Logic Puzzle - Full Adder

In this puzzle, you will learn how to perform basic arithmetic using a logic circuit called a full adder.


Why are adders important? Lots of basic computation and math is dependent on addition and multiplication, making adders a crucial circuit for computers. And while doing basic arithmetic may seem to be a solved problem, there is actually lots of ongoing research on how to optimize adders. When so many of a computer’s instructions rely on addition, even tiny improvements can make a big difference!

So how are adders built? Let’s first think about the problem:

  • Can’t we just use an OR gate for addition? An OR gate will add two numbers, e.g. 0 and 0, 0 and 1, or 1 and 0, but what if we have 1 and 1? If we use just an OR gate for addition, we would lose important information – in other words, the answer should be 2, but the OR gate says the answer is 1. Generating this extra bit is called generating a carry. If our circuit cannot generate a carry, we end up with overflow, which amounts to lost information.
    • Overflow happening unknowingly is BAD. A register overflowing was at the source of the Ariane 5 rocket crash.
  • As we build more complex circuits, we will want to add more than two bits together. To do so, we need a way to carry in bits from a previous adder and carry out bits to the adder.


So let’s check out the full adder design below. This circuit implements a two-bit adder, where each input has one bit.

A + B + Cin = S with Cout

While there is an OR gate in the design, there are several other logic gates too. Using the truth table below, can you make sense of the other logic gates?

Switch #
1 A
2 B
3 Cin

Trouble viewing the simulation? Check it out on Wokwi.

Cin A B S Cout
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1

Extra credit: using this circuit, can you build a four-bit full adder? This circuit needs to accept two two-bit values, A and B, and a carry-in bit. It needs to output a two-bit value, S, and a carry-out bit.

Feel free to play around to try to figure it out. You can always return to the logic gate tutorial if you need a refresher.

You can also check out the solution.

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