## A Multiplication Based Logic Puzzle

### 671 is the Magic Sum of an 11 x 11 Magic Square

6 – 7 + 1 = 0 so 671 is divisible by 11.

671 is the sum of the fifteen prime numbers from 17 to 73.

Because 61 is one of its factors, 671 is the hypotenuse of the Pythagorean triple 121-660-671. The greatest common factor of those three numbers practically jumps out at me. Does it do the same thing to you?

Best of all 671 is the magic sum of an 11 x 11 magic square. (That link from wikipedia helped me construct this square. I’ll give directions so you can do it, too!)

Notice how every row, column, and diagonal on the square sums to 671. The reason it is the magic sum is because the sum of all the numbers from 1 to 121 can be computed and then divided by 11 (the number of rows). Here is the equation:

• 671 = 121 x 120/2/11

Because 11 is an odd number there are simple directions to complete the entire square:

The number 1 is located in the exact center of the top row.

Find the number 2 on the square. (It’s located on the bottom row just right of the exact center square.) Notice that the numbers 3, 4, 5, and 6 are on the same diagonal. If you imagine the diagonal wrapping around the square, you can continue to follow it for numbers 7, 8, 9, 10, and 11. We can’t put the number 12 along the same diagonal because the number 1 is already in that spot, so we put the 12 UNDER the 11 and begin working on a new diagonal.

Anytime a number already occupies a space on a diagonal, put the next number under the preceding number and continue making a new diagonal. When a diagonal reaches the edge of the square, imagine that edge is connected to the opposite edge and continue the diagonal from the opposite edge.

I found it to be the trickiest placing the numbers 67 and 68, but other than that it was rather easy to know where to put the numbers.

Notice that the difference between any smaller number and the larger number just below it is either 12 or 1.

If you have excel on your computer, click on 12 Factors 2015-11-02, select the magic square tab, and then you can make this 11 x 11 magic square yourself. As you type in numbers, the columns, rows, and diagonals will automatically keep a running sum.

Once you get the square to give the magic sum in each direction, you can try doing the same thing with the 13 x 13 magic square that I’ve included on the same page. Its magic sum is 1105 which can be also be computed:

• 1105 = 169 x 170/22/13.

There is actually many more possible and probably more complicated 11 x 11 and 13 x 13 magic squares. I hope you enjoy making some with this easy method.

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• 671 is a composite number.
• Prime factorization: 671 = 11 x 61
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 671 has exactly 4 factors.
• Factors of 671: 1, 11, 61, 671
• Factor pairs: 671 = 1 x 671 or 11 x 61
• 671 has no square factors that allow its square root to be simplified. √671 ≈ 25.903668.

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### 320 Christmas Factor Trees

• 320 is a composite number.
• Prime factorization: 320 = 2 x 2 x 2 x 2 x 2 x 2 x 5, which can be written 320 = (2^6) x 5
• The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 x 2 = 14. Therefore 320 has exactly 14 factors.
• Factors of 320: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 160, 320
• Factor pairs: 320 = 1 x 320, 2 x 160, 4 x 80, 5 x 64, 8 x 40, 10 x 32, or 16 x 20
• Taking the factor pair with the largest square number factor, we get √320 = (√5)(√64) = 8√5 ≈ 17.889

In these factor trees for 320, we can also see factor trees for 4, 8, 10, 16, 20, 32, 40, 64, 80, and 160. The tops of each of those factor trees are in brown. No matter which factor pairs we use to build the tree, we always get the same prime factors which are indicated in red.

Now try finding the factors in this Christmas tree puzzle that make it work like a multiplication table:

Print the puzzles or type the factors on this excel file: 10 Factors 2014-12-08

A Logical Approach to FIND THE FACTORS: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.