To solve this problem, let's consider the dimensions of the blocks and the requirement that one tower must be 2 cm taller than the other.

Step 1: Understanding the Dimensions

Each block has dimensions $11 \times 11 \times 18$. Since the blocks are stacked to form towers, the height of each block is 18 cm.

Step 2: Define the Heights of the Towers

Let $n_1$ be the number of blocks in the first tower, and $n_2$ be the number of blocks in the second tower. The height of the first tower is therefore $18 \times n_1$ cm, and the height of the second tower is $18 \times n_2$ cm.

Step 3: Express the Problem Mathematically

We need the first tower to be 2 cm taller than the second tower, so: $18 \times n_1 = 18 \times n_2 + 2$

Step 4: Solve for $n_1$ and $n_2$

To make the first tower 2 cm taller, the equation becomes: $18n_1 - 18n_2 = 2$ $18(n_1 - n_2) = 2$ $n_1 - n_2 = \frac{2}{18} = \frac{1}{9}$ However, $n_1$ and $n_2$ must be integers, and $\frac{1}{9}$ is not an integer, meaning that with whole blocks, we cannot achieve a 2 cm difference between the towers.

Step 5: Conclusion

Since it's impossible to achieve a 2 cm height difference using whole blocks of 18 cm height, the problem as stated has no solution with these block dimensions. You would need to alter the block dimensions or the required height difference to make the problem feasible.

Would you like to explore alternative problems or solutions based on this?

Here are some related questions:

1. What would be the minimum height difference achievable with these blocks?
2. If the height difference needed to be 18 cm instead of 2 cm, how many blocks would be required?
3. What if you were allowed to cut a block to achieve the 2 cm difference?
4. How would the problem change if the block height were a divisor of 2 cm?
5. What other combinations of block dimensions could result in a 2 cm difference?

Tip: When dealing with problems involving integer constraints, consider the divisibility of the numbers involved to check if a solution is possible.