## Questions, Suggestions, & References

### by Neil Abrahams

Back to Franklin Square

#### Questions

Here are some ideas for questions:

Is it possible to make a Franklin Cube with some or all of the properties of the square? Which properties?

Can some of the observations be combined to discover new magical or half-magical shapes?

Can the construction algorithm and some observations be used to say more about the value of specific cells on the square?

Can Observation 17 be changed into an if, only if, statement regarding where on-their-side Vs must be located in order to be magic? Can you prove it?

Can the locations of the median value cells in each column be predicted? If so, can the prediction be proven?

What is the mean value of the range of all of the columns of a Franklin Square?

Which columns of a Franklin Square have the least range? How can Observation 13 be extended?

Devise a proof for observation 2.

If you are an experienced user of spreadsheets, you can probably devise a way of easily generating Franklin Squares. If you aren't, this might be helpful.

After placing the values in the seed columns manually, use logical tests in the formula for a cell's value to determine the odd/even value of the column and of the row it is in. Based upon that information, you can devise an expression using both the row's seed value and the value of the previous cell to determines the cell's value.

Spreadsheets expedite the testing of conjectures. Say a student suspects that some checkerboard-like pattern of n cells is magical. He or she can select a cell well off the square, click on auto-sum, and then select cells falling into that pattern on the sheet using the Control key with the Windows version of Excel (your spreadsheet may differ), then press Enter when done. If the sum is the magic constant, he or she can then reselect the cell and drag the mouse so that many other cells are selected and use the Edit Fill command to obtain the values of the same shaped groups of cells on the square which are in the same positions relative to the initial group of cells as the cells just filled are relative to the initial off-square cell. One caution in this is that some of the new groups evaluated may include empty cells, but students would probably recognize this if it occurred.

#### References

Olivastro, Dominic. 1993. Ancient Puzzles: Classic Brainteasers and Other Timeless

Mathematical Games of the Last 10 Centuries. New York: Bantam Books. Henrich, Christopher J. "Magic Squares and Linear Algebra." American Mathematical Monthly 98 (June-July 1991): 481-88.

Patel, Lalbhai D. "The secret of Franklin's 8 X 8 'magic' square. (Benjamin Franklin)."

Journal of Recreational Mathematics 23 (Fall 1991):175-83.

Questions? Comments? Write to Neil Abrahams