Logic Gates Puzzle 1

There are two inputs A and B, each with two bits (bit 0 and 1).

Also there is an output X with four bits (0, 1, 2, 3).

The input A and B is somehow combined into output X.

X = A ??? B

Create a truth table with all possible inputs and the resulting outputs, write all inputs and outputs also as decimal numbers. Do you see the pattern now?

Truth table:

A  B  X     A B X
00 00 ????  0 0 ??
01 00 ????  1 0 ??
....

The circuit shows a two bit multiplication circuit.

If you create a truth table for the circuit you see it easily.

A  B  X
00 00 0000   0 x 0 = 0 
01 00 0000   1 x 0 = 0
10 00 0000   2 x 0 = 0
11 00 0000   3 x 0 = 0
00 01 0000   0 x 1 = 0
01 01 0001   1 x 1 = 1
10 01 0010   2 x 1 = 2
11 01 0011   3 x 1 = 3
00 10 0000   0 x 2 = 0
01 10 0010   1 x 2 = 2
10 10 0100   2 x 2 = 4
11 10 0110   3 x 2 = 6
00 11 0000   0 x 3 = 0
01 11 0011   1 x 3 = 3
10 11 0110   2 x 3 = 6
11 11 1001   3 x 3 = 9

You can separate the circuit into four parts.

1. Pass the A value to the output if B0 is one

2. Pass the A value, shifted one bit (x2) to the output, if B1 is one.

3. Detect an “overflow”, if A = 11 and B = 11.

4. “Add” the results of part 1, 2 and 3.

In a programming language, e.g. Python, you could write the four parts like this:

a = 3
b = 2

# Part 1
ab0 = (a if (b & 0b1) else 0)
# Part 2
ab1 = ((a << 1) if (b & 0b10) else 0)
# Part 3
ab_overflow = (0b1100 if (a == 0b11 and b == 0b11) else 0)
# Part 4
x = ab0 ^ ab1 ^ ab_overflow

It is no exact logic representation of the circuit, because the input and output bits are kept combined in the variables.