Introduction:
In this scenario, we have a big box at 100 degrees Celsius connected to an ice cube through identical metal rods. Initially, the ice cube is melting at a rate of Q1 gram per second. The rods are then rearranged in series between the box and the ice cube, and we need to determine the new rate of melting, Q2, in terms of Q1.

Explanation:
To analyze this situation, we need to understand how heat flows through the system. Heat transfer occurs from the higher temperature (the box) to the lower temperature (the ice cube) until thermal equilibrium is reached.

1. Initial Configuration:
Initially, when the rods are connected parallel between the box and the ice cube, the heat flows through both rods simultaneously. This means that the rate of heat transfer through each rod is equal, resulting in an equal rate of melting for the ice cube.

2. Rearranged Configuration:
When the rods are rearranged in series, the heat transfer occurs sequentially through each rod. This means that the rate of heat transfer through each rod is different. The first rod in contact with the box will transfer heat to the second rod, and so on until the final rod in contact with the ice cube.

3. Analysis:
In the initial configuration, where the rods are parallel, the rate of heat transfer is divided equally between the rods. Let's assume this rate as "x" gram per second for each rod. Therefore, the total rate of heat transfer, and hence the rate of melting, is 2x gram per second.

In the rearranged configuration, where the rods are in series, the rate of heat transfer through each rod is different. Let's assume the rate for the first rod as "y" gram per second. As heat flows sequentially through the rods, the rate of heat transfer decreases. Let's assume the rate for the second rod as "z" gram per second.

Therefore, the total rate of heat transfer in the rearranged configuration, and hence the rate of melting, is y + z gram per second.

4. Determining the Ratio:
From the conservation of energy, we know that the total amount of heat transferred in both configurations should be the same. Therefore, we can equate the total rate of heat transfer in both configurations:

2x = y + z

To determine the ratio Q2/Q1, we need to compare the rate of melting in the rearranged configuration (Q2) with the initial rate of melting (Q1).

Q2/Q1 = (y + z)/(2x)

Conclusion:
The ratio Q2/Q1 depends on the specific values of y, z, and 2x. Without additional information about these values, we cannot determine the exact ratio. However, we have established the relationship between the rate of melting in the rearranged configuration (Q2) and the initial rate of melting (Q1) as (y + z)/(2x).

It will probably depend on the number of rods taken.