Let's solve the problem by carefully analyzing the blocks' dimensions and how we can stack them to achieve two towers where one is exactly 2 cm taller than the other.

Step 1: Dimensions of the Blocks

Each block has dimensions of 11 cm × 11 cm × 18 cm. When stacking, we can orient them in different ways:

1. Stacking with the height of the block as 11 cm.
2. Stacking with the height of the block as 18 cm.

We'll try different orientations to minimize the number of blocks while ensuring one tower is 2 cm taller than the other.

Step 2: Defining the Heights of the Towers

Let’s define the total height of the first tower as $H_1$ and the second tower as $H_2$. According to the problem:

$H_1 = H_2 + 2 \, \text{cm}$

Step 3: Exploring Possible Configurations

1. All blocks with 18 cm height:

   • If both towers use only blocks oriented with their 18 cm side as the height, the difference in height between any two towers will always be a multiple of 18 cm (since 18 cm is the block height). This won't work, since 18 cm isn't divisible by 2 cm.

2. Mixing orientations (18 cm and 11 cm):

   • If we stack some blocks with 18 cm height and some with 11 cm height, we can achieve more flexibility in adjusting the height difference between the two towers.

Step 4: Trial and Error to Find Minimum Number of Blocks

Let’s start by constructing two towers where the difference in height is exactly 2 cm. One way to do this is by trying combinations of the two block orientations (18 cm and 11 cm).

• First tower (18 cm blocks + 11 cm blocks): Let's try using one 18 cm block and multiple 11 cm blocks.

• Second tower (18 cm blocks + 11 cm blocks): Similarly, we’ll use one 18 cm block and some number of 11 cm blocks.

Trial 1:

• For the first tower, use:

  • 1 block of 18 cm + 3 blocks of 11 cm. $H_1 = 18 \, \text{cm} + 3 \times 11 \, \text{cm} = 18 + 33 = 51 \, \text{cm}$

• For the second tower, use:

  • 4 blocks of 11 cm. $H_2 = 4 \times 11 \, \text{cm} = 44 \, \text{cm}$

In this case: $H_1 - H_2 = 51 \, \text{cm} - 44 \, \text{cm} = 7 \, \text{cm}$ This difference is too large.

Trial 2:

• For the first tower, use:

  • 2 blocks of 18 cm + 1 block of 11 cm. $H_1 = 2 \times 18 \, \text{cm} + 1 \times 11 \, \text{cm} = 36 + 11 = 47 \, \text{cm}$

• For the second tower, use:

  • 1 block of 18 cm + 2 blocks of 11 cm. $H_2 = 1 \times 18 \, \text{cm} + 2 \times 11 \, \text{cm} = 18 + 22 = 40 \, \text{cm}$

In this case: $H_1 - H_2 = 47 \, \text{cm} - 40 \, \text{cm} = 7 \, \text{cm}$ The difference is still not 2 cm.

Trial 3 (Successful):

Let’s now attempt another combination.

• First tower: 3 blocks of 18 cm. $H_1 = 3 \times 18 \, \text{cm} = 54 \, \text{cm}$

• Second tower: 1 block of 18 cm + 3 blocks of 11 cm. $H_2 = 18 \, \text{cm} + 3 \times 11 \, \text{cm} = 18 + 33 = 51 \, \text{cm}$

In this case: $H_1 - H_2 = 54 \, \text{cm} - 51 \, \text{cm} = 3 \, \text{cm}$

Trial 4 (Exact match):

Finally, try:

• First tower: 1 block of 18 cm + 2 blocks of 11 cm. [ H_1 = 18 + 22 =