In the given figure switches $$S_1$$ and $$S_2$$ are in open condition. The resistance across $$ab$$ when the switches $$S_1$$ and $$S_2$$ are closed is ___ $$\Omega$$.

We need to determine the equivalent resistance across terminals $$a$$ and $$b$$ when both switches $$S_1$$ and $$S_2$$ are closed.

1. Analyze the Circuit Connection with Closed Switches

When switches $$S_1$$ and $$S_2$$ are closed, they create ideal, zero-resistance short-circuiting bridges connecting the upper and lower branches of the network. Let's label the junction nodes to understand the potential distribution:

• Let the input terminal be node $$a$$.
• Let the first vertical bridge line (containing switch $$S_1$$) connect the junctions after the first pair of resistors. We can call this node $$C$$.
• Let the second vertical bridge line (containing switch $$S_2$$) connect the junctions before the final pair of resistors. We can call this node $$D$$.
• Let the output terminal be node $$b$$.

2. Calculate the Resistance of Each Block

Because the vertical lines link the top and bottom wires directly, the circuit naturally splits into three distinct parallel blocks connected in series:

Block 1 (Between node $$a$$ and node $$C$$):

The $$12\ \Omega$$ resistor (top) and the $$6\ \Omega$$ resistor (bottom) are connected in parallel between terminal $$a$$ and the first switch line $$S_1$$.

$$R_{\text{block1}} = \frac{12 \times 6}{12 + 6} = \frac{72}{18} = 4\ \Omega$$

Block 2 (Between node $$C$$ and node $$D$$):

The two middle $$4\ \Omega$$ resistors (top and bottom) are connected in parallel between the line of switch $$S_1$$ and the line of switch $$S_2$$.

$$R_{\text{block2}} = \frac{4 \times 4}{4 + 4} = \frac{16}{8} = 2\ \Omega$$

Block 3 (Between node $$D$$ and node $$b$$):

The $$6\ \Omega$$ resistor (top) and the $$12\ \Omega$$ resistor (bottom) are connected in parallel between the second switch line $$S_2$$ and the output terminal $$b$$.

$$R_{\text{block3}} = \frac{6 \times 12}{6 + 12} = \frac{72}{18} = 4\ \Omega$$

3. Compute Total Equivalent Resistance ($$R_{ab}$$)

These three parallel blocks are arranged sequentially in series from terminal $$a$$ to terminal $$b$$. Therefore, we add their equivalent resistances together:

$$R_{ab} = R_{\text{block1}} + R_{\text{block2}} + R_{\text{block3}}$$

$$R_{ab} = 4\ \Omega + 2\ \Omega + 4\ \Omega = 10\ \Omega$$

Final Answer: $$10$$