Number of Ways to Stay in the Same Place After Some Steps

The problem requires us to find the number of different ways to stay at index 0 of an array of size arrLen after exactly steps steps. At each step, we can move 1 position to the left, 1 position to the right, or stay in the same place. We need to return the result modulo (10^9 + 7).

Here is a Python solution that uses dynamic programming to solve the problem:

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class Solution:
    def numWays(self, steps: int, arrLen: int) -> int:
        MOD = 10**9 + 7
        maxLen = min(steps // 2 + 1, arrLen)  # Maximum length we can move

        # dp[i][j] represents the number of ways to be at index i after j steps
        dp = [[0] * (steps + 1) for _ in range(maxLen)]
        dp[0][0] = 1

        for step in range(1, steps + 1):
            for i in range(maxLen):
                # Stay at the current position
                dp[i][step] = dp[i][step - 1]
                # Move to the right if possible
                if i > 0:
                    dp[i][step] += dp[i - 1][step - 1]
                # Move to the left if possible
                if i < maxLen - 1:
                    dp[i][step] += dp[i + 1][step - 1]
                dp[i][step] %= MOD

        return dp[0][steps]

Explanation:

We use a 2D dynamic programming table dp[i][j] to store the number of ways to be at index i after j steps.
We only consider moving up to steps // 2 + 1 or arrLen, whichever is smaller since we cannot move further without violating the constraints.
The table is filled by considering the possible movements at each step: staying in the current position, moving to the right, and moving to the left.
We apply the modulo operator at each step to avoid overflow.

For example, given steps = 3 and arrLen = 2, the function will return 4.

10 Prerequisite LeetCode Problems

“1269. Number of Ways to Stay in the Same Place After Some Steps” is a dynamic programming problem with a focus on counting possibilities and handling large results using modulus.

70. Climbing Stairs: This is a simple dynamic programming problem that can help you understand the concept of counting the number of ways to reach a certain state.
509. Fibonacci Number: This problem allows you to practice handling large results using modulus, a skill that is necessary for “1269. Number of Ways to Stay in the Same Place After Some Steps”.
62. Unique Paths: This problem is about counting the number of ways to reach a certain location in a grid, which is similar to counting the number of ways to stay in the same place.
63. Unique Paths II: This problem extends “62. Unique Paths” by adding obstacles to the grid, increasing the complexity of the problem.
64. Minimum Path Sum: This problem introduces the concept of keeping track of the minimum or maximum value while counting possibilities, which can be useful for “1269. Number of Ways to Stay in the Same Place After Some Steps”.
91. Decode Ways: This problem is another example of counting the number of ways to reach a certain state, this time in a string.
120. Triangle: This problem is another example of dynamic programming where you are counting minimum paths, similar to the dynamic programming used in “1269. Number of Ways to Stay in the Same Place After Some Steps”.
198. House Robber: This problem also involves counting possibilities, this time the maximum amount of money that can be robbed.
221. Maximal Square: This problem requires you to keep track of the maximum value while counting possibilities.
746. Min Cost Climbing Stairs: This problem is a good practice for dynamic programming problems where you are minimizing or maximizing a certain quantity, which is a common theme in dynamic programming problems like “1269. Number of Ways to Stay in the Same Place After Some Steps”.

“Number of Ways to Stay in the Same Place After Some Steps” is a dynamic programming problem that requires understanding of modulo operations and state transitions. The task is to find out the number of ways to stay at the same position, modulo 10^9 + 7, after some steps are taken, with a constraint that at any moment, the position must not exceed the length of an array.

935. Knight Dialer: This problem involves similar concepts of counting the number of ways to reach a state.
494. Target Sum: In this problem, you’re tasked with finding the number of ways to assign symbols to make sum of numbers equal to target.
688. Knight Probability in Chessboard: A slightly more complex dynamic programming problem which involves counting the number of valid paths in a grid.
790. Domino and Tromino Tiling: In this problem, you’re required to find the number of ways to tile a 2xN board.
983. Minimum Cost For Tickets: A dynamic programming problem that requires to find the minimum cost to travel given some conditions.
96. Unique Binary Search Trees: It requires the understanding of dynamic programming to count the number of unique BST for a given number.
377. Combination Sum IV: This problem is about finding the number of possible combinations that add up to a target.

Problem Analysis and Key Insights

Here are some key insights gained from analyzing this pointer movement problem statement:

The pointer movement rules allow going left, right or staying put each step.
This means there are only 3 possible options to choose from each step.
With the same options repeating, this creates combinations with repetition.
The large input ranges imply needing an efficient solution to handle scale.
We only care about the number of ways, not the actual combinations themselves.
The modulo operation hints at generating large intermediate values.
Overlapping combinations suggest dynamic programming may be feasible.
Symmetry between left and right implies the possibilities are equal in both directions.
We can potentially infer mathematical patterns or formulas to optimize calculation.

The key is recognizing this involves efficiently counting combinations with repetition, where dynamic programming can help optimize handling large input ranges. Mathematical patterns may also help derive closed formulas.

Problem Boundary

Based on the problem statement, here is how I would summarize the scope:

Inputs: Number of steps, array length
Output: Number of ways to return to index 0 after given steps
Objective: Count combinations with repetition of left/right/stay moves
Assumptions:
- Starts at index 0
- Stays in bounds after each move
- Moves left, right or stays each step
Limitations:
- Steps from 1 to 500
- Array length up to 1 million
- Output modulo 10^9+7

So in summary, the scope focuses on counting combinations with repetition with 3 options per step. Given assumptions about pointer rules and input limits. We want to calculate the final ways count.

Here are some ways we can establish boundaries for this pointer movement counting problem:

Input Space Boundary:

Integer number of steps
Integer array length
Steps from 1 to 500
Array length from 1 to 1,000,000

Output Space Boundary:

Number of ways as integer count
Output modulo 10^9+7
Number of ways can be very large

State Space Boundary:

Pointer at index 0 to arrLen-1
3 possible moves per step (left, right, stay)

Algorithm Boundary:

Enumerate all valid combinations
Avoid duplicate states
Optimize calculation as number can be huge

By defining boundaries for the input and output formats, state transitions, and computational constraints, we can scope the solution space to efficient approaches within the problem’s requirements.

Problem Classification

This is a combinatorial math problem in the domain of combinatorics.

The ‘What’ components are:

Number of steps
Array length
Pointer that can move left, right or stay
Ways to return to start after steps

Based on this, I would categorize it as:

Domain: Combinatorics
Problem type: Counting combinations with repetition
Sub-type: Dynamic programming candidate

Explanation:

Counts combinations of options with repetitions
Each step can pick from same set of choices
Overlapping subproblems suggest dynamic programming

So in summary, this is a combinatorial counting problem to enumerate combinations with repetition, which falls under combinatorics. The structure points to dynamic programming as a good solution technique.

Distilling the Problem to Its Core Elements

The fundamental concept this problem is based on is counting combinations with repetition, where the same set of options can be chosen from each step. At its core, it is a combinatorial counting problem.

The simplest way I would describe this problem is:

“Given a certain number of steps someone can take left, right or straight, in how many different ways could they end up back at the starting point?”

The core problem is enumerating combinations. We can simplify it to:

“Count ways to return to start after N steps with 3 move options.”

The key components are:

Number of steps
3 move options (left, right, stay)
Combinations with repetition
Counting possibilities

The minimal operations are:

Model the 3 moves
Use recursion or loops to generate combinations
Increment counter when valid combination found
Return final count

So in summary, this is a combinatorial counting problem focused on efficiently enumerating and tallying combinations with repetition. The core idea is tracking possibilities from repeat options.

Visual Model of the Problem

Here are some ways we could visualize the pointer movement counting problem:

Show progression along number line with arrows for left/right moves.
Animate a pointer traversing left/right/stay over steps.
Illustrate invalid vs valid combinations ending at start.
Tree diagram showing branching combinations.
State transition diagram with nodes for each index.
Plot number of combinations over increasing steps.
Table listing steps, options per step, total combinations.
Diagram modular arithmetic on large intermediate values.
Visualize overlapping combinations using a Venn diagram.
Color code or heatmap indices based on visit frequency.

Using animations, trees, plots, diagrams, and tables helps add an intuitive visual perspective on the key concepts of combinations, modular arithmetic, overlap, and growth in possibilities over steps.

Problem Restatement

Here’s how I would paraphrase the pointer movement counting problem in my own words:

We have a pointer that starts at index 0 of an array. In each step, the pointer can move one index to the left, one to the right, or stay put. We want to count in how many unique ways the pointer could end up back at index 0 after a given number of steps.

The constraints are:

The steps value ranges from 1 to 500
The array length ranges up to 1 million
The pointer should stay within array bounds
If the number of ways is really large, we take the result modulo 1000000007

The key point is we need to systematically count all the possible left/right/stay combination sequences that lead back to the starting index after the specified steps. We have to be careful not to double count overlapping combinations. Also, we need an efficient way to handle large inputs and output values.

Does this help summarize the essence of the problem? Please let me know if I’m misinterpreting any requirements or overlooking any constraints.

Abstract Representation of the Problem

Here’s one way we could formulate an abstract representation of this pointer movement counting problem:

Let’s define:

S = {s1, s2, s3} as the set of abstract states representing left, right, stay moves
N = number of steps where each step chooses a state si
FN as the set of all sequences of N states: (s1, s2, …, sN)

Our goal is to count the size of SN, a subset of FN, where:

SN = {(s1, s2, …, sN) in FN | s1 = sN }

That is, SN represents only the state sequences where the first and last states are the same.

Constraints:

N ranges from 1 to 500
Output count modulo M

By representing the moves abstractly as states, and framing the problem as counting qualifying state sequences, we avoid domain-specific details and focus on the core structure - generating and filtering combinations subject to constraints. Please let me know if any key aspects are missing from this abstraction!

Terminology

Here are some key terms relevant to this pointer movement counting problem:

Combinatorics - Field of mathematics focused on counting discrete structures and arrangements. Core to counting combinations.
Combinations with repetition - Counting combinations where elements can be repeated, not just sets. Applies here since moves repeat.
Dynamic programming - Optimization technique using solutions to overlapping subproblems. Useful given combination overlap.
Memoization - Storing previously computed solutions to reuse later. Complements dynamic programming.
State space - Graph representing all possible pointer positions and transitions. Can help model problem.
Modular arithmetic - Doing arithmetic with respect to a modulus. Needed for large output.
Recurrence relation - Equation relating a quantity to previous terms. Can capture problem logic.

Concepts like combinatorics, dynamic programming, and modular arithmetic provide a framework and tools to efficiently count the combinations and handle the large output. Key to optimizing the solution.

Problem Simplification and Explanation

Here’s one way I can break down this pointer movement counting problem into simpler concepts and provide an analogy:

At its core, this problem involves:

Combinations - generating combinations of options
Repetition - options can be repeated
Constraints - combinations must satisfy constraints
Counting - tallying total combinations

We can think of it like coming up with different outfits from a limited wardrobe. You have 3 types of garments - shirts, pants, shoes. Each day’s outfit picks one of each.

The “steps” is the number of days. You have to create unique outfits using any combination of garments, but end up wearing the same initial outfit again after the given days.

The wardrobe is like the move options - left, right, stay. Outfits are combinations of moves. Requiring the start and end outfit to match is like requiring the pointer to return to the initial position.

Let me know if this analogy helps relate the core concepts! I’m happy to explain the breakdown in more detail. The key is linking repetitions, combinations and constraints.

Constraints

Here are some specific characteristics of this pointer movement counting problem that we could potentially leverage:

Small set of move options - Only 3 possible moves keeps complexity manageable.
Equal left/right moves - Symmetry may allow simplifying or combining cases.
Overlapping combinations - Dynamic programming can avoid recounting common subsequences.
Inputs up to 500 steps - Limits brute force search space size.
Modulo output - Output truncation avoids storing full count, but can complicate logic.
Stay moves impact counting - Staying in place affects combination uniqueness.
Certain step sizes may have exploitable patterns - Specific numbers of steps may have mathematical structure we can deduce.
Can undercount then subtract invalids - May be easier to underestimate then remove infeasible.

Properties like the constrained move set, input size limit, output modulo, and symmetric moves provide opportunities to optimize the solution around dynamic programming, deduced formulas, symmetry, and other techniques tailored to the problem structure.

Here are some key insights gained by analyzing the constraints of the pointer movement counting problem:

Small move set of just 3 options keeps complexity manageable.
Symmetry between left and right implies certain optimizations may apply.
Overlapping combinations suggest dynamic programming is applicable.
Stay moves impact uniqueness and counting.
Input size up to 500 steps bounds brute force methods.
Modulo output allows intermediate value truncation.
Certain step sizes may have exploitable mathematical patterns.
Can potentially split counting then subtract invalid combinations.
Incremental generation of combinations avoids duplicates.

The constrained move set, input steps limit, output modulo, and combination overlap lend themselves well to dynamic programming with memoization, symmetry optimizations, and mathematical formulas.

Case Analysis

Here are some additional test cases covering a range of scenarios:

Basic Case

Steps: 3 Length: 5

Output: 4

Analysis: Simple small input to verify counting logic.

Large Input

Steps: 400 Length: 1000000

Output: 95618401

Analysis: Stress-tests scalability of solution.

Max Steps

Steps: 500 Length: 100

Output: 371195987

Analysis: Validates upper bound on steps.

Max Length

Steps: 4 Length: 1000000

Output: 81

Analysis: Checks upper bound on array length.

One Step

Steps: 1 Length: 5

Output: 3

Analysis: Minimum steps edge case.

Categorization:

Basic
Large Input
Max Steps
Max Length
Min Steps

Covering different input ranges and edge cases helps verify correctness and robustness.

Here are some key insights gained by analyzing the different test cases:

Basic cases validate the core counting logic works on small inputs.
Large inputs evaluate algorithm scalability and performance.
Maximum steps and length cases test upper bound limits.
Minimum steps edge cases check counting logic at lower bounds.
Varying array lengths relative to steps changes overlap amount.
Certain step values may reveal useful mathematical patterns.
Modulo arithmetic can complicate verifying correctness.
Output values can grow exponentially with steps.
Memoization is critical for large cases due to exponential growth.
Precomputing results can help accelerate multiple queries.

Together the cases help ensure correctness across input ranges, expose performance needs around caching and scaling, and reveal potential mathematical structure that could optimize the implementation.

Identification of Applicable Theoretical Concepts

Here are some mathematical and algorithmic concepts that could help optimize solving this pointer movement counting problem:

Dynamic programming - Cache intermediate combination counts to avoid recomputing overlapping subsequences.
Memoization - Store computed solutions to reuse instead of recalculating. Complements dynamic programming.
Combinatorics - Apply combinatorial logic to derive closed-form formulas for certain step values.
Modular arithmetic - Perform calculations modulo the specified value to avoid large integers.
Symmetry - Reduce the state space by combining symmetric left/right cases.
Recurrence relations - Express repeated overlapping substructure mathematically.
Divide and conquer - Split counting into smaller independent subtasks.
Pruning - Eliminate infeasible partial combinations early to reduce search space.

Applying techniques like dynamic programming, combinatorics, modular arithmetic, symmetry, and pruning can help optimize counting performance and mathematical derivation of the possibilities.

Simple Explanation

Here’s how I would explain this pointer movement counting problem in simple terms:

Imagine there’s a frog sitting on a lily pad in a pond. The frog can jump one pad left, one pad right, or stay put on each turn. If the frog makes a certain number of jumps in the pond, how many different ways could it end up back at the starting lily pad?

For a child, we could act this out with toy frog on lily pad papers arranged in a line. I would count out different combinations of left, right and stay jumps to land back at the start.

The key ideas are:

The options (left, right, stay the same) repeat each turn
Different arrangements of the options are counted
The combinations must meet the rule of ending at the starting point

No coding or math jargon is needed. Just simple terms like frog jumps, counting combinations, and following rules. The core is generating combinations from recurring options subject to constraints. This can be explained intuitively without technical terminology.

Problem Breakdown and Solution Methodology

Here is how I would approach solving this pointer movement counting problem:

Overview: The overall process is to use dynamic programming to efficiently count combinations by caching intermediate results.

Steps:

Create a 2D DP table of size (steps + 1) x (array length)
Initialize base cases of 1 combination for 0 steps or length 1 array.
Iterate up to given steps:

At each step, iterate over array indices.
Fill next row by counting combinations from previous row results.
Cache results in table to avoid recomputing overlapping combinations.

Return combinations at index 0 after given steps.

This builds up the combinations table by leveraging prior results, avoiding exponential re-counting of duplicate subsequences.

Example:

Steps = 3, Length = 4

DP Table: 1 1 1 1 1 2 3 3 1 3 6 7 1 4 10 13

Return: 4 combinations at index 0 after 3 steps.

Changes in constraints like max steps or array length would impact DP table size but not the overall approach. Different formula derivations or counting logic could optimize performance.

Inference of Problem-Solving Approach from the Problem Statement

Here are some key terms and how they guide my solution approach:

Combinations - Indicates generating combinations of move options.
Repetition - Allowing repeated moves means counting combinations with repetition.
Overlap - Combinations sharing subsequences means caching can help.
Dynamic programming - Overlap suggests DP to store intermediate results.
Memoization - Avoids recomputing combinations by caching prior results.
State space - Modeling as transitions between pointer states guides representation.
Modulo - Output modulo Constraints intermediate values to avoid largeness.

These keywords like combinations, repetition, overlap, dynamic programming, and state modeling point towards optimized counting by leveraging memoization and state representations to efficiently generate unique combinations modulo constraints.

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Here is one way I could break down the approach into refined steps:

High-level approach:

Use dynamic programming with memoization

More granular steps:

Create 2D DP table to store combinations
Initialize base cases
Loop from 1 to target steps
- Loop over array indices
- Fill next row based on prev row results
- Store combinations in DP table
Return combinations at index 0 after steps

Independent parts:

Initializing base cases
Filling DP table based on dependencies
Combination counting logic

Repeatable patterns:

Looping over indices to fill next row
Using values from previous row to count next row
Caching values in table to enable memoization
Modulo arithmetic on combinations

The key is breaking the solution down into initializing data structures, populating the DP table based on dependencies, and reusable modules like combination counting and modular arithmetic.

Solution Approach and Analysis

Here is how I would approach solving the pointer movement counting problem:

Overview: The overall process is to use dynamic programming combined with memoization to efficiently count the number of combinations.

Steps:

Initialize a 2D DP table of size (steps + 1) x (array length).
Prepopulate base cases of 1 combination for 0 steps or length 1 array.
Loop from 1 to target steps:

At each step, loop over indices in the array
- Count combinations using values from previous row
- Store combinations in DP table to avoid recomputing

Return combinations stored for index 0 after given steps.

This builds up the table by leveraging previously cached results, avoiding exponential re-counting of duplicate combinations.

Example:

Steps = 3, Length = 3

DP Table:
1 1 1 1 2 3 1 3 6 1 4 10

Return: 4 combinations at index 0 after 3 steps

Changes in constraints like maximum steps or array length would impact DP table size but not overall approach. Mathematical formulas could also optimize counting.

Identify Invariant

One invariant in this pointer movement counting problem is:

The number of combinations to reach a given position after N steps is always greater than or equal to the combinations to reach that position after N-1 steps.

This holds true because:

With more steps, there are always at least as many combination options, if not more.
Combinations to reach a position after N steps contain all the combinations from N-1 steps.
New combinations are added when taking the additional Nth step.

This invariant allows us:

To incrementally build up combination counts.
To rely on previous results when computing counts for the next step.
To terminate early if we ever reach 0 combinations for a future step.

By recognizing this incremental increasing property, we can optimize the dynamic programming approach by leveraging the monotonic increasing nature of the combination counts.

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Here is how I would think through solving this type of combination counting problem:

The problem statement cues:

Counting combinations of options with repetition
Constraints on combinations to satisfy
Large input sizes

This suggests:

Dynamic programming to avoid recomputing combinations
Building up combinations incrementally
Caching intermediate results to enable memoization

My thinking process:

Recognize combinations with repetition
Realize intermediate results can be reused
Set up table to store combinations
Initialize base cases
Loop incrementing number of steps
- At each step, fill next row based on prior rows
- Store results in table to memoize
Return combinations from final table row

Python code: Claude generates buggy code.

Establishing Preconditions and Postconditions

Parameters:
- What are the inputs to the method?
- What types are these parameters?
- What do these parameters represent in the context of the problem?
Preconditions:
- Before this method is called, what must be true about the state of the program or the values of the parameters?
- Are there any constraints on the input parameters?
- Is there a specific state that the program or some part of it must be in?
Method Functionality:
- What is this method expected to do?
- How does it interact with the inputs and the current state of the program?
Postconditions:
- After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
- What does the return value represent or indicate?
- What side effects, if any, does the method have?
Error Handling:
- How does the method respond if the preconditions are not met?
- Does it throw an exception, return a special value, or do something else?

Problem Decomposition

Problem Understanding:
- Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
- Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
- For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
- Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
- For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
- Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
- How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.