The Lost Squares of Dr. Franklin: Ben Franklin's Missing Squares and the Secret of the Magic Circle

Pasles, Paul C.

doi:10.1080/00029890.2001.11919777

Cited by 11 publications

(6 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An example of one of our non-pandiagonal (semi-magic) natural Franklin squares (the 81st produced by the program) follows. Until recently, very few natural Franklin bent diagonal squares were known (Pasles 2001); one of 8th order and one of 16th order have been known from Franklin's writings for a long time; a second square of each order were lost for a long time (Pasles 2001). Recently, Ahmed (2004a,b) gave three more 8th order natural Franklin squares.…”

Section: Resultsmentioning

confidence: 99%

“…A resurgence of interest in the handful of Benjamin Franklin's bent diagonal squares of 8th and 16th order which he constructed in 1736-1737 (Pasles 2001(Pasles , 2003Ahmed 2004a,b) may be related to the tercentenary of his birth which occurred on 17 January 2006. The same year also marks the 250th anniversary of Franklin's election as a Fellow of the Royal Society of London.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS

2006

View full text Add to dashboard Cite

All 8th order Franklin bent diagonal squares with distinct elements 1, …, 64 have been constructed by an exact backtracking method. Our count of 1, 105, 920 dramatically increases the handful of known examples, and is some eight orders of magnitude less than a recent upper bound. Exactly one-third of these squares are pandiagonal, and therefore magic. Moreover, these pandiagonal Franklin squares have the same population count as the eighth order ‘complete’, or ‘most-perfect pandiagonal magic’, squares. However, while distinct, both types of squares are related by a simple transformation. The situation for other orders is also discussed.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS

2006

View full text Add to dashboard Cite

show abstract

“…The tasks of counting and constructing classical most-perfect squares were first approached by McClintock [6] and culminate in the work of Ollerenshaw and Bree [8], which gives a count of the classical most-perfect squares for any doubly even order n, along with a construction method for all such squares. Also, classical most-perfect squares are useful in constructing Franklin magic squares (see [11] and [7], and [9] for historical background).…”

Section: Introductionmentioning

confidence: 99%

Linear Type-p Most-Perfect Squares

Lorch

2017

Preprint

View full text Add to dashboard Cite

We describe a generalization of most-perfect magic squares, called type-p mostperfect squares, and in prime-power orders we give a linear construction of these squares reminiscent of de la Loubère's classical magic square construction method. Type-p mostperfect squares can be used to construct other interesting squares (e.g., generalized Franklin squares) and our linear construction may have implications for counting type-p most-perfect squares.

show abstract

“…In fact, these squares have innumerable fascinating properties. See [1], [3], and [4] for a detailed study of these squares.…”

mentioning

confidence: 99%