Eulerian Composition of Certain Franklin Squares

TL;DR

This paper investigates Franklin squares by decomposing them into quotient and remainder components, providing evidence that Eulerian composition was used in their construction and demonstrating how to create new Franklin squares using this method.

Contribution

It offers a decomposition approach supporting the Eulerian composition method as a construction technique for Franklin squares and introduces a way to generate new squares.

Findings

Decomposition of Franklin squares into quotient and remainder squares.

Support for Eulerian composition as a construction method.

Method to construct new Franklin squares.

Abstract

Several specific Franklin squares and magic squares are decomposed into their quotient and remainder squares. The results support the conjecture that Franklin used the Eulerian composition method to construct many of his squares. This method also can be used to construct new Franklin squares as illustrated herein.

Equations

$$m=\frac{n}{2}\left(n^2+1\right).$$

$$\begin{tabular}[]{|cccccc|}\hline\cr$\!\spadesuit\!\!$&$\!\!{\mathbf{\Diamond}}\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!$\\ $\!\triangledown\!\!$&$\!\!\spadesuit\!\!$&$\!\!{\mathbf{\Diamond}}\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!$\\ $\!\blacktriangle\!\!$&$\!\!\triangledown\!\!$&$\!\!\spadesuit\!\!$&$\!\!\Diamond\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!$\\ $\!\blacktriangle\!\!$&$\!\!\triangledown\!\!$&$\!\!\spadesuit\!\!$&$\!\!\Diamond\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!$\\ $\!\triangledown\!\!$&$\!\!\spadesuit\!\!$&$\!\!\Diamond\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!$\\ $\!\spadesuit\!\!$&$\!\!\Diamond\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!$\\ \hline\cr\end{tabular}\ \ \ \begin{tabular}[]{|cccccc|}\hline\cr$\!\spadesuit\!\!$&$\!\!{\mathbf{\Diamond}}\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!$\\ $\!{\mathbf{\Diamond}}\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!\!$&$\!\!\spadesuit\!$\\ $\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!\!$&$\!\!\spadesuit\!\!$&$\!\!{\mathbf{\Diamond}}\!$\\ $\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!\!$&$\!\!\spadesuit\!\!$&$\!\!{\mathbf{\Diamond}}\!$\\ $\!\Diamond\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!\!\!$&$\!\!\spadesuit\!$\\ $\!\spadesuit\!\!$&$\!\!\Diamond\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!$\\ \hline\cr\end{tabular}\ \ \ \begin{tabular}[]{|cccccc|}\hline\cr$\!\spadesuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\clubsuit\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\spadesuit\!$\\ $\!\heartsuit\!\!$&$\!\!\clubsuit\!\!$&$\!\!\Diamond\!\!$&$\!\!\Diamond\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!$\\ $\!\clubsuit\!\!$&$\!\!\Diamond\!\!$&$\!\!\blacktriangle\!\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\Diamond\!\!$&$\!\!\clubsuit\!$\\ $\!{\mathbf{\Diamond}}\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!\!$&$\!\!\triangledown\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\Diamond\!$\\ $\!\blacktriangle\!\!\!$&$\!\!\triangledown\!\!$&$\!\!\spadesuit\!\!$&$\!\!\spadesuit\!\!$&$\!\!\triangledown\!\!$&$\!\!\blacktriangle\!$\\ $\!\triangledown\!\!$&$\!\!\spadesuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\spadesuit\!\!$&$\!\!\triangledown\!$\\ \hline\cr\end{tabular}\ \ \ \begin{tabular}[]{|cccccc|}\hline\cr$\!\spadesuit\!\!$&$\!\!\triangledown\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!\!$&$\!\!\spadesuit\!$\\ $\!\Diamond\!\!$&$\!\!\spadesuit\!\!$&$\!\!\triangledown\!\!$&$\!\!\triangledown\!\!$&$\!\!\spadesuit\!\!\!$&$\!\!\!\Diamond\!$\\ $\!\clubsuit\!\!$&$\!\!\Diamond\!\!$&$\!\!\spadesuit\!\!$&$\!\!\spadesuit\!\!$&$\!\!\Diamond\!\!$&$\!\!\clubsuit\!\!$\\ $\!\heartsuit\!\!$&$\!\!\clubsuit\!\!$&$\!\!\Diamond\!\!$&$\!\!\Diamond\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!$\\ $\!\blacktriangle\!\!\!$&$\!\!\heartsuit\!\!$&$\!\!\clubsuit\!\!$&$\!\!\clubsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!$\\ $\!\triangledown\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\heartsuit\!\!$&$\!\!\heartsuit\!\!$&$\!\!\blacktriangle\!\!$&$\!\!\triangledown\!$\\ \hline\cr\end{tabular}\,.$$

$$M_n=n{Q}_n+R_n+U_n,$$

$$R_n=\left(M_n-U_n\right)\mathrm{mod}n,Q_n=\frac{1}{n}\left(M_n-U_n-R_n\right)$$

$$M_{6}=\left[\begin{array}[]{cccccc}2&9&4&29&36&31\vspace{-0.01in}\\ 34&32&30&7&5&3\vspace{-0.01in}\\ 6&1&8&33&28&35\vspace{-0.01in}\\ 20&27&22&11&18&13\vspace{-0.01in}\\ 25&23&21&16&14&12\vspace{-0.01in}\\ 24&19&26&15&10&17\vspace{-0.01in}\end{array}\right].$$

$$Q_{6}=\left[\begin{array}[]{cccccc}0&1&0&4&5&5\vspace{-0.01in}\\ 5&5&4&1&0&0\vspace{-0.01in}\\ 0&0&1&5&4&5\vspace{-0.01in}\\ 3&4&3&1&2&2\vspace{-0.01in}\\ 4&3&3&2&2&1\vspace{-0.01in}\\ 3&3&4&2&1&2\vspace{-0.01in}\end{array}\right],\quad R_{6}=\left[\begin{array}[]{cccccc}1&2&3&4&5&0\vspace{-0.01in}\\ 3&1&5&0&4&2\vspace{-0.01in}\\ 5&0&1&2&3&4\vspace{-0.01in}\\ 1&2&3&4&5&0\vspace{-0.01in}\\ 0&4&2&3&1&5\vspace{-0.01in}\\ 5&0&1&2&3&4\vspace{-0.01in}\end{array}\right].$$

$$M_{6}=\left[\begin{array}[]{cccccc}3&36&30&4&11&27\vspace{-0.01in}\\ 22&13&35&12&14&15\vspace{-0.01in}\\ 16&18&8&31&17&21\vspace{-0.01in}\\ 28&20&6&29&19&9\vspace{-0.01in}\\ 32&23&25&2&24&5\vspace{-0.01in}\\ 10&1&7&33&26&34\vspace{-0.01in}\end{array}\right]$$

$$Q_{6}=\left[\begin{array}[]{cccccc}0&5&4&0&1&4\vspace{-0.01in}\\ 3&2&5&1&2&2\vspace{-0.01in}\\ 2&2&1&5&2&3\vspace{-0.01in}\\ 4&3&0&4&3&1\vspace{-0.01in}\\ 5&3&4&0&3&0\vspace{-0.01in}\\ 1&0&1&5&4&5\vspace{-0.01in}\end{array}\right],\quad R_{6}=\left[\begin{array}[]{cccccc}2&5&5&3&4&2\vspace{-0.01in}\\ 3&0&4&5&1&2\vspace{-0.01in}\\ 3&5&1&0&4&2\vspace{-0.01in}\\ 3&1&5&4&0&2\vspace{-0.01in}\\ 1&4&0&1&5&4\vspace{-0.01in}\\ 3&0&0&2&1&3\vspace{-0.01in}\end{array}\right].$$

$$M_{6}=\left[\begin{array}[]{cccccc}28&4&3&31&35&10\vspace{-0.01in}\\ 36&18&21&24&11&1\vspace{-0.01in}\\ 7&23&12&17&22&30\vspace{-0.01in}\\ 8&13&26&19&16&29\vspace{-0.01in}\\ 5&20&15&14&25&32\vspace{-0.01in}\\ 27&33&34&6&2&9\vspace{-0.01in}\end{array}\right]$$

$$Q_{6}=\left[\begin{array}[]{cccccc}4&0&0&5&5&1\vspace{-0.01in}\\ 5&2&3&3&1&0\vspace{-0.01in}\\ 1&3&1&2&3&4\vspace{-0.01in}\\ 1&2&4&3&2&4\vspace{-0.01in}\\ 0&3&2&2&4&5\vspace{-0.01in}\\ 4&5&5&0&0&1\vspace{-0.01in}\end{array}\right],\quad R_{6}=\left[\begin{array}[]{cccccc}3&3&2&0&4&3\vspace{-0.01in}\\ 5&5&2&5&4&0\vspace{-0.01in}\\ 0&4&5&4&3&5\vspace{-0.01in}\\ 1&0&1&0&3&4\vspace{-0.01in}\\ 4&1&2&1&0&1\vspace{-0.01in}\\ 2&2&3&5&1&2\vspace{-0.01in}\end{array}\right].$$

$$F_{8}=\left[\begin{array}[]{cccccccc}52&61&4&13&20&29&36&45\vspace{-0.01in}\\ 14&3&62&51&46&35&30&19\vspace{-0.01in}\\ 53&60&5&12&21&28&37&44\vspace{-0.01in}\\ 11&6&59&54&43&38&27&22\vspace{-0.01in}\\ 55&58&7&10&23&26&39&42\vspace{-0.01in}\\ 9&8&57&56&41&40&25&24\vspace{-0.01in}\\ 50&63&2&15&18&31&34&47\vspace{-0.01in}\\ 16&1&64&49&48&33&32&17\vspace{-0.01in}\end{array}\right]$$

$$Q_{8}=\left[\begin{array}[]{cccccccc}\mathbf{6}&\mathbf{7}&\mathbf{0}&\mathbf{1}&\mathbf{2}&\mathbf{3}&\mathbf{4}&\mathbf{5\vspace{-0.01in}}\\ 1&0&7&6&5&4&3&2\vspace{-0.01in}\\ 6&7&0&1&2&3&4&5\vspace{-0.01in}\\ 1&0&7&6&5&4&3&2\vspace{-0.01in}\\ 6&7&0&1&2&3&4&5\vspace{-0.01in}\\ 1&0&7&6&5&4&3&2\vspace{-0.01in}\\ 6&7&0&1&2&3&4&5\vspace{-0.01in}\\ 1&0&7&6&5&4&3&2\vspace{-0.01in}\end{array}\right],\quad R_{8}=\left[\begin{array}[]{cccccccc}\mathbf{3}&4&3&4&3&4&3&4\vspace{-0.01in}\\ 5&\mathbf{2}&5&2&5&2&5&2\vspace{-0.01in}\\ 4&3&\mathbf{4}&3&4&3&4&3\vspace{-0.01in}\\ 2&5&2&\mathbf{5}&2&5&2&5\vspace{-0.01in}\\ 6&1&6&1&\mathbf{6}&1&6&1\vspace{-0.01in}\\ 0&7&0&7&0&\mathbf{7}&0&7\vspace{-0.01in}\\ 1&6&1&6&1&6&\mathbf{1}&6\vspace{-0.01in}\\ 7&0&7&0&7&0&7&\mathbf{0}\vspace{-0.01in}\end{array}\right].$$

$$F_{8}=\left[\begin{array}[]{cccccccc}2&57&6&61&8&63&4&59\vspace{-0.01in}\\ 7&64&3&60&1&58&5&62\vspace{-0.01in}\\ 49&10&53&14&55&16&51&12\vspace{-0.01in}\\ 56&15&52&11&50&9&54&13\vspace{-0.01in}\\ 42&17&46&21&48&23&44&19\vspace{-0.01in}\\ 47&24&43&20&41&18&45&22\vspace{-0.01in}\\ 25&34&29&38&31&40&27&36\vspace{-0.01in}\\ 32&39&28&35&26&33&30&37\vspace{-0.01in}\end{array}\right].$$

$$Q_{8}=\left[\begin{array}[]{cccccccc}0&7&0&7&0&7&0&7\vspace{-0.01in}\\ 0&7&0&7&0&7&0&7\vspace{-0.01in}\\ 6&1&6&1&6&1&6&1\vspace{-0.01in}\\ 6&1&6&1&6&1&6&1\vspace{-0.01in}\\ 5&2&5&2&5&2&5&2\vspace{-0.01in}\\ 5&2&5&2&5&2&5&2\vspace{-0.01in}\\ 3&4&3&4&3&4&3&4\vspace{-0.01in}\\ 3&4&3&4&3&4&3&4\vspace{-0.01in}\end{array}\right],\quad R_{8}=\left[\begin{array}[]{cccccccc}1&0&5&4&7&6&3&2\vspace{-0.01in}\\ 6&7&2&3&0&1&4&5\vspace{-0.01in}\\ 0&1&4&5&6&7&2&3\vspace{-0.01in}\\ 7&6&3&2&1&0&5&4\vspace{-0.01in}\\ 1&0&5&4&7&6&3&2\vspace{-0.01in}\\ 6&7&2&3&0&1&4&5\vspace{-0.01in}\\ 0&1&4&5&6&7&2&3\vspace{-0.01in}\\ 7&6&3&2&1&0&5&4\vspace{-0.01in}\end{array}\right].$$

$$F_{8}=\left[\begin{array}[]{cccccccc}17&47&30&36&21&43&26&40\vspace{-0.01in}\\ 32&34&19&45&28&38&23&41\vspace{-0.01in}\\ 33&31&46&20&37&27&42&24\vspace{-0.01in}\\ 48&18&35&29&44&22&39&25\vspace{-0.01in}\\ 49&15&62&4&53&11&58&8\vspace{-0.01in}\\ 64&2&51&13&60&6&55&9\vspace{-0.01in}\\ 1&63&14&52&5&59&10&56\vspace{-0.01in}\\ 16&50&3&61&12&54&7&57\vspace{-0.01in}\end{array}\right]$$

$$Q_{8}=\left[\begin{array}[]{cccccccc}2&5&3&4&2&5&3&4\vspace{-0.01in}\\ 3&4&2&5&3&4&2&5\vspace{-0.01in}\\ 4&3&5&2&4&3&5&2\vspace{-0.01in}\\ 5&2&4&3&5&2&4&3\vspace{-0.01in}\\ 6&1&7&0&6&1&7&0\vspace{-0.01in}\\ 7&0&6&1&7&0&6&1\vspace{-0.01in}\\ 0&7&1&6&0&7&1&6\vspace{-0.01in}\\ 1&6&0&7&1&6&0&7\vspace{-0.01in}\end{array}\right],\quad R_{8}=\left[\begin{array}[]{cccccccc}0&6&5&3&4&2&1&7\vspace{-0.01in}\\ 7&1&2&4&3&5&6&0\vspace{-0.01in}\\ 0&6&5&3&4&2&1&7\vspace{-0.01in}\\ 7&1&2&4&3&5&6&0\vspace{-0.01in}\\ 0&6&5&3&4&2&1&7\vspace{-0.01in}\\ 7&1&2&4&3&5&6&0\vspace{-0.01in}\\ 0&6&5&3&4&2&1&7\vspace{-0.01in}\\ 7&1&2&4&3&5&6&0\vspace{-0.01in}\end{array}\right].$$

$$F_{8}=\left[\begin{array}[]{cccccccc}6&55&12&57&2&51&16&61\vspace{-0.01in}\\ 11&58&5&56&15&62&1&52\vspace{-0.01in}\\ 53&8&59&10&49&4&63&14\vspace{-0.01in}\\ 60&9&54&7&64&13&50&3\vspace{-0.01in}\\ 21&40&27&42&17&36&31&46\vspace{-0.01in}\\ 28&41&22&39&32&45&18&35\vspace{-0.01in}\\ 38&23&44&25&34&19&48&29\vspace{-0.01in}\\ 43&26&37&24&47&30&33&20\vspace{-0.01in}\end{array}\right]$$

$$Q_{8}=\left[\begin{array}[]{cccccccc}0&6&1&7&0&6&1&7\vspace{-0.01in}\\ 1&7&0&6&1&7&0&6\vspace{-0.01in}\\ 6&0&7&1&6&0&7&1\vspace{-0.01in}\\ 7&1&6&0&7&1&6&0\vspace{-0.01in}\\ 2&4&3&5&2&4&3&5\vspace{-0.01in}\\ 3&5&2&4&3&5&2&4\vspace{-0.01in}\\ 4&2&5&3&4&2&5&3\vspace{-0.01in}\\ 5&3&4&2&5&3&4&2\vspace{-0.01in}\end{array}\right],\quad R_{8}=\left[\begin{array}[]{cccccccc}5&6&3&0&1&2&7&4\vspace{-0.01in}\\ 2&1&4&7&6&5&0&3\vspace{-0.01in}\\ 4&7&2&1&0&3&6&5\vspace{-0.01in}\\ 3&0&5&6&7&4&1&2\vspace{-0.01in}\\ 4&7&2&1&0&3&6&5\vspace{-0.01in}\\ 3&0&5&6&7&4&1&2\vspace{-0.01in}\\ 5&6&3&0&1&2&7&4\vspace{-0.01in}\\ 2&1&4&7&6&5&0&3\vspace{-0.01in}\end{array}\right].$$

$$F_{16}=\!\left[\!\begin{array}[]{cccccccccccccccc}200&217&232&249&8&25&40&57&72&89&104&121&136&153&168&185\\ 58&39&26&7&250&231&218&199&186&167&154&135&122&103&90&71\\ 198&219&230&251&6&27&38&59&70&91&102&123&134&155&166&187\\ 60&37&28&5&252&229&220&197&188&165&156&133&124&101&92&69\\ 201&216&233&248&9&24&41&56&73&88&105&120&137&152&169&184\\ 55&42&23&10&247&234&215&202&183&170&151&138&119&106&87&74\\ 203&214&235&246&11&22&43&54&75&86&107&118&139&150&171&182\\ 53&44&21&12&245&236&213&204&181&172&149&140&117&108&85&76\\ 205&212&237&244&13&20&45&52&77&84&109&116&141&148&173&180\\ 51&46&19&14&243&238&211&206&179&174&147&142&115&110&83&78\\ 207&210&239&242&15&18&47&50&79&82&111&114&143&146&175&178\\ 49&48&17&16&241&240&209&208&177&176&145&144&113&112&81&80\\ 196&221&228&253&4&29&36&61&68&93&100&125&132&157&164&189\\ 62&35&30&3&254&227&222&195&190&163&158&131&126&99&94&67\\ 194&223&226&255&2&31&34&63&66&95&98&127&130&159&162&191\\ 64&33&32&1&256&225&224&193&192&161&160&129&128&97&96&65\end{array}\!\right]$$

$$Q_{16}=\left[\begin{array}[]{cccccccccccccccc}12&13&14&15&0&1&2&3&4&5&6&7&8&9&10&11\\ 3&2&1&0&15&14&13&12&11&10&9&8&7&6&5&4\\ 12&13&14&15&0&1&2&3&4&5&6&7&8&9&10&11\\ 3&2&1&0&15&14&13&12&11&10&9&8&7&6&5&4\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 12&13&14&15&0&1&2&3&4&5&6&7&8&9&10&11\\ 3&2&1&0&15&14&13&12&11&10&9&8&7&6&5&4\end{array}\right],$$

$$R_{16}=\left[\begin{array}[]{cccccccccccccccc}7&8&7&8&7&8&7&8&7&8&7&8&7&8&7&8\\ 9&6&9&6&9&6&9&6&9&6&9&6&9&6&9&6\\ 5&10&5&10&5&10&5&10&5&10&5&10&5&10&5&10\\ 11&4&11&4&11&4&11&4&11&4&11&4&11&4&11&4\\ 8&7&8&7&8&7&8&7&8&7&8&7&8&7&8&7\\ 6&9&6&9&6&9&6&9&6&9&6&9&6&9&6&9\\ 10&5&10&5&10&5&10&5&10&5&10&5&10&5&10&5\\ 4&11&4&11&4&11&4&11&4&11&4&11&4&11&4&11\\ 12&3&12&3&12&3&12&3&12&3&12&3&12&3&12&3\\ 2&13&2&13&2&13&2&13&2&13&2&13&2&13&2&13\\ 14&1&14&1&14&1&14&1&14&1&14&1&14&1&14&1\\ 0&15&0&15&0&15&0&15&0&15&0&15&0&15&0&15\\ 3&12&3&12&3&12&3&12&3&12&3&12&3&12&3&12\\ 13&2&13&2&13&2&13&2&13&2&13&2&13&2&13&2\\ 1&14&1&14&1&14&1&14&1&14&1&14&1&14&1&14\\ 15&0&15&0&15&0&15&0&15&0&15&0&15&0&15&0\end{array}\right].$$

$$F_{16}=\left[\!\begin{array}[]{cccccccccccccccc}16&255&2&241&14&253&4&243&12&251&6&245&10&249&8&247\\ 1&242&15&256&3&244&13&254&5&246&11&252&7&248&9&250\\ 240&31&226&17&238&29&228&19&236&27&230&21&234&25&232&23\\ 225&18&239&32&227&20&237&30&229&22&235&28&231&24&233&26\\ 223&48&209&34&221&46&211&36&219&44&213&38&217&42&215&40\\ 210&33&224&47&212&35&222&45&214&37&220&43&216&39&218&41\\ 63&208&49&194&61&206&51&196&59&204&53&198&57&202&55&200\\ 50&193&64&207&52&195&62&205&54&197&60&203&56&199&58&201\\ 80&191&66&177&78&189&68&179&76&187&70&181&74&185&72&183\\ 65&178&79&192&67&180&77&190&69&182&75&188&71&184&73&186\\ 176&95&162&81&174&93&164&83&172&91&166&85&170&89&168&87\\ 161&82&175&96&163&84&173&94&165&86&171&92&167&88&169&90\\ 159&112&145&98&157&110&147&100&155&108&149&102&153&106&151&104\\ 146&97&160&111&148&99&158&109&150&101&156&107&152&103&154&105\\ 127&144&113&130&125&142&115&132&123&140&117&134&121&138&119&136\\ 114&129&128&143&116&131&126&141&118&133&124&139&120&135&122&137\end{array}\!\right]$$

$$Q_{16}=\left[\begin{array}[]{cccccccccccccccc}0&15&0&15&0&15&0&15&0&15&0&15&0&15&0&15\vspace{-0.01in}\\ 0&15&0&15&0&15&0&15&0&15&0&15&0&15&0&15\vspace{-0.01in}\\ 14&1&14&1&14&1&14&1&14&1&14&1&14&1&14&1\vspace{-0.01in}\\ 14&1&14&1&14&1&14&1&14&1&14&1&14&1&14&1\vspace{-0.01in}\\ 13&2&13&2&13&2&13&2&13&2&13&2&13&2&13&2\vspace{-0.01in}\\ 13&2&13&2&13&2&13&2&13&2&13&2&13&2&13&2\vspace{-0.01in}\\ 3&12&3&12&3&12&3&12&3&12&3&12&3&12&3&12\vspace{-0.01in}\\ 3&12&3&12&3&12&3&12&3&12&3&12&3&12&3&12\vspace{-0.01in}\\ 4&11&4&11&4&11&4&11&4&11&4&11&4&11&4&11\vspace{-0.01in}\\ 4&11&4&11&4&11&4&11&4&11&4&11&4&11&4&11\vspace{-0.01in}\\ 10&5&10&5&10&5&10&5&10&5&10&5&10&5&10&5\vspace{-0.01in}\\ 10&5&10&5&10&5&10&5&10&5&10&5&10&5&10&5\vspace{-0.01in}\\ 9&6&9&6&9&6&9&6&9&6&9&6&9&6&9&6\vspace{-0.01in}\\ 9&6&9&6&9&6&9&6&9&6&9&6&9&6&9&6\vspace{-0.01in}\\ 7&8&7&8&7&8&7&8&7&8&7&8&7&8&7&8\vspace{-0.01in}\\ 7&8&7&8&7&8&7&8&7&8&7&8&7&8&7&8\vspace{-0.01in}\end{array}\right],$$

$$R_{16}=\left[\begin{array}[]{cccccccccccccccc}15&14&1&0&13&12&3&2&11&10&5&4&9&8&7&6\vspace{-0.01in}\\ 0&1&14&15&2&3&12&13&4&5&10&11&6&7&8&9\vspace{-0.01in}\\ 15&14&1&0&13&12&3&2&11&10&5&4&9&8&7&6\vspace{-0.01in}\\ 0&1&14&15&2&3&12&13&4&5&10&11&6&7&8&9\vspace{-0.01in}\\ 14&15&0&1&12&13&2&3&10&11&4&5&8&9&6&7\vspace{-0.01in}\\ 1&0&15&14&3&2&13&12&5&4&11&10&7&6&9&8\vspace{-0.01in}\\ 14&15&0&1&12&13&2&3&10&11&4&5&8&9&6&7\vspace{-0.01in}\\ 1&0&15&14&3&2&13&12&5&4&11&10&7&6&9&8\vspace{-0.01in}\\ 15&14&1&0&13&12&3&2&11&10&5&4&9&8&7&6\vspace{-0.01in}\\ 0&1&14&15&2&3&12&13&4&5&10&11&6&7&8&9\vspace{-0.01in}\\ 15&14&1&0&13&12&3&2&11&10&5&4&9&8&7&6\vspace{-0.01in}\\ 0&1&14&15&2&3&12&13&4&5&10&11&6&7&8&9\vspace{-0.01in}\\ 14&15&0&1&12&13&2&3&10&11&4&5&8&9&6&7\vspace{-0.01in}\\ 1&0&15&14&3&2&13&12&5&4&11&10&7&6&9&8\vspace{-0.01in}\\ 14&15&0&1&12&13&2&3&10&11&4&5&8&9&6&7\vspace{-0.01in}\\ 1&0&15&14&3&2&13&12&5&4&11&10&7&6&9&8\vspace{-0.01in}\end{array}\right].$$

$$F_{16}=\left[\!\begin{array}[]{cccccccccccccccc}2&241&6&245&10&249&14&253&16&255&12&251&8&247&4&243\\ 15&256&11&252&7&248&3&244&1&242&5&246&9&250&13&254\\ 225&18&229&22&233&26&237&30&239&32&235&28&231&24&227&20\\ 240&31&236&27&232&23&228&19&226&17&230&21&234&25&238&29\\ 210&33&214&37&218&41&222&45&224&47&220&43&216&39&212&35\\ 223&48&219&44&215&40&211&36&209&34&213&38&217&42&221&46\\ 49&194&53&198&57&202&61&206&63&208&59&204&55&200&51&196\\ 64&207&60&203&56&199&52&195&50&193&54&197&58&201&62&205\\ 66&177&70&181&74&185&78&189&80&191&76&187&72&183&68&179\\ 79&192&75&188&71&184&67&180&65&178&69&182&73&186&77&190\\ 161&82&165&86&169&90&173&94&175&96&171&92&167&88&163&84\\ 176&95&172&91&168&87&164&83&162&81&166&85&170&89&174&93\\ 146&97&150&101&154&105&158&109&160&111&156&107&152&103&148&99\\ 159&112&155&108&151&104&147&100&145&98&149&102&153&106&157&110\\ 113&130&117&134&121&138&125&142&127&144&123&140&119&136&115&132\\ 128&143&124&139&120&135&116&131&114&129&118&133&122&137&126&141\end{array}\!\right].$$

$$F_{16}=\left[\!\begin{array}[]{cccccccccccccccc}65&191&94&164&69&187&90&168&73&183&86&172&77&179&82&176\\ 96&162&67&189&92&166&71&185&88&170&75&181&84&174&79&177\\ 97&159&126&132&101&155&122&136&105&151&118&140&109&147&114&144\\ 128&130&99&157&124&134&103&153&120&138&107&149&116&142&111&145\\ 129&127&158&100&133&123&154&104&137&119&150&108&141&115&146&112\\ 160&98&131&125&156&102&135&121&152&106&139&117&148&110&143&113\\ 161&95&190&68&165&91&186&72&169&87&182&76&173&83&178&80\\ 192&66&163&93&188&70&167&89&184&74&171&85&180&78&175&81\\ 193&63&222&36&197&59&218&40&201&55&214&44&205&51&210&48\\ 224&34&195&61&220&38&199&57&216&42&203&53&212&46&207&49\\ 225&31&254&4&229&27&250&8&233&23&246&12&237&19&242&16\\ 256&2&227&29&252&6&231&25&248&10&235&21&244&14&239&17\\ 1&255&30&228&5&251&26&232&9&247&22&236&13&243&18&240\\ 32&226&3&253&28&230&7&249&24&234&11&245&20&238&15&241\\ 33&223&62&196&37&219&58&200&41&215&54&204&45&211&50&208\\ 64&194&35&221&60&198&39&217&56&202&43&213&52&206&47&209\end{array}\!\right].$$

$$F_{24}$$

$$\displaystyle\left.\begin{array}[]{cccccccccccc}156&181&204&229&252&277&300&325&348&373&396&421\\ 422&395&374&347&326&299&278&251&230&203&182&155\\ 154&183&202&231&250&279&298&327&346&375&394&423\\ 424&393&376&345&328&297&280&249&232&201&184&153\\ 152&185&200&233&248&281&296&329&344&377&392&425\\ 426&391&378&343&330&295&282&247&234&199&186&151\\ 157&180&205&228&253&276&301&324&349&372&397&420\\ 419&398&371&350&323&302&275&254&227&206&179&158\\ 159&178&207&226&255&274&303&322&351&370&399&418\\ 417&400&369&352&321&304&273&256&225&208&177&160\\ 161&176&209&224&257&272&305&320&353&368&401&416\\ 415&402&367&354&319&306&271&258&223&210&175&162\\ 163&174&211&222&259&270&307&318&355&366&403&414\\ 413&404&365&356&317&308&269&260&221&212&173&164\\ 165&172&213&220&261&268&309&316&357&364&405&412\\ 411&406&363&358&315&310&267&262&219&214&171&166\\ 167&170&215&218&263&266&311&314&359&362&407&410\\ 409&408&361&360&313&312&265&264&217&216&169&168\\ 150&187&198&235&246&283&294&331&342&379&390&427\\ 428&389&380&341&332&293&284&245&236&197&188&149\\ 148&189&196&237&244&285&292&333&340&381&388&429\\ 430&387&382&339&334&291&286&243&238&195&190&147\\ 146&191&194&239&242&287&290&335&338&383&386&431\\ 432&385&384&337&336&289&288&241&240&193&192&145\end{array}\right]$$

$$Q_{24}$$

$$\displaystyle\left.\begin{array}[]{cccccccccccc}6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\\ 6&7&8&9&10&11&12&13&14&15&16&17\\ 17&16&15&14&13&12&11&10&9&8&7&6\end{array}\right]$$

$$R_{24}$$

Full text

Eulerian Composition of Certain Franklin Squares

RONALD P. NORDGREN

Rice University

Houston, TX 77251

nordgren@rice.edu

Abstract. Several specific Franklin squares and magic squares are decomposed into their quotient and remainder squares. The results support the conjecture that Franklin used the Eulerian composition method to construct many of his squares. This method also can be used to construct new Franklin squares as illustrated herein.

Introduction

Franklin squares were created by Benjamin Franklin and two of them were first published in 1769 according to Pasles [10, p. 127]. They differ from ordinary magic squares by the imposition of bent-diagonal sum conditions instead of the two diagonal sum conditions. Franklin squares have received considerable attention over the years as reviewed by Pasles [10]. A general treatment of magic squares and circles is given by Pickover [12].

Unfortunately, Franklin did not reveal his construction methods and there is controversy on this question. His order-8 and order-16 squares of 1769, as given in [10], can be constructed by a direct method due to Jacobs [4] which he applies to construct an order-24 Franklin square. This construction involves sequential placement of integer elements in five prescribed steps. Jacobs’ construction also yields the order-32 Franklin square of Behrforooz [1]. However, a proof of the validity of Jacobs’ construction for order-$8k$ Franklin squares is lacking.

Franklin squares also can be constructed by the addition of two orthogonal auxiliary squares according to a composition formula first published by Euler [2] in 1776. According to Pasles [10], the composition construction was used by Youle in 1813 to construct Franklin’s original two squares and perhaps by Franklin himself (before Euler). However, it is not clear how Franklin formed his two auxiliary squares, if indeed he used this method.

Nordgren [6] gives a systematic construction for Franklin squares of order $8k$. He obtains two types of formulas for the elements of the two auxiliary squares that allow straightforward numerical formation of order-$8k$ Franklin squares. These formulas are used to verify that the auxiliary squares are suitable for construction of Franklin squares. Since this construction leads to Franklin’s original 1769 squares for $k=1,2,$ it may be regarded as an extension of his methodology and may have been used by him. This speculation is supported by an examination of the Eulerian composition of three order-8 Franklin squares by Pasles [10] and three order-16 Franklin squares in what follows. The auxiliary squares for these Franklin squares follow regular patterns. The Eulerian composition of some of these squares and others is given by Morris [5] who also contends that Franklin used this method. It is difficult to see how many of these Franklin squares could be constructed by direct methods.

The formulas given in [6] also generate the order-24 Franklin square of Jacobs [4] (Figures 1,2,3) and the order-32 Franklin square of Behrforooz [1]. A new order-40 Franklin square, constructed by Nordgren’s method, is shown in Figure 4. It also can be constructed by Jacob’s method. These constructions lend support to Jacobs’ construction but a direct connection between the two methods is lacking.

Numerical generation of all natural order-8 Franklin squares is carried out by Schindel, et.al. [13]. The basic 4320 of these squares were made available to the author and one of them is decomposed into auxiliary squares in what follows.

Definitions

For completeness, we review the definitions of magic and Franklin squares. All rows and columns of a semi-magic square matrix must sum to an index number $m.$ If its main diagonal and the cross diagonal also sum to $m,$ then the square is magic. Natural $n\times n$ (order-$n)$ magic and semi-magic squares have elements $1,2,\ldots,n^{2}$ for which

[TABLE]

Instead of the two diagonal sum conditions, Franklin defined four families of bent diagonals which must sum to $m$. In the following families of order-6 squares, elements on the six bent diagonals (with wraparound) have the same symbol:

[TABLE]

Also, Franklin required that the elements on all left and right half-rows and all upper and lower half-columns of his squares must sum to $m/2$ which makes his squares semi-magic. This requirement forces natural Franklin squares to be of doubly-even order $\left(n=4k\right).$ In addition, he required that elements in all $2\times 2$ subsquares (including broken ones) of his order-$n$ square sum to $4m/n.$ Matrix formulas for the three Franklin square conditions are given by Nordgren [7]. Hurkens [3] shows by exhaustive search that no natural order-12 Franklin squares exist. Also, Pasles [11] shows that there are no natural order-4 Franklin squares.

In a pandiagonal square, the elements on all diagonals in both directions (including broken ones) sum to $m.$ Some Franklin squares also are pandiagonal, including one-third of the order-8 natural ones [13]. Nordgren [6] shows that a Franklin square of order $8k$ can be transformed to a pandiagonal magic square in two ways but the converse is not true in general. Also, Nordgren [7] shows that an order-$8k$ pandiagonal Franklin magic square can be obtained from transformation of a complete (or most-perfect) magic square which are studied and enumerated by Ollerenshaw and Brée [9].

For a square matrix $M_{n}$ of order-$n,$ Euler’s composition formula [2] can be written as

[TABLE]

where $Q_{n}$ and $R_{n}$ are order-$n$ orthogonal matrices and $U_{n}$ is the unity matrix with all elements $1.$ Orthogonal matrices are defined as having each ordered pair of elements in the same position in the two matrices occurring once and only once. A natural Franklin matrix can be constructed by requiring that the quotient matrix $Q_{n}$ and the remainder matrix $R_{n}$ have elements $0,1,\ldots,n-1$ repeated $n$ times. If such $Q_{n}$ and $R_{n}$ are orthogonal and satisfy the three Franklin sum conditions with $m$ replaced by $\hat{m}=n\left(n-1\right)/2,$ then $M_{n}$ from (3) is a Franklin matrix.

Given $M_{n},$ then $R_{n}$ and $Q_{n}$ are obtained from

[TABLE]

which are used in what follows.

Order-6 Squares

Pasles [10, p. 196] gives the following square published by Franklin in 1769:

[TABLE]

It is natural, semi-magic, and its four main bent-diagonals (starting at the corners) sum to $m=111$ as do the four bent diagonals that bend at the middle of the four sides. In addition, other sets of squares add to $m$ [10, p. 197]. However, it does not satisfy all the Franklin square sum conditions. Its auxiliary matrices from (4) are

[TABLE]

These auxiliary matrices are not semi-magic since their second and fifth columns do not sum to $\hat{m}=15$ as do all other columns and all rows. This counterexample indicates that semi-magic conditions on $Q_{n}$ and $R_{n}$ are not necessary conditions for a square to be semi-magic. Also, some of the bent diagonals of $Q_{n}$ and $R_{n}$ mentioned above do not sum to $\hat{m}=15$. However, a counterexample of a full Franklin square has not been found and it is an open question whether Franklin conditions on $Q_{n}$ and $R_{n}$ are necessary for $M_{n}$ from (3) to be a Franklin square.

Here is the order-6 natural magic square given by Euler [2]:

[TABLE]

with auxiliary matrices from (4):

[TABLE]

This time $Q_{6}$ and $R_{6}$ are magic with $\hat{m}=15$.

Here is the order-6 natural magic square from the Historical Museum in Xian, China (photograph in [8]):

[TABLE]

with auxiliary matrices from (4):

[TABLE]

Again $Q_{6}$ and $R_{6}$ are magic with $\hat{m}=15$.

The method of constructing the above three $M_{6}$ squares is not apparent and it does not appear to make use of auxiliary squares.

Order-8 Squares

We consider the Eulerian composition of three Franklin squares given by Pasles [10, pp. 236-237]. Several of these squares also have been decomposed by Morris [5]. Here is Franklin’s original order-8 square published in 1769 [10, p. 127]:

[TABLE]

with auxiliary matrices from (4) and Pasles [10, p. 236]:

[TABLE]

These auxiliary matrices satisfy Franklin’s three sum conditions with $\tilde{m}=28$. They follow from formulas given by Nordgren [6] based on the form of the first row of $Q_{8}$ and the main diagonal of $R_{8}.$ Also, this $F_{8}$ can be constructed by the direct method of Jacobs [4].

Here is Franklin’s order-8 pandiagonal magic square given by Pasles [10, p. 207]:

[TABLE]

It satisfies Franklin’s bent diagonal sum conditions and his $2\times 2$ subsquare sum conditions but not his half-row/column sum conditions. Its auxiliary matrices from (4) and Pasles [10, p. 237] are:

[TABLE]

These auxiliary matrices also satisfy two of Franklin’s three sum conditions with $\tilde{m}=28$ and they are pandiagonal.

Here is another of Franklin’s order-8 Franklin square given by Pasles [10, p. 169]:

[TABLE]

with auxiliary matrices from (4) and Pasles [10, p. 237]:

[TABLE]

These auxiliary matrices also satisfy Franklin’s three sum conditions with $\tilde{m}=28.$ This $Q_{8}$ is pandiagonal but $R_{8}$ is not and so $F_{8}$ of (15) is not pandiagonal.

The elements of all of the above auxiliary matrices follow a distinctive pattern which may be how Franklin constructed them. A direct construction of the two Franklin matrices (13,15) appears to be rather difficult. The reader is invited to try!

The following order-8 pandiagonal Franklin magic square is given by Schindel, et.al. [13, Supplement #2574]:

[TABLE]

with auxiliary matrices from (4):

[TABLE]

These auxiliary matrices also satisfy Franklin’s three sum conditions with $\tilde{m}=28$ and they are pandiagonal. Although there is a pattern to their elements, it is not clear how they could be systematically constructed. Auxiliary matrices of other Franklin matrices from [13] show similar patterns.

Order-16 Squares

Here is Franklin’s original order-16 square published in1769 according to Pasles [10, p. 135]:

[TABLE]

with auxiliary matrices from (4) and [5]:

[TABLE]

[TABLE]

These auxiliary matrices satisfy Franklin’s three sum conditions with $\tilde{m}=120$. They can be constructed by Nordgren’s formulas [6]. The Franklin square $F_{16}$ also is constructed by Jacobs’ direct method [4].

Here is Franklin’s pandiagonal square according to Pasles [10, p. 202]:

[TABLE]

with auxiliary matrices from (4) and [5]:

[TABLE]

[TABLE]

These auxiliary matrices satisfy Franklin’s three sum conditions with $\tilde{m}=120$ and they are pandiagonal. Again their elements follow a distinct pattern.

Here is a new order-16 natural, pandiagonal, Franklin square constructed by generalizing the order-8 auxiliary squares (14):

[TABLE]

Further generalizations should lead to higher order-$8k$ squares of this same type.

Here is a new order-16 natural Franklin square constructed by generalizing the order-8 auxiliary squares (16):

[TABLE]

Further generalizations should lead to higher order-$8k$ squares of this same type. Several other squares of the Franklin type are constructed by Morris [5] using Eulerian composition. Also, Hurkens [3] constructs several Franklin squares of various orders by direct methods.

Conclusion

In the author’s opinion and that of others [5, 10], the distinct pattern of elements of the auxiliary matrices $Q_{n}$ and $R_{n}$ considered here indicate that Franklin probably used the Eulerian composite method of construction of his squares of orders 8 and 16. It is difficult to see how many of his squares and others presented here and in [5, 10] could be constructed by a direct method. However, the referees of Nordgren’s article [6] contend that Franklin used a direct method which shows that the question of Franklin’s construction method may never be resolved to everyone’s satisfaction.

**Acknowledgement   **I thank Peter Loly and Donald Morris for helpful discussions and Adam Rogers for sending me the 4320 basic order-8 Franklin squares from [13].