4x4 Magic Square Calculator

Input Your 4x4 Grid

Enforce normal square (use each number 1 to 16 once)

Use normal enforcement for classic puzzles; disable it to explore generalized magic-style constructions.

Evaluation Summary

Enter values, generate a square, or load the example, then run the checker.

Magic constant for 4x4: 34

CSV exports the current grid values. PDF uses browser print-to-file for quick sharing.

Example 4x4 Magic Square (Normal, Using 1 to 16)

This classic layout satisfies equal sums of 34 across all rows, columns, and diagonals.

Row	Values				Row Sum
R1	16	2	3	13	34
R2	5	11	10	8	34
R3	9	7	6	12	34
R4	4	14	15	1	34
Column Sum	34	34	34	34
Diagonal Sums					Main = 34, Secondary = 34

Key Features of This 4x4 Magic Square Calculator

Instant validation of all rows, columns, and both diagonals.
Switchable normal-mode check using integers 1 to 16 once.
Random valid 4x4 normal magic square generator for practice.
Grid clearing, example loading, and export-ready reporting layout.

Normal and General Magic Square Analysis

Use the checkbox to constrain solutions to normal 1–16 layouts or explore generalized configurations.

The tool reports uniqueness, range coverage, and whether chosen values satisfy stricter puzzle conditions.

Diagnostic Insights for Teaching and Debugging

Per-line deviations from 34 highlight problematic rows or columns.
Total sum comparison indicates consistency with standard constructions.
Messages guide learners from near-miss grids to fully valid solutions.

Exports, Documentation, and Classroom Usage

CSV output supports archiving attempts and side-by-side comparisons.
Printable PDF reports fit worksheets, assignments, or puzzle booklets.
Clean white layout integrates smoothly into educational or research portals.

Formula Used

For a square of order n, the magic constant is:

M = n(n^2 + 1) / 2

For a 4x4 grid, M = 4(4^2 + 1)/2 = 4(17)/2 = 34.

A valid 4x4 magic square requires:

Every row sum equals 34.
Every column sum equals 34.
Both main diagonals sum to 34.

A "normal" 4x4 magic square also uses each integer from 1 to 16 exactly once.

How to Use This Calculator

Enter sixteen numbers into the 4x4 grid cells.
Optionally enable Enforce normal square to require numbers 1–16 uniquely.
Use Load Classic Example to display a standard normal magic square.
Use Generate Random Normal Variant for alternate valid configurations.
Click Check Magic Properties to compute sums and diagnostics.
Review deviations, uniqueness notes, and evaluation messages for detailed feedback.
Export your configuration via CSV or browser-based PDF for documentation.

Ideal for classrooms, puzzle builders, and researchers exploring structured numeric patterns.

Frequently Asked Questions

What is a 4x4 magic square?

A 4x4 magic square is a four-by-four grid where every row, column, and both main diagonals share the same magic constant, here 34. Normal versions also use each integer from 1 to 16 exactly once.

How does this calculator check if my grid is magic?

It sums each row, each column, and both diagonals, then compares them with 34. It also reports deviations, optional 1–16 uniqueness, and whether your grid qualifies as a valid magic square.

What does “Enforce normal square” mean?

When enabled, results are evaluated against strict normal rules: you must use all integers from 1 to 16 exactly once. Any duplicates or missing values are highlighted in the evaluation messages.

Can I explore non-standard magic-like patterns?

Yes. Leave normal enforcement unchecked to test grids using repeated numbers or other ranges while still checking whether lines reach the 34 magic constant. It is helpful for experimentation and teaching near-miss structures.

How is the random normal variant generated?

The tool starts from a known 4x4 magic square and applies random rotations and reflections. These symmetry operations preserve sums and uniqueness, guaranteeing every generated grid remains a valid normal magic square.

Why do I see red badges or deviations?

Red badges indicate rows, columns, or diagonals whose sums differ from 34. Use the deviation list to adjust entries step by step until all indicators turn green and the grid becomes fully magic.

How do CSV and PDF exports help me?

CSV lets you store or compare multiple grids in spreadsheets. The PDF option uses your browser’s print dialog, enabling quick sharing in assignments, presentations, puzzle collections, or lesson notes.