For every 6 blue diamonds, I will get 1 red diamond. Also, after converting to red diamonds, the remaining blue diamonds will directly convert to red diamonds. For instance, for 15 blue diamonds, I will get 5 red diamonds. 2 red diamonds for 12 blue diamonds and 3 from remaining blue diamonds. Now, the question is, for how many least amount of blue diamonds, I will get 81 red diamonds.

Let's solve this step by step.

### The process of conversion is described as follows:
1. For every 6 blue diamonds, you get 1 red diamond.
2. After the conversion, any remaining blue diamonds are converted directly to red diamonds.

### Goal: To find the minimum number of blue diamonds needed to get exactly 81 red diamonds.

### Step 1: First conversion
For every 6 blue diamonds, you get 1 red diamond.

Let the initial number of blue diamonds be denoted as \( x \).

1. From the first step of conversion, the number of red diamonds is:
   \[
   \text{Red diamonds after first step} = \frac{x}{6}
   \]
   (integer part, as it's a ratio).

After this, \( r \) blue diamonds remain, where \( r \) is the remainder when dividing \( x \) by 6, i.e., \( r = x \mod 6 \).

### Step 2: Converting remaining blue diamonds
According to the problem, after the first conversion, the remaining blue diamonds are directly converted to red diamonds. So, the total number of red diamonds will be the sum of the red diamonds obtained from the first conversion plus the red diamonds converted from the remainder.

### Example:
For 15 blue diamonds:
- Divide 15 by 6:
  \[
  15 \div 6 = 2 \text{ (quotient)}, \, 3 \text{ (remainder)}
  \]
  You get 2 red diamonds from the first conversion, and the remaining 3 blue diamonds convert directly to 3 red diamonds, giving you a total of \( 2 + 3 = 5 \) red diamonds.

### Step 3: Generalization to find 81 red diamonds
Let's solve this iteratively or algebraically. To find the least number of blue diamonds needed to get exactly 81 red diamonds, we need to account for the conversions.

To reach 81 red diamonds, we need to solve for \( x \) in the equation:
\[
\frac{x}{6} + (x \mod 6) = 81
\]

We can compute this numerically by trying different values of \( x \).

Let me calculate that for you.The least number of blue diamonds required to get exactly 81 red diamonds is **461**.

Would you like more details on how this result was reached or have any other questions?

Here are 5 related questions to expand on this:

1. What if I need 100 red diamonds? How many blue diamonds would I need?
2. How would the result change if the conversion ratio was 5 blue diamonds for 1 red diamond?
3. What if I could convert remaining blue diamonds at a different ratio than 1-to-1?
4. Can this process be generalized for any number of target red diamonds?
5. How would the answer change if there was a penalty for leftover blue diamonds that don’t convert?

**Tip:** When solving problems like this, breaking them into smaller steps and setting up equations can help organize your approach.

Learn how to calculate the minimum number of blue diamonds needed to obtain exactly 81 red diamonds through a conversion process. This problem involves division, modulo operations, and solving linear equations. Suitable for Grades 8-10 students.

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