Mathemagical Properties of 1105

1105 is the magic sum of a 13 × 13 magic square. Why?
Because 13×13 = 169 and 169×170÷2÷13 = 13×85 = 1105. If you follow the location of the numbers 1, 2, 3, 4, all the way to 169 in the magic square, you will see the pattern that I used to make that magic square. If you click on 10-factors-1102-1110  and go to the magic squares tab, you can use the same pattern or try another to create an 11 × 11, 13 × 13, or 15 × 15 magic square. The sums on the rows, columns, and diagonals will automatically populate as you write in the numbers so you can verify that you have indeed created a magic square.

1105 tiny squares can be made into a decagon so we say it is a decagonal number: Those 1105  tiny squares can also be arranged into a centered square: Why is 1105 the 24th Centered Square Number? Because it is the sum of consecutive square numbers:
24² + 23² = 1105

But that’s not all! 1105 is the smallest number that is the sum of two squares FOUR different ways:

24² + 23² = 1105
31² + 12² = 1105
32² + 9² = 1105
33² + 4² = 1105

1105 is also the smallest number that is the hypotenuse of THIRTEEN different Pythagorean triples. Yes, THIRTEEN! (Seven was the most any previous number has had.) It is also the smallest number to have FOUR of its Pythagorean triplets be primitives (Those four are in blue type.):

47-1104-1105 calculated from 24² – 23², 2(24)(23), 24² + 23²
105-1100-1105 which is 5 times (21-220-221)
169-1092-1105 which is 13 times (13-84-85)
264-1073-1105 calculated from 2(33)(4), 33² – 4², 33² + 4²
272-1071-1105 which is 17 times (16-63-65)
425-1020-1105 which is (5-12-13) times 85
468-1001-1105 which is 13 times (36-77-85)
520-975-1105 which is (8-15-17) times 65
561-952-1105 which is 17 times (33-56-85)
576-943-1105 calculated from 2(32)(9), 32² – 9², 32² + 9²
663-884-1105 which is (3-4-5) times 221
700-855-1105 which is 5 times (140-171-221)
744-817-1105 calculated from 2(31)(12), 31² – 12², 31² + 12²

Why is it the hypotenuse more often than any previous number? Because of its factors! 1105 = 5 × 13 × 17, so it is the smallest number that is the product of THREE different Pythagorean hypotenuses.

It gets 1 triple for each of its three individual factors: 5, 13, 17, 2 triples for each of the three ways the factors can pair up with each other: 65, 85, 221, and four primitive triples for the one way they can all three be together: 1105. Thus it gets 2º×3 + 2¹×3 + 2²×1 = 3 + 6 + 4 = 13 triples.

Speaking of factors, let’s take a look at 1105’s factoring information:

• 1105 is a composite number.
• Prime factorization: 1105 = 5 × 13 × 17
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1105 has exactly 8 factors.
• Factors of 1105: 1, 5, 13, 17, 65, 85, 221, 1105
• Factor pairs: 1105 = 1 × 1105, 5 × 221, 13 × 85, or 17 × 65
• 1105 has no square factors that allow its square root to be simplified. √1105 ≈ 33.24154 1105 is also a palindrome in four different bases, and I also like the way it looks in base 8:
It’s 10001010001 in BASE 2 because 2¹º + 2⁶ + 2⁴ + 2º = 1105,
101101 in BASE 4 because 4⁵ + 4³ + 4² + 4º = 1105,
2121 in BASE 8 because 2(8³) + 1(8²) + 2(8) + 1(1) = 1105,
313 in BASE 19 because 3(19²) + 1(19) + 3(1) = 1105
1M1 in BASE 24 (M is 22 base 10) because 24² + 22(24) + 1 = 1105

Last, but certainly not least, you wouldn’t think 1105 is a prime number, but it is a pseudoprime: the second smallest Carmichael number. Only Carmichael number 561 is smaller than it is.

A Carmichael number is a composite number that behaves like a prime number by giving a false positive to all of certain quick prime number tests:
1105 passes the test p¹¹⁰⁵ Mod 1105 = p for all prime numbers p < 1105. Here is an image of my computer calculator showing 1105 passing the first five tests! Only a prime number should pass all these tests. 1105 is indeed a number with amazing mathemagical properties!

880 Flip the Diagonals!

There are 880 different 4 × 4 Magic Squares. (In addition, each of those can be rotated and/or reflected to make 880 × 8 Magic Squares.) I have found a few that start with some simple ways to order the numbers 1 to 16 in the boxes and then flip the diagonals to create a perfect Magic Square.

For this first one I began by putting the numbers 1-4 in one box with the numbers 13-16 in the cater-corner box. Then 5-8 and 9-12 occupy the other spaces. Then I flipped the diagonals and got a Magic Square! If we flip the locations of the numbers 5-8 and the numbers 9-12, it still works: This third Magic Square begins with the most common way to order the numbers from 1 to 16: With just a little tweaking of that most common way to order the numbers, I was able to make the following three Magic Squares. Begin by switching the first two rows with each other and the last two rows with each other. Notice in every beginning square the diagonals already equal 34. We can also switch the first two rows with the last two rows, but that just gives us a reflection of the one just above it. (The colors don’t match, but it is still a reflection!) Try switching the first two columns with the last two columns. You’ll get another different one. There are MANY more ways to do a 4 × 4 Magic Square.

Last week’s excel file, 12 factors 864-874, included some Magic Square templates including the 4 × 4 one. You can use that file to find more Magic Squares. Go ahead give it a try!

Here’s a little more about the number 880:

880 is the hypotenuse of a Pythagorean triple:

• 528-704-880, which is (3-4-5) times 176.

880 is the sum of the twelve prime numbers from 47 to 101.

• 880 is a composite number.
• Prime factorization: 880 = 2 × 2 × 2 × 2 × 5 × 11, which can be written 880 = 2⁴ × 5 × 11
• The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 880 has exactly 20 factors.
• Factors of 880: 1, 2, 4, 5, 8, 10, 11, 16, 20, 22, 40, 44, 55, 80, 88, 110, 176, 220, 440, 880
• Factor pairs: 880 = 1 × 880, 2 × 440, 4 × 220, 5 × 176, 8 × 110, 10 × 88, 11 × 80, 16 × 55, 20 × 44 or 22 × 40
• Taking the factor pair with the largest square number factor, we get √880 = (√16)(√55) = 4√55 ≈ 29.66479. 870 If You Can Solve a 3×3 and a 4×4 Magic Square, Then You Can Solve a 12×12 Magic Square

It’s simple multiplication, folks!

3×4 = 12, so if you can solve a 3×3 Magic Square and a 4×4 Magic Square, then you can solve a 12×12 Magic Square, too.

Seriously. Really. It’s true! You really can solve a 12×12 Magic Square. I did it. Twice. You’ll see. You can do it, too. That solution consists of sixteen 3×3 Magic Squares that use all the numbers from 1 to 144. The one in the corner is the most familiar one. The other ones are just that familiar Magic Square plus 9, 18, 27 and a few other multiples of 9. I used the 4×4 Magic Square below as a guide as I placed the sixteen 3×3 Magic Squares on the excel file I made, 12 factors 864-874. make science GIFs like this at MakeaGif

There is 1 solution times 8 orientations (right-side-up, upside-down, etc.) for the 3×3 Magic Square, and there are 880 solutions times 8 orientations for the 4×4 Magic Square. That’s a lot of different possible solutions!

I also solved the 12×12 Magic Square by using nine 4×4 Magic Squares and placing them on the second grid in 12 factors 864-874. This arrangement also has many more possibilities than the one shown here. You can easily solve your own 12×12 Magic Square by opening this excel file, 12 factors 864-874, going to the 12×12 Magic Square tab, and following the directions.

Here’s more about the awesome number 870:

29 × 30 = 870. That means 870 is 29 × (29+1), so 870 is the sum of the first 29 even numbers.

870 is the sum of the ten prime numbers from 67 to 107.

Because 5 and 29 are its factors, 870 is the hypotenuse of four Pythagorean triples: ( Note that 5 × 29 = 145)

• 102-864-870, which is 6 times (17-144-145)
• 144-858-870, which is 6 times (24-143-145)
• 522-696-870, which is (3-4-5) times 174
• 600-630-870, which is (20-21-29) times 30

870 has some interesting representations in other bases:

• 606 in BASE 12, because 6(12²) + 0(12¹) + 6(12º) = 6(145) = 870
• 303 in BASE 17, because 3(17²) + 0(17¹) + 3(17º) = 3(290) = 870
• 110 in BASE 29, because 1(29²) + 1(29¹) + 0(29º) = 870
• T0 in BASE 30 (T is 29 base 10), because 29(30¹) + 0(30º) = 29(30) = 870

8 + 0 + 7 + 8³ + 0³ + 7³ = 870. Thank you Stetson.edu for that fun fact.

• 870 is a composite number.
• Prime factorization: 870 = 2 × 3 × 5 × 29
• The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 870 has exactly 16 factors.
• Factors of 870: 1, 2, 3, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, 435, 870
• Factor pairs: 870 = 1 × 870, 2 × 435, 3 × 290, 5 × 174, 6 × 145, 10 × 87, 15 × 58, or 29 × 30
• 870 has no square factors that allow its square root to be simplified. √870 ≈ 29.495762 22 Factor Trees

22 is a composite number. 22 = 1 x 22 or 2 x 11. Factors of 22: 1, 2, 11, 22. Prime factorization: 22 = 2 x 11. When 22 is a clue in the FIND THE FACTORS 1 – 12 puzzles, use 2 and 11 as the factors.

‘Tis the season to use factor trees to find all the factors of a number. Here is a factor tree that shows all the prime factors of 560. Next to it is a factor tree for 324. Do you see all of their prime factors clearly?

Because sometimes one can’t “see the factors for the trees,” I recommend circling the prime factors or doing something else to make them more distinct.  Here are the trees again with every prime factor clearly visible:

I make logic puzzles based on the multiplication table. The puzzles for today’s post could also be called factor trees because you have to factor the clues to find the solution, and the puzzles are shaped like evergreen trees in whole or in part. I’ve even included a tree hanging from the ceiling, of all places! It may seem like a modern novelty, but people have been hanging Christmas trees upside down in Eastern Europe for centuries. To solve the puzzles either cut and paste the puzzles into a document to print or click 12 Factors 2013-12-12.