To add three n bit numbers, there is a simple method.
Take n one-bit full adders. Each full adder takes one bit of each of the three numbers as input, and generates a sum and a carry.
You create one n+1 number by shifting all the carrys one position to the left and filling with a zero, and one n+1 number by taking all the sums and adding a zero bit. Now you just have two n+1 bit adders to add.
Full-adder 0: x0, y0, z0 -> s0, c0
Full-adder 1: x1, y1, z1 -> s1, c1
Full-adder n-1: x(n-1), y(n-1), z(n-1) -> s(n-1), c(n-1)
Now add (0, s(n-1), s(n-2), ..., s2, s1, s0) and (c(n-1), c(n-2), ..., c1, c0, 0).
You use the same principle to add k numbers quickly: For every 3 n-bit numbers, you use a line of full adders to create two n+1 bit numbers. For example, given k = 10 numbers, you use three lines of full numbers to reduce this to 7 numbers (by turning three times 3 numbers into three times 2 numbers), then reduce 7 to 5, 5 to 4, 4 to 3, and 3 to 2. And then comes your n-bit adder.
The same principle can be used at a massive scale to build a 64 x 64 bit multiplier: You multiply a 64 bit number by 1 bit using 64 AND gates. Now you have 64 64-bit numbers to add. A massive number of full adders converts 21 sums of 3 numbers into 42 numbers and so on.