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Franklin Square: Introduction

by Neil Abrahams

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[Note: N is the order (number of cells per side) of the square. For our purposes, it's best to think of the square as assuming the shape of a torus. Consequently, the left and rightmost columns and the top and bottom rows are considered to be adjacent. Magic squares having this property are called pan-diagonal.]

Background:

Franklin Squares are a variant of magic squares created by Benjamin Franklin. He could write his order-8 square very quickly and enjoyed astounding his friends with it. As with other magic squares, the sum of the numbers (values) in each of its columns and rows equals the magic constant of (n/2) * (n2 + 1). (We will say that the columns and rows are magical, for short.) Unlike true magic squares, however, the diagonals are not magical. That's a small price to pay, however, for some unusual attributes of Franklin Squares:
  1. many V-shaped groups of n cells - whether facing right, left, up, or down - are magical, and;
  2. any mini-square of n/2 or n cells, respectively - whichever of the two is a square numbe - is either magical or half-magical. If you look for them, you may find that other groups of n or n/2 cells - not necessarily adjacent to each other - are also magical or half-magical.

Format:

The materials presented here can be used in a variety of ways. They can be the basis of:
  1. construction activities,
  2. practice with finding patterns and using statistical terminology, and even
  3. creating informal or formal algebraic proofs.
Below you will find:
  1. instructions for constructing Franklin squares;
  2. observations about the squares using statistical terminology with fairly rigorous proofs of most;
  3. a sheet with several 8 X 8 grids (some with examples of a key observation), and
  4. sample questions that may be asked of students.
Students often enjoy out-of-the-ordinary activities and may be energized by them. Even if all that they do is construct Franklin squares (order-16 for highly-motivated students) by hand and verify some of their properties, they have accomplished something on their own and out of the ordinary. If your purpose is to have students identify patterns, you can shorten the time spent by supplying the completed square or a spreadsheet with it to facilitate analysis (advice is given on how to generate one).

However you intend to proceed, I suggest that you start the lesson by providing some historical background about magic squares (information can be found in some of the many books and articles about mathematical recreations; see references below for some specifically about Franklin squares). After the students have their squares they can look for patterns using statistical terms and/or try to discover some magical groups. The grid sheet will be useful here. Encourage students not only to generalize their observations, but to look for counterexamples to their conjectures as well. (Maybe some students in each group can be designated official debunkers.)

Without question, the two most challenging things you can ask students to do are:
  1. to create an algorithm to generate a Franklin square of any order (assuming they did not use the one provided), and
  2. to provide proofs of their observations of the square's properties.
I suspect that only highly-motivated students will want to take on either of these tasks. However, it might still be useful to ask some students to attempt to complete some only partially-completed proofs.

You might end the unit with a class discussion of everyone's findings. Each group can be assigned a different topic to investigate thoroughly and explain to the rest of the class while learning as much else about the square in general as they can. Greater emphasis should be placed on the validity and precision of findings than on their quantity. I hope that your students and you enjoy this activity. Please let me know how it goes.

Grade Level:

Anywhere from middle school to undergraduate.

Possible Lesson Objectives

  1. To construct a Franklin Square from an algorithm.
  2. To identify patterns involving relatively low numbers and basic arithmetic operations on an 8 X 8 grid.
  3. To describe accurately the patterns on the square in the most general terms possible by using statistical terminology and the variable n, and/or to provide illustrations of them.
  4. To create proofs of the properties of a Franklin Square and/or to create an algorithm to generate one, either by hand or machine.

Materials and Equipment Needed:

  1. A copy of Franklin's Order-8 Square or instructions for constructing one for every student or group of students. (If students are asked to create an algorithm to construct one, order-16 squares should also be provided).
  2. Four-function calculators.
  3. Computers with installed spreadsheets (optional).

Prerequisite Skills and Knowledge:

  1. Ability to follow detailed instructions.
  2. Knowledge of basic statistical terms.
  3. Experience writing about mathematical concepts and/or proof-writing experience (optional).

Next Page: Construction Algorithm



Questions? Comments? Write to Neil Abrahams


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