After I had seen a couple of calculators/ALU videos, I decided to design a full adder myself but wanted to make it as narrow as possible and stackable (when placed directly next to each other, the carry out of the first has to be connected directly to the carry in of the second one).

When I started building, 3 blocks wide seemed to me to be the narrowest way to construct the the entire circuit, adding the one block of air to isolate each full adder, which lead to a total width of 4 blocks.
At first I designed one which followed the commonly used circuit of XOR, AND and OR gates which can be seen in the figure down below.

However due to the serial XOR gates the "calculation time" was a bit high (6 redstone ticks or more) but when you search on planetminecraft.com for "full adder" one of the first results would be: "Efficient full adder" by CX gamer which is only 2 ticks. There fore I took a look at the truth table:

After which I realized how it could be done in minecraft in such way it would only take 2 ticks. Because the Carry out can be done using three parallel AND gates (A&B, A&Cin and B&Cin) where all the outputs are connected together and all the AND gates have two ticks delay. And the Sum can be done using a three input XOR and AND gate, where the XOR gate only will output an high signal when only one input is high (how to do this can be seen in the last image) and the AND gate will only be high when all the three inputs are high.

Add some time building, making a lot of use of the properties of glowstone, and it is done. The end result uses the following materials:

• 74 Redstone Wires
  
• 14 Redstone Torches
  
• 15 Glowstone
  
• 3 Repeaters
  
• 84 Blocks (opaque)
  

Where the repeaters manly are used for increasing the range from where the input can come from, however it does not effect the "calculation time". The red wool blocks represent the inputs and the blue one(s) the output (however the the glowstone block which I labelled "S" could not be replaced with blue wool).