This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.

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Answer
18 \text{ mm}$.
Step 1: Identify the known values and the formula for an uncalibrated thermometer. The length of the mercury thread at (melting ice) is . The length of the mercury thread at (steam) is . The length of the mercury thread in the hot liquid is .
The temperature on a linear scale can be found using the formula:
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Identify the known values and the formula for an uncalibrated thermometer. The length of the mercury thread at 0^ (melting ice) is L_0 = 18 mm.
This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.