Magic Squares

Magic squares are simply delightful and fun to make. They have very special properties which I am sure will interest you.

An example of a 3 by 3 magic square is:

$\begin{array} {c|c|c} 4 & 3 & 8\\ \hline 9 & 5 & 1\\ \hline 2 & 7 & 6 \end{array}$

Can you see anything special? Well, each row, column and diagonal have the same sum. In this case, it is 15. For a 4 by 4 square, the sum is 34. For a 5 by 5 square it is 65.

Problem 1: Can you find a rule for the sum of an $n$ by $n$ magic square?

Give a solution to the problem in the comments below.

How may different magic squares which are 3 by 3 squares? 4 by 4? 5 by 5? If you want to answer these questions, answer them in the problems posted with this set.

Once you solve those, try to solve this question. How many unique magic squares are possible which have a side length of $n$ where $n > 5$ ?

Solve this and you may become one of the most famous mathematicians of all time. This is because no one has solved it.

Here is a 4 by 4 magic square but it is a special magic square.

$\begin{array} {c|c|c|c} 7 & 12 & 1 & 14\\ \hline 2 & 13 & 8 & 11\\ \hline 16 & 3 & 10 & 5\\ \hline 9 & 6 & 15 & 4 \end{array}$

Each row, column and diagonal have a sum of 34 but all the 2 by 2 squares have a sum of 34. These sort of magic squares are called most-perfect magic squares.

Problem 2: Determine how many different 36 by 36 most-perfect magic squares are possible.

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Math

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Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

\sin \theta

\boxed{123}

\boxed{123}

Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Great explanation!!

milind prabhu - 7 years, 2 months ago

Please define what are considered different magic squares in your problems.

It would also be helpful to provide a link from this problems back to this note.

Calvin Lin Staff - 7 years, 2 months ago

For 4 by 4 : S-34=4q+r ..... I think it is just like a Sudoku that n appears once and only once in each row, column and section. Hmmmm Am I right ? please correct me . :D I'm interested in Magic Squares. Thankssss

rein esgue - 6 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$