Given information:
Total weight of the wood = 6 kg
1/3 part of the wood remains inside the water in the floating state

To determine the maximum weight that can be added to the floating solid such that the entire piece of wood is submerged, we need to consider the concept of buoyancy.

Buoyancy is the upward force exerted by a fluid on an object immersed in it. According to Archimedes' principle, the buoyant force experienced by an object is equal to the weight of the fluid displaced by the object.

Let's analyze the given situation step by step:

1. Weight of the wood:
The total weight of the wood is given as 6 kg.

2. Buoyant force:
In the floating state, 1/3 of the wood remains inside the water. This means that the remaining 2/3 of the wood is above the water surface. The buoyant force acting on the wood is equal to the weight of the water displaced by the submerged part of the wood.

The buoyant force (B) can be calculated using the formula:
B = ρ × V × g
where ρ is the density of water, V is the volume of the submerged part of the wood, and g is the acceleration due to gravity.

3. Weight of the water displaced:
The weight of the water displaced is equal to the buoyant force acting on the wood. This weight is given by the formula:
Weight of water displaced = ρ × V × g

4. Maximum weight to be added:
To submerge the entire piece of wood, the weight of the water displaced should be equal to the weight of the wood. Therefore, we need to add an additional weight (W) to the floating solid.

The maximum weight that can be added without sinking the wood is given by:
Weight of water displaced + W = Weight of wood

Solving for W, we get:
W = Weight of wood - Weight of water displaced

Let's put the values into the equation:

Weight of wood = 6 kg
Weight of water displaced = ρ × V × g

Now, let's analyze the options:

a) 12 kg:
If we add a weight of 12 kg, the total weight becomes 6 + 12 = 18 kg.
Since the weight of the water displaced is less than 18 kg, the wood will not sink completely. Therefore, option A is not correct.

b) 10 kg:
If we add a weight of 10 kg, the total weight becomes 6 + 10 = 16 kg.
Since the weight of the water displaced is less than 16 kg, the wood will not sink completely. Therefore, option B is not correct.

c) 14 kg:
If we add a weight of 14 kg, the total weight becomes 6 + 14 = 20 kg.
Since the weight of the water displaced is less than 20 kg, the wood will not sink completely. Therefore, option C is not correct.

d) 15 kg:
If we add a weight of 15 kg, the total weight becomes 6 + 15 = 21 kg.
Since the weight of the water displaced is greater than 21 kg, the wood will sink completely. Therefore, option D is not correct.

From the analysis, we can conclude that none of the given options are correct. The correct answer is not provided in the given options.