixon has been adamant about it in the past. To me, if you haven't drained the mash when you add that water, you're simply doing a mash infusion, not a sparge. Other people's definitions (especially Dixon's!) may differ...but they'd be wrong!

Right, it's not sparging because you have one runoff of a single gravity! Batch sparging is two (or more) discrete runoffs, each of lesser gravity. Fly sparging is one continuous runoff with asymptotically decreasing gravity (think back to calculus :-) ). In the practical sense, the reason one or more sparges gets you a higher efficiency is because the wort held back by the grain (a fixed amount based on grain mass) contains less sugar.

Assuming that the sugar is in solution and distributed evenly in the liquid in the mash tun, your lauter efficiency will be proportional to the amount of liquid you can drain from the tun, versus the liquid left behind in the grain, for each sparge.

Define:

Q = quarts of water

L = pounds of grain

R is mash thickness in quarts/lb = Q/L

Assume grain absorption of 0.5 qts/lb. For a single sparge with R = 4 qts/lb, the proportion of water (and therefore sugar) left behind is:

(0.5 * L) / Q = 0.5 / R = 1 / 2*R

So 1/8 (12.5%) of the water is left behind, meaning sparge efficiency is 87.5 %.

Now, assume you mash with 2 qts/lb of water, drain the tun, and batch sparge with another 2 qts/lb of water.

Sugar lost in the first drain of the tun is:

(1 / 2*2) = 1/4 or 25% of sugar left behind, giving you 75% lauter efficiency.

Now, assuming that stirring in the second batch of water redistributes the sugars equally in the entire volume of the mash, you will be able to extract another fraction of these left behind sugars. The total water in the tun will be 2qts/lb + the 0.5 qts/lb left behind from the mash, giving you 2.5 qts/lb.

Sugars left behind after the sparge:

(1/4) * (1/2*2.5) = 1/20, or 5%, giving you 95% lauter efficiency.

Whew! I think that's (mostly) right, but it should at least give an example of why a sparge gives higher lauter efficiency than no-sparge. With fly sparging, you should theoretically be able to approach 100% lauter efficiency, but it doesn't make too much difference since a lot of efficiency has to do with mash conversion. Kai did a more complete analysis of this, including factors such as the changes to volume due to addition of sugars to water, and he even put together an efficiency analysis spreadsheet that's on his site somewhere.