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Franklin Square: Introduction
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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:
- many V-shaped groups of n cells - whether facing right, left, up, or down - are magical, and;
- 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:
- construction activities,
- practice with finding patterns and using
statistical terminology, and even
- creating informal or formal algebraic proofs.
Below you will find:
- instructions for constructing Franklin squares;
- observations about the squares
using statistical terminology with fairly rigorous proofs of most;
- a sheet with several 8 X 8 grids (some with examples of a key observation), and
- 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:
- to create an algorithm to generate a Franklin square of any order (assuming they
did not use the one provided), and
- 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
- To construct a Franklin Square from an algorithm.
- To identify patterns
involving relatively low numbers and basic arithmetic operations
on an 8 X 8 grid.
- 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.
- 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:
- 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).
- Four-function calculators.
- Computers with installed spreadsheets (optional).
Prerequisite Skills and Knowledge:
- Ability to follow detailed instructions.
- Knowledge of basic statistical terms.
- Experience writing about mathematical concepts and/or proof-writing
experience (optional).
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