---
title: "Hard magic square: free printable with answers"
description: "Free printable hard magic squares with answers. A 4 by 4 grid using 1 to 16, each line summing to 34. One completion, printed with the answer."
source: "https://www.oncebooks.com/magic-square/hard"
---

1. [Home](https://www.oncebooks.com/)
2. [Magic square](https://www.oncebooks.com/magic-square)
3. Hard

Free printable

# Hard magic square

A hard magic square steps up to a 4 by 4 grid using the numbers 1 to 16, where every row, column and diagonal adds to 34. More cells means more to balance, and each puzzle still has exactly one completion.

PuzzleDesign

Difficulty

Larger grids with fewer givens are harder. Every puzzle has one completion.

4×4 grid. Fill it with 1 to 16 so every row, column and diagonal sums to 34.

Theme

ClassicBoldSoftShadedNavyIndigoSkyTealEmeraldForestPlumBerryCrimsonEmberAmberCocoa

**Customise colours**

Numbers

Cell lines

Box lines

Page background

Style

StandardClean

Clean drops the title band and footer, leaving the grid.

Generate puzzlePrint / save PDF

Two pages: the puzzle and its completed answer. Every puzzle has one solution, checked by the same engine as our KDP studio.

## Want the whole book?

This makes one puzzle. Sign up to turn it into a full Magic Square book for Amazon KDP: hundreds of graded puzzles at any trim, as print-ready PDFs.

[Build the full book](https://app.oncebooks.com/publisher-studio/magic-square/edit?from=magic-square&v=2&s=eyJkaWZmaWN1bHR5IjoiSEFSRCIsImNvbG9yVGhlbWUiOnsidGhlbWVJZCI6ImNsYXNzaWMiLCJvdmVycmlkZXMiOnt9fX0)

Free to build · pay once to export

Preview

The puzzle and its completed answer, as they print.

Hide the preview

- Magic Square Hard

  Every row, column and diagonal sums to 34

  1

  Puzzle
- Magic Square Hard · Solution

  Every row, column and diagonal sums to 34

  2

  Solution

Generate puzzlePrint / save PDF

## How to solve a magic square

Fill the grid with each number from 1 to N² once so every row, column and both diagonals add up to the same total.

### The method

1. Find the magic total. For an N by N grid the total is N × (N² + 1) ÷ 2, printed under the grid, so a 3 by 3 is 15 and a 4 by 4 is 34.
2. Start where a line is nearly full. A row, column or diagonal with only one empty square is forced: the missing number is the total minus the others.
3. Use each number once. Every value from 1 to N² appears exactly once, so cross off numbers as you place them and you narrow the choices fast.
4. Cross rows with columns. A square has to satisfy its row, its column and any diagonal it sits on. Where only one unused number fits all of them, lock it in.

**The one rule**

Use 1 to N² once each so every row, column and both diagonals share the same sum.

**What the answer page shows**

The fully completed square is printed at the back.

**How the grades differ**

Bigger grids and fewer given numbers mean more to deduce before the grid settles.

**Good for**

Solvers who enjoy arithmetic and a tidy, self-checking grid.

## Making a whole magic square book?

This page makes one puzzle. To publish a book on Amazon KDP, the [KDP magic square maker](https://www.oncebooks.com/kdp-magic-square-maker) generates hundreds of graded puzzles at any trim, as a print-ready 300 DPI interior and solutions PDF. Generate once, export unlimited sizes.

## More printable magic squares

- [Easy magic square](https://www.oncebooks.com/magic-square/easy)
- [Medium magic square](https://www.oncebooks.com/magic-square/medium)
- [Large magic square](https://www.oncebooks.com/magic-square/large)
- [Magic squares for kids](https://www.oncebooks.com/magic-square/for-kids)
- [Magic squares for beginners](https://www.oncebooks.com/magic-square/for-beginners)

[See every printable magic square](https://www.oncebooks.com/magic-square)

## Questions

**Are these hard magic square free to print?**

Yes. Generate the puzzle above and print it with its answer, at no cost and with no sign-up. It is free for classroom, home and personal use.

**Does every puzzle have one solution?**

Yes. Every puzzle is generated and checked so the given numbers fix exactly one completion, verified before it prints. The answer is on the second page.

**Can I make a magic square book to sell on KDP?**

Yes. The free tool makes one puzzle. The KDP magic square maker generates hundreds of graded puzzles at any KDP trim, as a print-ready 300 DPI interior and solutions PDF, so you can publish a book.