Math 104 - Cooley Math For Elementary Teachers I OCC

Activity #3 – Magic Squares

California State Content Standard – Mathematical Reasoning – Grade Six
1.0 Students make decisions about how to approach problems:
1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying
missing information, sequencing and prioritizing information, and observing patterns.

J Exercises:

1) A magic square is a square where all rows, columns, and diagonals have the same sum (or

difference, if mentioned). Using each of the following digits in the sequence, 1, 2, 3, 4, 5, 6, 7, 8, 9, once and only once, fill in magic square A. Then use the following digits in the sequence,

2, 5, 8, 11, 14, 17, 20, 23, 26, to fill in magic square B.

Magic Square A Magic Square B

2) Can you make any observations on the placement of the numbers as you compare Magic Square A to

Magic Square B?

3) A magic square where all rows, columns, and diagonals have the same sum, can always be

constructed, provided the nine numbers designed are of the form of an arithmetic sequence. An

arithmetic sequence is where any two consecutive numbers in a sequence always share the same

common difference or distance. Notice the sequence used in Magic Square A is an arithmetic

sequence, because the common difference between any two consecutive numbers is +1. Also, the

sequence used in Magic Square B is also an arithmetic sequence, because the common difference

between any two consecutive numbers is +3. Create an arithmetic sequence of your own and use that

sequence to construct a magic square.


4) The following magic squares have “generic order”. This means that any arithmetic sequence of

9 numbers, 16 numbers, or 25 numbers can be placed in magic squares of size 3 x 3, 4 x 4, and 5 x 5,

respectively, by following the order of the numbers contained in the squares below. For example, if

you had an arithmetic sequence of 23, 27, 31, 35, 39, 43, 47, 51, 55, you can simply follow the order

of the 3 x 3 magic square. Place the initial number 23 where the 1 is, place the next number 27,

where the 2 is, and so on. The magic square will then be complete!

Order 3 / Order 4 / Order 5
8 / 1 / 6 / 1 / 15 / 14 / 4 / 17 / 24 / 1 / 8 / 15
3 / 5 / 7 / 12 / 6 / 7 / 9 / 23 / 5 / 7 / 14 / 16
4 / 9 / 2 / 8 / 10 / 11 / 5 / 4 / 6 / 13 / 20 / 22
13 / 3 / 2 / 16 / 10 / 12 / 19 / 21 / 3
/ 11 / 18 / 25 / 2 / 9

Generate arithmetic sequences to make magic squares of size 4 and size 5 by following the generic

orders above. Verify that the sums of the rows, columns, and diagonals are all correct.

Arithmetic sequence 4 x 4:

Arithmetic sequence 5 x 5:

4 x 4 / 5 x 5

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