Are you able to recreate the mathematical formula or formulas/ equations? Or rather reverse engineer the formula? 

Reverse-engineering a precise mathematical formula from Samuel Benner's empirical observations and predictions involves analyzing the patterns he identified and trying to fit them into a mathematical model. Given Benner's original work was more heuristic than formulaic, we can still attempt to create a model based on observed cycles.

Let’s start by defining the cycles in a mathematical context:

### Step 1: Identifying Patterns
From the data, we can see periodic intervals in the years for different types of economic conditions:
- **Panics (A)** occur approximately every 18-20 years.
- **Good times (B)** occur in cycles of approximately 8-10 years within a broader 20-year cycle.
- **Hard times (C)** seem to follow a similar pattern but are offset.

### Step 2: Creating Functions
1. **Panic Years (A):**
   \[
   A(n) = A_0 + 18n \quad \text{for} \quad n \in \mathbb{Z}
   \]
   Where \(A_0\) is the base year (e.g., 1927).

2. **Good Times (B):**
   \[
   B(n) = B_0 + 9n \quad \text{within} \quad 20n \quad \text{cycles} \quad \text{for} \quad n \in \mathbb{Z}
   \]
   Where \(B_0\) is the base year (e.g., 1926).

3. **Hard Times (C):**
   \[
   C(n) = C_0 + 11n \quad \text{within} \quad 20n \quad \text{cycles} \quad \text{for} \quad n \in \mathbb{Z}
   \]
   Where \(C_0\) is the base year (e.g., 1924).

### Step 3: Applying Modulo for Cycles
To fit these into a cyclical model:
- For **Panics (A)**:
  \[
  A(t) = 1927 + 18 \times t \quad \text{mod} \quad 80
  \]

- For **Good Times (B)**:
  \[
  B(t) = 1926 + 9 \times t \quad \text{mod} \quad 20
  \]

- For **Hard Times (C)**:
  \[
  C(t) = 1924 + 11 \times t \quad \text{mod} \quad 20
  \]

### Step 4: Combining into One Model
A combined model might look something like this:
\[
E(t) = 
\begin{cases}
\text{Panic} & \text{if} \quad t \equiv 1927 + 18k \mod 80 \\
\text{Good Times} & \text{if} \quad t \equiv 1926 + 9k \mod 20 \\
\text{Hard Times} & \text{if} \quad t \equiv 1924 + 11k \mod 20 \\
\end{cases}
\]
Where \( k \in \mathbb{Z} \) and \( t \) is the year.

### Reverse-Engineering Example Calculation
To see if a specific year is a panic year:
- **For 1981:**
  \[
  1981 - 1927 = 54 \quad \text{which is not divisible by 18, thus not a panic year.}
  \]

- **For Good Times:**
  \[
  1980 - 1926 = 54 \quad \text{mod 20} \equiv 14 \quad \text{(which fits in the 9-year cycle)}
  \]

- **For Hard Times:**
  \[
  1985 - 1924 = 61 \quad \text{mod 20} \equiv 1 \quad \text{(which fits in the 11-year cycle)}
  \]

This approach offers a way to model Benner's cycles mathematically. However, note that this reverse engineering assumes perfect regularity in the cycles, which may not fully capture Benner's original heuristic observations.

Would you like more details or have any questions? Here are some questions you might consider:
1. How can we validate the accuracy of this model?
2. What are the limitations of using such a model for economic predictions?
3. How did Benner’s predictions hold up historically?
4. Can we integrate modern economic indicators into this model?
5. What are alternative models for predicting economic cycles?

**Tip:** Mathematical models based on historical patterns can provide insights but should be used cautiously, considering the complexity and many variables influencing economies.

Explore how to reverse-engineer Samuel Benner's empirical observations into a mathematical model. Learn about cyclic economic patterns including panics, good times, and hard times, with detailed formulas and calculations. Ideal for advanced learners and economists.

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