We are tasked with determining the least total number of blocks required to construct two towers such that one tower is exactly 2 cm taller than the other. Each block has dimensions $11 \, \text{cm} \times 11 \, \text{cm} \times 18 \, \text{cm}$ and can be stacked in any orientation.

Key considerations:

1. The blocks can be stacked with one of three possible heights: 11 cm, 11 cm, or 18 cm, depending on which side is placed vertically.
2. We need to construct two towers, such that the height difference between them is exactly 2 cm.

Approach:

Let the height of the first tower be $h_1$ and the height of the second tower be $h_2$. The problem asks us to ensure that: $h_1 = h_2 + 2 \, \text{cm}.$ Both $h_1$ and $h_2$ must be formed by stacking a combination of blocks in any orientation, which means each height must be a sum of multiples of 11 cm and 18 cm.

We need to explore combinations of these heights to find the smallest total number of blocks that satisfies the condition.

Let me calculate and find the minimal solution.The least total number of blocks required is 13 blocks. Here's how the towers are constructed:

• First tower: 5 blocks with a total height of 90 cm.
• Second tower: 8 blocks with a total height of 88 cm.

The first tower is exactly 2 cm taller than the second tower, satisfying the problem's condition.

Would you like more details or further clarification?

Here are 5 related questions for practice:

1. How would the solution change if the required height difference was 5 cm instead of 2 cm?
2. What is the impact of changing block dimensions on the total number of blocks needed?
3. How would you generalize the method used to solve this problem to other height differences?
4. What if the blocks were not allowed to be placed in any orientation?
5. Could we determine an algebraic formula for the minimum number of blocks?

Tip: When solving combinatorial problems like this, breaking down the problem into possible configurations and minimizing the solution helps save time and effort!