Differences
This shows you the differences between two versions of the page.
| Next revision | Previous revision | ||
| fuss:puzzles [2014/01/15 20:02] – external edit 127.0.0.1 | fuss:puzzles [2022/04/19 08:28] (current) – external edit 127.0.0.1 | ||
|---|---|---|---|
| Line 1: | Line 1: | ||
| + | ====== Magic Square Generation ====== | ||
| + | A magic square is an arrangement of numbers in a square grid, where the numbers in each row, column, and the numbers on the big diagonals, all add up to the same number. We can determine that number. Suppose $M$ is the number that each row, column or big diagonal must add up to. Since there are $n$ rows, the sum of all the numbers in the magic square must be $n*M$. Now, the numbers being added give the series $1, | ||
| + | |||
| + | $$ | ||
| + | \sum_{i=1}^{n^{2}}{i}=n*M | ||
| + | $$ | ||
| + | |||
| + | Knowing that: | ||
| + | |||
| + | $$ | ||
| + | 1+2+3+...+n^{2}=\frac{n^2(n^2+1)}{2} | ||
| + | $$ | ||
| + | |||
| + | then it follows that: | ||
| + | |||
| + | $$ | ||
| + | n*M=\frac{n^{2}(n^{2}+1)}{2} | ||
| + | $$ | ||
| + | |||
| + | reducing, we obtain: | ||
| + | |||
| + | $$ | ||
| + | M=\frac{n(n^{2}+1)}{2} | ||
| + | $$ | ||
| + | |||
| + | Thus, for a $3 \times 3$ magic square, M=$15$, for a $4 \times 4$ magic square $M=34$, for a $5 \times 5$ magic square $M=65$, etc... | ||
| + | |||
| + | ===== de la Loubère' | ||
| + | |||
| + | - Start in the middle column and place the number $x=1$. | ||
| + | - Go up and right and place $x+1$. | ||
| + | - If you exceed the square bounds, wrap around and place $x+1$. | ||
| + | - If you meet an occupied square, move down and place $x+1$. | ||
| + | - Repeat at '' | ||
| + | |||
| + | {{puzzles_magicsquareupright.gif}} | ||
| + | |||
| + | ==== Generalizing Loubère Algorithm ==== | ||
| + | |||
| + | The same concept applies, for all $90^\circ$ counter-clockwise rotations of the movement direction, going back each time a square is occupied. | ||
| + | |||
| + | {{puzzles_generalizedloubere.png}} | ||
| + | |||
| + | Every movement sequence will yield a magic square, all the columns, lines and big diagonals adding up to $15$. The squares generated by all four sequences are: | ||
| + | |||
| + | $$ | ||
| + | \begin{align} | ||
| + | MS_{1} = | ||
| + | \begin{pmatrix} | ||
| + | 8 & 1 & 6 \\ | ||
| + | 3 & 5 & 7 \\ | ||
| + | 4 & 9 & 2 \\ | ||
| + | \end{pmatrix} | ||
| + | & | ||
| + | MS_{2} = | ||
| + | \begin{pmatrix} | ||
| + | 6 & 7 & 2 \\ | ||
| + | 1 & 5 & 9 \\ | ||
| + | 8 & 3 & 4 \\ | ||
| + | \end{pmatrix} | ||
| + | & | ||
| + | MS_{3} = | ||
| + | \begin{pmatrix} | ||
| + | 2 & 9 & 4 \\ | ||
| + | 7 & 5 & 3 \\ | ||
| + | 6 & 1 & 8 \\ | ||
| + | \end{pmatrix} | ||
| + | & | ||
| + | MS_{4} = | ||
| + | \begin{pmatrix} | ||
| + | 4 & 3 & 8 \\ | ||
| + | 9 & 5 & 1 \\ | ||
| + | 2 & 7 & 6 \\ | ||
| + | \end{pmatrix} | ||
| + | \end{align} | ||
| + | $$ | ||
| + | |||
| + | The same applies to any odd square (with a central tile), although not all movements will necessarily build a magic square. | ||
| + | |||
| + | {{puzzles_loubereodd.png}} | ||
| + | |||
| + | In this case, two of the paths do not generate magic squares. All the rest though do generate magic squares. The results in a $5 \times 5$ case are the following squares: | ||
| + | $$ | ||
| + | \begin{align} | ||
| + | \color{green} { | ||
| + | MS_{1} = | ||
| + | \begin{pmatrix} | ||
| + | 17 & 24 & 1 & 8 & 15 \\ | ||
| + | 23 & 5 & 7 & 14 & 16 \\ | ||
| + | 4 & 6 & 13 & 20 & 22 \\ | ||
| + | 10 & 12 & 19 & 21 & 3 \\ | ||
| + | 11 & 18 & 25 & 2 & 9 | ||
| + | \end{pmatrix} | ||
| + | } | ||
| + | & | ||
| + | \color{green} { | ||
| + | MS_{2} = | ||
| + | \begin{pmatrix} | ||
| + | 11 & 18 & 25 & 2 & 9 \\ | ||
| + | 17 & 24 & 1 & 8 & 15 \\ | ||
| + | 23 & 5 & 7 & 14 & 16 \\ | ||
| + | 4 & 6 & 13 & 20 & 22 \\ | ||
| + | 10 & 12 & 19 & 21 & 3 | ||
| + | \end{pmatrix} | ||
| + | } | ||
| + | & | ||
| + | \color{green} { | ||
| + | MS_{3} = | ||
| + | \begin{pmatrix} | ||
| + | 15 & 9 & 3 & 22 & 16 \\ | ||
| + | 8 & 2 & 21 & 20 & 14 \\ | ||
| + | 1 & 25 & 19 & 13 & 7 \\ | ||
| + | 24 & 18 & 12 & 6 & 5 \\ | ||
| + | 17 & 11 & 10 & 4 & 23 | ||
| + | \end{pmatrix} | ||
| + | } | ||
| + | \end{align} | ||
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | \begin{align} | ||
| + | \color{green} { | ||
| + | MS_{4} = | ||
| + | \begin{pmatrix} | ||
| + | 16 & 15 & 9 & 3 & 22 \\ | ||
| + | 14 & 8 & 2 & 21 & 20 \\ | ||
| + | 7 & 1 & 25 & 19 & 13 \\ | ||
| + | 5 & 24 & 18 & 12 & 6 \\ | ||
| + | 23 & 17 & 11 & 10 & 4 | ||
| + | \end{pmatrix} | ||
| + | } | ||
| + | & | ||
| + | \color{green} { | ||
| + | MS_{5} = | ||
| + | \begin{pmatrix} | ||
| + | 9 & 2 & 25 & 18 & 11 \\ | ||
| + | 3 & 21 & 19 & 12 & 10 \\ | ||
| + | 22 & 20 & 13 & 6 & 4 \\ | ||
| + | 16 & 14 & 7 & 5 & 23 \\ | ||
| + | 15 & 8 & 1 & 24 & 17 | ||
| + | \end{pmatrix} | ||
| + | } | ||
| + | & | ||
| + | \color{red} { | ||
| + | MS_{6} = | ||
| + | \begin{pmatrix} | ||
| + | 3 & 21 & 19 & 12 & 10 \\ | ||
| + | 22 & 20 & 13 & 6 & 4 \\ | ||
| + | 16 & 14 & 7 & 5 & 23 \\ | ||
| + | 15 & 18 & 1 & 24 & 17 \\ | ||
| + | 9 & 2 & 25 & 18 & 11 | ||
| + | \end{pmatrix} | ||
| + | } | ||
| + | \end{align} | ||
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | \begin{align} | ||
| + | \color{green} { | ||
| + | MS_{7} = | ||
| + | \begin{pmatrix} | ||
| + | 11 & 10 & 4 & 23 & 17 \\ | ||
| + | 18 & 12 & 6 & 5 & 24 \\ | ||
| + | 25 & 19 & 13 & 7 & 1 \\ | ||
| + | 2 & 21 & 20 & 14 & 8 \\ | ||
| + | 9 & 3 & 22 & 16 & 15 | ||
| + | \end{pmatrix} | ||
| + | } | ||
| + | & | ||
| + | \color{red} { | ||
| + | MS_{8} = | ||
| + | \begin{pmatrix} | ||
| + | 10 & 4 & 23 & 17 & 11 \\ | ||
| + | 12 & 6 & 5 & 24 & 18 \\ | ||
| + | 19 & 13 & 7 & 1 & 25 \\ | ||
| + | 21 & 20 & 14 & 8 & 2 \\ | ||
| + | 3 & 22 & 16 & 15 & 9 | ||
| + | \end{pmatrix} | ||
| + | } | ||
| + | \end{align} | ||
| + | $$ | ||
| + | |||
| + | where the green squares are magic squares all the lines, columns and big diagonals adding up to $65$ while the red squares are not magic squares. | ||
fuss/puzzles.txt · Last modified: 2022/04/19 08:28 by 127.0.0.1
For the contact, copyright, license, warranty and privacy terms for the usage of this website please see the contact, license, privacy, copyright.