The problem asks for the least number of rectangular blocks (with dimensions $11 \, \text{cm} \times 11 \, \text{cm} \times 18 \, \text{cm}$) needed to make two towers such that one tower is exactly 2 cm taller than the other.

Step 1: Understanding the Problem

• The blocks can be stacked in any orientation.
• The goal is to find two towers where the height difference is exactly 2 cm, using the least number of blocks.

Each block can be oriented in three ways, giving the following possible heights:

• 11 cm (using the 11 cm face as the height),
• 18 cm (using the 18 cm face as the height).

Step 2: Representing the Towers

Let:

• $h_1$ be the height of the first tower.
• $h_2$ be the height of the second tower.
• The number of blocks in the first tower be $n_1$.
• The number of blocks in the second tower be $n_2$.

Thus, we want: $h_1 = h_2 + 2 \text{ cm}$

Step 3: Analyzing the Block Heights

Let's explore two orientations for simplicity:

1. If each block is 18 cm tall:

   • The first tower can have $n_1$ blocks, giving $h_1 = 18 \times n_1$.
   • The second tower can have $n_2$ blocks, giving $h_2 = 18 \times n_2$. Thus, we want: $18n_1 = 18n_2 + 2$ However, this is not possible since $18n_1 - 18n_2$ will always be a multiple of 18, so we can't get exactly a 2 cm difference.

2. If each block is 11 cm tall:

   • Similarly, this approach doesn't work since the difference between heights will always be a multiple of 11.

Step 4: Mixing Heights

The best way to approach the problem is by mixing orientations of blocks. If we can alternate between blocks of 11 cm and 18 cm heights, we might find a solution.

Let's now attempt to find the least number of blocks for the towers.

Give me a moment to compute thisThe least total number of blocks required to make the two towers, where the first tower is exactly 2 cm taller than the second, is 13 blocks.

Here’s how the towers are constructed:

• First tower: 5 blocks, each oriented with a height of 18 cm, giving a total height of $5 \times 18 = 90 \, \text{cm}$.
• Second tower: 8 blocks, each oriented with a height of 11 cm, giving a total height of $8 \times 11 = 88 \, \text{cm}$.

Thus, the first tower is 90 cm tall, and the second is 88 cm tall, which satisfies the condition that the first tower is 2 cm taller than the second.

Would you like further clarification or details?

Here are 5 related questions:

1. What if the height difference between the towers needed to be 3 cm instead of 2 cm?
2. Can the towers be constructed using a different mix of orientations?
3. How would the solution change if we had different block dimensions?
4. What is the maximum possible height difference with exactly 13 blocks?
5. Can we achieve the same height difference with fewer blocks by using fractions of blocks?

Tip: When solving optimization problems, consider brute force only if the range is small; otherwise, look for patterns or mathematical shortcuts.