Waste less time on Facebook — follow Brilliant.
×

Magic Hexagon

Most of us are familiar with Magic Squares and its wonders. But little is known about Magic Hexagons. This post is attributed for describing "Magic Hexagons".

A magic hexagon of order n is an arrangement of numbers in a centered hexagonal pattern with n cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant. A normal magic hexagon contains the consecutive integers from 1 to 3n^2 − 3n + 1. It turns out that normal magic hexagons exist only for n = 1 (which is trivial) and n = 3. Moreover, the solution of order 3 is essentially unique. These hexagons are shown for your reference. The first magic hexagon has Magic sum 1 and second has Magic Sum 38.

Note by Shivanand Pandey
3 years, 9 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. 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"
MathAppears 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

Sort by:

Top Newest

This is interesting. Do you know why no other magic hexagon of higher orders exist?

With \(n=2 \), why can't we have

\[ \begin{array} { l l l l } & 1 & & 2 \\ 5 & & 4 & & 3 \\ & 6 & & 7 \\ \end{array} \]

The entries in each row sum to 12. Is there something that I'm missing?

Calvin Lin Staff - 3 years, 9 months ago

Log in to reply

Higher order magic hexagons do exist, but cannot be 'normal'.

Jonathan Wong - 3 years, 9 months ago

Log in to reply

I believe what you wrote down isn't exactly correct. Although a few of the rows sum to 12, not all of them do. I'll go through a few of the rows.

\(1 + 2 \neq 12 \)

\(5 + 6 \neq 12 \)

\(6 + 7 \neq 12 \)

and so on.

Milly Choochoo - 3 years, 9 months ago

Log in to reply

Ah, even 'partial' rows count. That makes sense now. Thanks!

Calvin Lin Staff - 3 years, 9 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...