Multiples of Reciprocals with Maximum Period form Magic Square/Examples/19
Example of Multiples of Reciprocals with Maximum Period form Magic Square
We have that:
- $\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
Hence:
- $\begin{array}{|r|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline \frac 1 {19} & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 \\ \hline \frac 2 {19} & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 \\ \hline \frac 3 {19} & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 \\ \hline \frac 4 {19} & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 \\ \hline \frac 5 {19} & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 \\ \hline \frac 6 {19} & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 \\ \hline \frac 7 {19} & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 \\ \hline \frac 8 {19} & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 \\ \hline \frac 9 {19} & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 \\ \hline \frac {10} {19} & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 \\ \hline \frac {11} {19} & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 \\ \hline \frac {12} {19} & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 \\ \hline \frac {13} {19} & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 \\ \hline \frac {14} {19} & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 \\ \hline \frac {15} {19} & 7 & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 \\ \hline \frac {16} {19} & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 & 9 & 4 & 7 & 3 & 6 \\ \hline \frac {17} {19} & 8 & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 \\ \hline \frac {18} {19} & 9 & 4 & 7 & 3 & 6 & 8 & 4 & 2 & 1 & 0 & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 8 \\ \hline \end{array}$
As can be seen, each of the rows and columns add to $81$.
As a bonus, so do the numbers along each diagonal.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $052,631,578,947,368,421$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $052,631,578,947,368,421$