To solve this problem, we need to find the minimum number of blocks required to construct two towers with a specific height difference. The first tower needs to be exactly 2 cm taller than the second tower.

Step-by-Step Solution:

Step 1: Understand the Block Dimensions and Orientation

We have an unlimited supply of blocks with dimensions:

• $11 \, \text{cm} \times 11 \, \text{cm} \times 18 \, \text{cm}$

Each block can be stacked in one of three orientations:

1. Height = 11 cm (if the 11 cm x 18 cm face is the base)
2. Height = 11 cm (if the 11 cm x 18 cm face is the base)
3. Height = 18 cm (if the 11 cm x 11 cm face is the base)

For our problem, we can use either 11 cm or 18 cm as the height of each block.

Step 2: Represent the Heights of Two Towers

Let:

• $x$ = Number of blocks used for the first tower.
• $y$ = Number of blocks used for the second tower.

The height of the first tower (in cm) can be expressed as:

$H_1 = 11a + 18b$

where $a$ and $b$ are the number of blocks oriented with heights 11 cm and 18 cm, respectively.

Similarly, the height of the second tower can be expressed as:

$H_2 = 11c + 18d$

where $c$ and $d$ are the number of blocks oriented with heights 11 cm and 18 cm, respectively.

Step 3: Set Up the Equations for Height Difference

The problem states that the first tower must be exactly 2 cm taller than the second tower:

$H_1 = H_2 + 2$

Substitute the height expressions:

$11a + 18b = 11c + 18d + 2$

This equation needs integer solutions for $a, b, c,$ and $d$.

Step 4: Explore Possible Combinations

We want to minimize the total number of blocks used, $x + y = (a + b) + (c + d)$.

Let's consider a few cases:

1. Case 1: All Blocks Oriented at 18 cm
   If all blocks are oriented to have a height of 18 cm, we have:

   $18b = 18d + 2 \implies b = d \, \text{(no integer solution for height difference)}$

   This case does not work since adding 2 cm is impossible with blocks oriented at 18 cm only.

2. Case 2: Mixing 11 cm and 18 cm Orientations
   Let's try combinations with both 11 cm and 18 cm orientations:

   We set:

   $H_1 = 18 \times 1 + 11 \times 1 = 18 + 11 = 29 \, \text{cm}$

   $H_2 = 11 \times 2 = 22 \, \text{cm}$

   Here, $H_1 = 29 \, \text{cm}$ and $H_2 = 22 \, \text{cm}$, so the difference is:

   $H_1 - H_2 = 29 - 22 = 7 \, \text{cm} \, (\text{not 2 cm})$

   This doesn't work either.

Step 5: Identify the Correct Combination

We aim to find the smallest configuration such that $H_1 - H_2 = 2$. By checking through combinations:

• First Tower: $18 \times 3 + 11 \times 1 = 54 + 11 = 65$
• Second Tower: $18 \times 3 = 54$

The difference is:

$H_1 - H_2 = 65 - 54 = 2 \, \text{cm}$

Thus, the first tower uses 4 blocks (3 blocks of 18 cm and 1 block of 11 cm), and the second tower uses 3 blocks (all 18 cm blocks).

Step 6: Conclusion

The least total number of blocks required is:

$4 + 3 = 7 \, \text{blocks.}$

Answer

The least total number of blocks required to make two towers such that the first is exactly 2 cm taller than the second is 7 blocks.