---
name: problem-solver
description: Step-by-step problem solving with detailed explanations. Uses structured approach (Identify, Plan, Execute, Verify) for math, physics, chemistry, and CS problems. Can provide hints before solutions. Use when solving academic problems, working through practice questions, or learning problem-solving methods. Triggers - solve problem, step by step, show work, explain solution, how to solve, problem help.
---

# Problem Solver

Structured problem-solving with step-by-step explanations.

## Problem-Solving Framework

```mermaid
flowchart LR
    A[Understand] --> B[Plan]
    B --> C[Execute]
    C --> D[Verify]
    D -->|Error| B
    D -->|Correct| E[Explain]
```

---

## Step 1: Understand (Identify)

### Questions to Ask
1. **What is given?** List all known information
2. **What is asked?** Clearly state what to find
3. **What type of problem?** Categorize (algebra, kinematics, etc.)
4. **What are the constraints?** Any restrictions or conditions

### Template
```markdown
**Given:**
- [List all known values with units]

**Find:**
- [What we need to calculate]

**Type:**
- [Problem category]
```

---

## Step 2: Plan (Strategy)

### Choose Approach
| Problem Type | Common Strategies |
|--------------|-------------------|
| Equation solving | Isolation, substitution, factoring |
| Word problems | Define variables, write equations |
| Physics | Draw diagram, identify forces/energies |
| Proofs | Direct, contradiction, induction |
| Optimization | Take derivative, find critical points |

### Write Steps (Before Executing)
1. First, I will...
2. Then, I will...
3. Finally, I will...

---

## Step 3: Execute (Solve)

### Show All Work
```markdown
**Step 1:** [Action]
[Mathematical work]
= [Result]

**Step 2:** [Action]
[Mathematical work]
= [Result]

**Step 3:** [Final calculation]
[Work]
= [Final answer with units]
```

### Best Practices
- One step at a time
- Include units throughout
- Box or highlight final answer
- Keep work organized vertically

---

## Step 4: Verify (Check)

### Verification Methods

| Method | How to Apply |
|--------|--------------|
| Plug back in | Substitute answer into original equation |
| Units analysis | Check dimensions match expected result |
| Estimate | Is the magnitude reasonable? |
| Alternate method | Solve differently, compare results |
| Edge cases | Does answer work at extremes? |

### Template
```markdown
**Check:**
- Substituting x = [answer] into original equation:
  [show substitution] = [expected value] ✓
- Units: [unit analysis]
- Reasonableness: [is the answer logical?]
```

---

## Teaching Modes

### Hint Mode
Give progressive hints, not full solutions:

**Hint 1:** "What type of problem is this?" (Identification)
**Hint 2:** "What formula relates these quantities?" (Direction)
**Hint 3:** "Try isolating variable X first." (Specific guidance)

### Socratic Mode
Guide through questions:
- "What do you notice about the given information?"
- "What principle applies here?"
- "What would happen if you tried...?"

### Full Solution Mode
Complete step-by-step solution with explanations.

---

## Common Problem Patterns

### Pattern 1: Solve for X
1. Simplify both sides
2. Collect X terms on one side
3. Isolate X

### Pattern 2: Word Problem
1. Define variables
2. Translate words to equations
3. Solve system of equations
4. Answer in context

### Pattern 3: Rate Problem
1. Identify rate (units per time)
2. Use: Amount = Rate × Time
3. Set up equation and solve

### Pattern 4: Optimization
1. Write objective function
2. Take derivative
3. Set equal to zero
4. Solve for critical points
5. Test for max/min

---

## Answer Template

```markdown
## Problem
[State the problem]

## Solution

### Understanding
- **Given:** [knowns]
- **Find:** [unknown]

### Approach
[Brief strategy explanation]

### Work
**Step 1:** [action]
[calculation]

**Step 2:** [action]
[calculation]

### Answer
**[Final answer with units]**

### Verification
[Check work]
```