Forum for Science, Industry and Business

Sponsored by:     3M 
Search our Site:


How Many Squares, Mr. Franklin?


This is a magic square created by Benjamin Franklin

When he wasn’t experimenting with lightning or overthrowing the British Empire, Benjamin Franklin found time to fool around with mathematics, inventing a variant of the magic square called Franklin’s squares. Now Maya Ahmed, a mathematics graduate student at UC Davis, has come up with a way to construct both Franklin’s own squares and others of the same type. The methods could have applications in computer programming for business.

A regular magic square is a table of numbers in which any row, column or diagonal adds up to the same number. Mathematicians around the world have studied them for thousands of years.

"They are classical, beautiful objects," said Jesus De Loera, associate professor of mathematics at UC Davis and Ahmed’s thesis supervisor.

Franklin’s squares are similar, but instead of diagonals adding to the magic number, the bent diagonals add to the magic number. The four corners and four center squares also add to the magic number.

Franklin himself created three such squares, two that are eight rows by eight columns and one of 16 by 16.

"No one knows how he did it," Ahmed said. "They’re very hard to construct."

Ahmed’s method turns what looks like an arithmetic problem into a geometry problem. The numbers in a Franklin square can be described by a series of equations -- 127 equations for an eight-by-eight square. Those equations also describe a cone-shaped object in multiple (more than three) dimensions. That yields the basic elements of a Franklin square.

Using this method, Ahmed could both reconstruct Franklin’s three original squares and create new ones that obey the same rules. She was also able to work out the maximum possible number of eight-by-eight Franklin squares: just over 228 trillion.

Franklin regarded his squares as "incapable of useful application." But Ahmed’s methods can also be used to find whole-number solutions to problems of linear equations. An example would be scheduling aircraft and crew members for an airline, De Loera said.

The paper is published in the May issue of the American Mathematical Monthly.

Andy Fell | UC Davis News
Further information:

More articles from Information Technology:

nachricht TIB’s Visual Analytics Research Group to develop methods for person detection and visualisation
19.03.2018 | Technische Informationsbibliothek (TIB)

nachricht Green Light for Galaxy Europe
15.03.2018 | Albert-Ludwigs-Universität Freiburg im Breisgau

All articles from Information Technology >>>

The most recent press releases about innovation >>>

Die letzten 5 Focus-News des innovations-reports im Überblick:

Im Focus: Tiny implants for cells are functional in vivo

For the first time, an interdisciplinary team from the University of Basel has succeeded in integrating artificial organelles into the cells of live zebrafish embryos. This innovative approach using artificial organelles as cellular implants offers new potential in treating a range of diseases, as the authors report in an article published in Nature Communications.

In the cells of higher organisms, organelles such as the nucleus or mitochondria perform a range of complex functions necessary for life. In the networks of...

Im Focus: Locomotion control with photopigments

Researchers from Göttingen University discover additional function of opsins

Animal photoreceptors capture light with photopigments. Researchers from the University of Göttingen have now discovered that these photopigments fulfill an...

Im Focus: Surveying the Arctic: Tracking down carbon particles

Researchers embark on aerial campaign over Northeast Greenland

On 15 March, the AWI research aeroplane Polar 5 will depart for Greenland. Concentrating on the furthest northeast region of the island, an international team...

Im Focus: Unique Insights into the Antarctic Ice Shelf System

Data collected on ocean-ice interactions in the little-researched regions of the far south

The world’s second-largest ice shelf was the destination for a Polarstern expedition that ended in Punta Arenas, Chile on 14th March 2018. Oceanographers from...

Im Focus: ILA 2018: Laser alternative to hexavalent chromium coating

At the 2018 ILA Berlin Air Show from April 25–29, the Fraunhofer Institute for Laser Technology ILT is showcasing extreme high-speed Laser Material Deposition (EHLA): A video documents how for metal components that are highly loaded, EHLA has already proved itself as an alternative to hard chrome plating, which is now allowed only under special conditions.

When the EU restricted the use of hexavalent chromium compounds to special applications requiring authorization, the move prompted a rethink in the surface...

All Focus news of the innovation-report >>>



Industry & Economy
Event News

Virtual reality conference comes to Reutlingen

19.03.2018 | Event News

Ultrafast Wireless and Chip Design at the DATE Conference in Dresden

16.03.2018 | Event News

International Tinnitus Conference of the Tinnitus Research Initiative in Regensburg

13.03.2018 | Event News

Latest News

Earlier flowering of modern winter wheat cultivars

20.03.2018 | Agricultural and Forestry Science

Smithsonian researchers name new ocean zone: The rariphotic

20.03.2018 | Life Sciences

Molecular doorstop could be key to new tuberculosis drugs

20.03.2018 | Life Sciences

Science & Research
Overview of more VideoLinks >>>