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Ben Franklin - A Math-a-Magician!

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Did you know that Benjamin Franklin created many inventions, including the Magic Square? (A magic square is a box of numbers arranged so that any line of numbers adds up to be the same number, including the diagonals!) Richard Walz has written a historical fiction book about this. It is fun for the students to read while at the same time it gives them a great deal of historical information. It also contains many activities that can be used along with the book.

History shows that Franklin served as clerk for the Pennsylvania Assembly. Uninterested in the meetings, Ben would doodle on a piece of paper to pass the time. In 1771, he stated, "I was at length tired with sitting there to hear debates, in which, as clerk, I could take no part, and which were often so un-entertaining that I was induc'd to amuse myself with making magic squares or circles" (Franklin 1793). So being bored, Ben wrote down numbers in a box divided into squares, and then pondered how the numbers added up in rows and columns...and thus the Magic Square was born. In fact, he studied and composed some amazing magical squares, even going so far as to declare one square “the most magically magical of any magic square made by any magician.”

I love to use Magic Squares in my classroom. The construction and analysis of magic squares provides practice in mental arithmetic, operations with numbers, geometry, and measurement plus it encourages logical reasoning and creativity, all in a game-like setting.  Furthermore, they are a powerful tool for teaching students basic addition skills since each row, column, and diagonal must add up to be the same sum.

One effective way to use a Magic Square is to omit a few of the numbers from the boxes, then have students try to figure out which numbers are missing. To find these numbers, first the students will have to calculate the magic sum. A Magic Square also provides an engaging way to develop mental math skills. Try using magic squares as a warm-up at the beginning of math class or as a math center activity. In addition, students might also want to create their own Magic Squares and then have their classmates solve them.

Below is a magic square for you to solve. You are to arrange the digits 1-9 in the squares below
so that each column, row and diagonal adds up to 15. Can you do it?

To find a solution to this magic square puzzle, look under
Answers to Problems at the top of the Home page.

Your students can make their own Magic Squares by following these steps.  Begin by using a box divided into nine squares.  (There are larger ones, but as they grow in size so does the difficulty.)

  1. The first numeral is placed in the top row, center column.
  2. An attempt is always make to place the next numeral in the square above and to the right of numeral last placed.  All the rest of the rules tell you what to do when rule #2 cannot be satisfied.
  3. If, in the placement of the next numeral according to rule #2, the numeral falls above the limits of the magic square, place the numeral in the bottom square of the next column to the right of the last placed numeral.
  4. If, in the normal placement of the next numeral, it falls to the right of the limits of the magic square, that numeral is placed in the left-hand square of the row above the last placed numeral.
  5. If the cell above and to the right is filled, place the numeral in the cell immediately below the last square filled.
  6. Using this method, filling the upper right-hand cell completes a sequence of moves. Then this happens, the next numeral is placed in the cell immediately below the upper right hand corner square.
Do these steps sound absolutely confusing?  Maybe the pictures below will help to clarify the rules.

Now have your students try this.  Using the blank nine squared Magic Square seen above, use the numerals 11, 12, 13, 14, 15, 16, 17, 18 and 19 to make each row (horizontal, diagonally & vertical) add up to 45. Ask your students if they see a pattern between this new Magic Square and the first one.  (Ten has simply been added to each digit.)  You might also try making your own at Make Your Own Magic Squares.

This post has only scratched the surface of Magic Squares, but isn't that like most things in math?  I trust your students will give Magic Squares a try while having fun doing it!

Check out my newest product entitled The A, B, C's of Number Tiles.  All of the 26 letter puzzles of this 42 page handout are solved in a similar way that magic squares are solved. The activities vary in levels of difficulty. Because the pages are arranged alphabetically, and are not in any particular order based on difficulty, the students are free to skip around in the book. Since the students do not write in the book, the math-a-magical puzzles can be copied and laminated so that they can be used from year to year.