Would this circuit work as a binary calculator?
Since you have no carry-in to worry about, what you need here is just a half-adder. The LED for the less significant bit will connect to its normal output, and the LED for the more significant bit will connect to its "carry".

Using normal logic gates, a half adder consists of an XOR to produce the normal output, and an AND gate to produce the carry:







simulate this circuit &ndash; Schematic created using CircuitLab

Now, that leaves only building an XOR and AND gates. If you're building this out of discrete parts so you don't care a lot about performance, and do care about simplicity, you might want to use DTL. An extremely simplified DTL AND gate might look something like this:







simulate this circuit

Note: in practical use, you rarely want to use DTL, because it's pretty slow. Also note that this excludes a couple of resistors that are normally included for level shifting to ensure that the output of one gate can drive the input of another gate. Since you're never cascading gates in this particular case, you can probably get away without that, but if you look up what a DTL AND gate looks like, chances are it'll be more complex than what I've shown here.

I'll leave the XOR for you, but the basic idea is pretty simple: start by designing at the logical level, then design the individual gates (or just use some pre-built logic ICs, of course).

Making things work tends to work in the reverse direction: once you've designed a gate, make it work in isolation. When you have all your gates working individually, connect them together and get a larger circuit to work. Repeat as needed until your whole circuit works.