Question

Magic squares. An \(n \times n\) matrix that is filled with the numbers \(1,2,3, \ldots, n^{2}\) is a magic square if the sum of the clements in each row, in cach column, and in the two diagonals is the same value. Write a program that reads in 16 values from the keyboard and tests whether they form a magic square when put into a \(4 \times 4\) table. You need to test two features: 1\. Does each of the numbers \(1,2, \ldots, 16\) occur in the user input? 2\. When the numbers are put into a square, are the sums of the rows, columns, and diagonals equal to each other?

Short answer

Read numbers, validate 1-16 sequence, compute sums, and confirm equality.

Step-by-step solution

Gather User Input

Create a function to read 16 integer values from the user. These values will be stored in a list named `numbers`.

Validate Unique Numbers

Check if the list `numbers` contains the integers from 1 to 16 with no repeats. You can achieve this by checking if sorting `numbers` gives the sequence [1, 2, 3, ..., 16].

Arrange Numbers into Matrix

Convert the list `numbers` into a 4x4 matrix. You can achieve this by slicing the list into four sublists, each containing four elements.

Calculate Row Sums

Compute the sum of each row in the matrix and store these sums in a list named `row_sums`.

Calculate Column Sums

Compute the sum of each column in the matrix, and store these sums in a list named `col_sums`. This can be done by iterating over column indices and summing values across rows.

Calculate Diagonal Sums

Calculate the sum of the main diagonal and the anti-diagonal. Store these values in `main_diag_sum` and `anti_diag_sum`, respectively. For the main diagonal, sum elements where the row index equals the column index. For the anti-diagonal, sum elements where the row index plus the column index is equal to n-1.

Check Magic Condition

Check if all values in `row_sums`, `col_sums`, `main_diag_sum`, and `anti_diag_sum` are equal. If so, the matrix is a magic square; otherwise, it is not.

Key concepts

Matrix Operations

Understanding matrix operations is crucial when working with magic squares. A matrix is essentially a two-dimensional array of numbers. In our problem, the matrix is represented as a 4x4 table holding 16 numbers. Operations like converting a list into a matrix and computing sums along the rows, columns, and diagonals are key steps in verifying a magic square.

To turn a list into a matrix, think of slicing the list into equal parts. For instance, a list with 16 numbers can be chopped into four smaller lists, each containing four numbers. Visualize this as four rows with four numbers in each. This makes handling and computing operations like sums more manageable.

For sums, iterate through the matrix by rows and columns. Calculating the row sums involves simply adding up the numbers in each row. Column sums are slightly more complex. Here, you iterate over each column index, summing values as you move across rows. This repetitive action gives us all necessary components to check if our matrix is a magic square.

Validation of Input

Input validation ensures that the matrix can potentially be a magic square. The first step is to confirm that all numbers from 1 to 16 are present without repetition. This is because a magic square uses each number only once. To validate this, sort the list of user input and compare it against the desired sequence [1, 2, 3, ..., 16].

If the sorted list matches, the numbers are unique and complete, suggesting the input is valid for further magic square checks. Without this initial validation, subsequent checks on sums could falsely mark a non-magic square as valid due to repeating or missing numbers. Thus, input validation is an essential line of defense before delving into deeper calculations.

Programming Logic

Solid programming logic is the backbone of solving the magic square problem. It guides the methodology used to gather, process, and verify the data. The logic is structured into clear, manageable steps, each building towards confirming whether a given set of numbers can form a magic square.

The logic begins with input collection. This is achieved by reading user-provided numbers. Following this, validation confirms that the input meets the basic criteria. The use of sequences and conditions helps dictate operations like looping through rows and columns, and if statements to verify sums.

Another key aspect is breaking down tasks into smaller functions, such as one for sum calculation and another for validation. This modular approach makes the program easier to develop, test, and debug. Each segment of code has a clear goal, simplifying the overall logic flow.

Algorithm Design

Designing an algorithm for checking magic squares requires careful thought and planning. An algorithm is a step-by-step procedure, ensuring tasks are systematically tackled. For validating magic squares, the algorithm excites from input handling through to final verification.

It begins with the single purpose of collecting and storing the numbers. Specifically, input must be handled in a way that lays the groundwork for building the matrix. The algorithm then segues into validation, ensuring completeness and uniqueness. This step helps guarantee that the remaining algorithm steps will not use faulty data.

Subsequently, matrix transformation and sum calculation form the heart of our algorithm. By iterating correctly, each part like row and column sums is calculated and compared as part of the final verification. The design culminates in checking the equality of these sums across different dimensions or aligning diagonals. This structured approach ensures that every potential grid is thoroughly tested for validity as a magic square.