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A, B and incoming C are bits. Find X = A XOR B; Sum = X XOR C; G = A AND B; P = X AND C; Carry = G OR P. XOR needs differing inputs, AND needs both ones and inclusive OR needs at least one. Check A + B + C = 2 x Carry + Sum. For two columns, begin at the ones place and pass its Carry to the twos column as C. Keep the last carry as the fours bit. These are ideal logical bits, not wiring instructions.
Worked example
With A = 0, B = 0, C = 1, X = 0, Sum = 1, G = 0, P = 0 and Carry = 0. Incoming C is 1, but outgoing Carry is 0. For binary 10 + 10 with initial C = 0: the ones column gives 0 with no carry; the twos column totals two, leaving a fours bit. The complete result is 100, meaning four.
