A Multiplication Based Logic Puzzle

Archive for the ‘Mathematics’ Category

951 is the 20th Centered Pentagonal Number

Since 951 is the 20th centered pentagonal number, I decided to make the following graphic with 20 concentric pentagons. I’ve outlined the pentagons in the center to make them clearer. The graphic also shows that 951 is one more than five times the 19th triangular number.

951 is also the hypotenuse of a Pythagorean triple:
225-924-951 which is 3 times (75-308-317)

As numbers get bigger, palindromes in base 2 get rarer, but 951 is one of them:
1110110111 in BASE 2 because 1(2⁹) + 1(2⁸) + 1(2⁷) + 0(2⁶) + 1(2⁵) + 1(2⁴) + 0(2³) + 1(2²) + 1(2¹) +1(2⁰) = 951
It is also 1D1 in BASE 25 (D is 14 base 10) because 1(25²) + 13(25¹) + 1(25⁰) = 951

  • 951 is a composite number.
  • Prime factorization: 951 = 3 × 317
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 951 has exactly 4 factors.
  • Factors of 951: 1, 3, 317, 951
  • Factor pairs: 951 = 1 × 951 or 3 × 317
  • 951 has no square factors that allow its square root to be simplified. √951 ≈ 30.83828789

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What Kind of Shape Is 946 In?

First of all, 946 is the sum of the numbers from 1 to 43, so it is the 43rd triangular number.

Every other triangular number is also a hexagonal number. Since 946 is the 43rd triangular number, and 43 is an odd number, 946 is also the 22nd hexagonal number. 946 is the 22nd hexagonal number because 22(2(22) – 1) = 22(43) = 946.

But that’s not all. 946 is different than any previous hexagonal number. 946 is the smallest hexagonal number that is also a hexagonal pyramidal number. It is, in fact, the 11th hexagonal pyramidal number. That means if you stack the hexagons in the graphic below in order from largest to smallest, you would get a hexagonal pyramid made with 946 tiny squares. That’s pretty cool, I think.

 

467 + 479 = 946 so 946 is the sum of two consecutive prime numbers.
946 is also the sum of the twenty prime numbers from 11 to 89.

946 is palindrome 181 in BASE 27 because
1(27²) + 8(27¹) + 1(27⁰) = 729 + 216 + 1 = 946

  • 946 is a composite number.
  • Prime factorization: 946 = 2 × 11 × 43
  • 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 946 has exactly 8 factors.
  • Factors of 946: 1, 2, 11, 22, 43, 86, 473, 946
  • Factor pairs: 946 = 1 × 946, 2 × 473, 11 × 86, or 22 × 43
  • 946 has no square factors that allow its square root to be simplified. √946 ≈ 30.75711

There’s Something Odd about the Number 945

945 = 1 × 3 × 5 × 7 × 9

The sum of the proper divisors of a number determines if the number is abundant, deficient, or perfect. If the sum is greater than the number, the number is abundant. If the sum is less than the number, the number is deficient. If the sum is equal to the number, the number is perfect.

What is a proper divisor? All the factors of a number except itself. Proper divisors are ALMOST the same thing as proper factors. (The number 1 is always a proper divisor, but NEVER a proper factor.)

The first 25 abundant numbers are 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, and 108. Notice that all those numbers are even.

OEIS informs us that 945 is the 232nd abundant number. The first 231 abundant numbers are all even numbers.

Wow, 945 is the smallest ODD abundant number. OEIS also lists the first 31 odd abundant numbers. Every one of the first 31 is divisible by 3 and ends with a 5, but if you scroll down the page you’ll see some that aren’t divisible by 3 or aren’t divisible by 5.

Since 1 × 3 × 5 × 7 × 9 = 945 is the smallest number on the list, you may be wondering about some other numbers:
1 × 3 × 5 × 7 × 9 × 11 = 10,395 made the list.
1 × 3 × 5 × 7 × 9 × 11 × 13 = 135,135 which is too big to be one of the first 31 odd abundant numbers. I was curious if it is also an abundant number, so I found its proper divisors and added them up:

945 is also the hypotenuse of a Pythagorean triple:
567-756-945 which is (3-4-5) times 189

945 looks interesting in a few other bases:
1661 in BASE 8 because 1(8³) + 6(8²) + 6(8¹) + 1(8⁰) = 945
RR in BASE 34 (R is 27 base 10), because 27(34¹) + 27(34⁰) = 27(35) = 945
R0 in BASE 35 because 27(35) + 0(1) = 945

  • 945 is a composite number.
  • Prime factorization: 945 = 3 × 3 × 3 × 5 × 7, which can be written 945 = 3³ × 5 × 7
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 945 has exactly 16 factors.
  • Factors of 945: 1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945
  • Factor pairs: 945 = 1 × 945, 3 × 315, 5 × 189, 7 × 135, 9 × 105, 15 × 63, 21 × 45, or 27 × 35
  • Taking the factor pair with the largest square number factor, we get √945 = (√9)(√105) = 3√105 ≈ 30.74085

Is There Anything Else Special about the Palindrome 939?

Yes, 939 is a palindrome in base 10, but also all of its factors (1, 3, 313, and 939) are palindromes. It is also palindrome
32223 in BASE 4 because 3(4⁴) + 2(4³) + 2(4²) + 2(4¹) + 3(4⁰) = 939

Okay, that’s nice. Is there anything else special about 939?

The first ten decimal places of the cube root of 939 contain ALL ten digits 0 to 9. That’s unusual, and a reason why 939 is a special number. I made this gif to highlight its uniqueness.
Cube Root 939

make science GIFs like this at MakeaGif

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Thank you Stetson.edu for informing us of that amazing fact about 939’s cube root.

939 is also the hypotenuse of a Pythagorean triple:
75-936-939 which is 3 times (25-312-313)

  • 939 is a composite number.
  • Prime factorization: 939 = 3 × 313
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 939 has exactly 4 factors.
  • Factors of 939: 1, 3, 313, 939
  • Factor pairs: 939 = 1 × 939 or 3 × 313
  • 939 has no square factors that allow its square root to be simplified. √939 ≈ 30.64310689

 

Why Is Prime Number 937 the 13th Star Number?

Even though 937 is a prime number, 937 tiny rectangles can be arranged into this beautiful star. Why?

937 is the 13th star number because 6(13)(13 – 1) + 1 = 937.

It is also the 13th star number because it is 12 times the 12th triangular number plus one: Look at this pattern:

The first star number is 12 times the 0th triangular number plus 1. Thus, 12(0) + 1 = 1 (1 yellow rectangle in the center)
The second star number is 12 times the 1st triangular number plus 1. Thus, 12(1) + 1 = 13 (12 green + 1 yellow rectangle in the center)
The third star number is 12 times the 2nd triangular number plus 1. Thus, 12(3) + 1 = 37 (24 blue + 12 green + 1 yellow rectangle in the center)
and so on. . .until
The thirteen star number is 12 times the 12th triangular number plus 1. Thus, 12(78) + 1 = 937 (144 yellow + 132 orange + 120 red + 108 purple + 96 blue + 84 green + 72 yellow + 60 orange + 48 red + 36 purple + 24 blue + 12 green + 1 yellow rectangle in the center)

I made the star so that it consists of one tiny rectangle in the center surrounded by 6 triangles with 78 (the 12th triangular number) rectangles each with another 6 triangles of the same size to form the 6 points of the star.

I very much enjoyed making this star. If you look closely you will see thirteen concentric stars in it following the pattern yellow, green, blue, purple, red, and orange repeated. I added star outlines to make the three smallest stars easier to see.

I think the graphic says a lot about the number 937 all by itself. I hope you enjoy looking at it.

Here’s a little more about the number 937:

24² + 19² = 937, so 937 is the hypotenuse of a Pythagorean triple:
215-912-937 which can be calculated from 24² – 19², 2(24)(19), 24² + 19²

937 is also a palindrome in three other bases:
1021201 in BASE 3 because 1(3⁶) + 0(3⁵) + 2(3⁴) + 1(3³) + 2(3²) + 0(3¹) + 1(3⁰) = 937
1F1 in BASE 24 (F is 15 in base 10) because 1(24²) + 15(24¹) + 1(24⁰) = 937
1A1 in BASE 26 (A is 10 in base 10) because 1(26²) + 10(26¹) + 1(26⁰) = 937

  • 937 is a prime number.
  • Prime factorization: 937 is prime.
  • The exponent of prime number 937 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 937 has exactly 2 factors.
  • Factors of 937: 1, 937
  • Factor pairs: 937 = 1 × 937
  • 937 has no square factors that allow its square root to be simplified. √937 ≈ 30.6104557

How do we know that 937 is a prime number? If 937 were not a prime number, then it would be divisible by at least one prime number less than or equal to √937 ≈ 30.6. Since 937 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 937 is a prime number.

Here’s another way we know that 937 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 24² + 19² = 937 with 24 and 19 having no common prime factors, 937 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √937 ≈ 30.6. Since 937 is not divisible by 5, 13, 17, or 29, we know that 937 is a prime number.

933 Negative and Positive Reflections

Magic mirror on the wall.
Am I teaching one and all?

Teachers reflect. They often ask themselves how their lesson went, what went well, and how they could improve.

Many years ago I taught an algebra class. The textbook suggested I use algebra tiles to teach a lesson on adding negative and positive integers. I had never heard of algebra tiles before. The school didn’t have any, and there wasn’t time to order some online. Later that September day, I looked at some Halloween candy in a store. When I saw a package of Pumpkin Mix m&m’s, I knew I had found the perfect algebra tiles. All the m&m’s in the package were brown or orange and had pumpkin faces with an “m” for the pumpkin’s nose, but this is how I saw them:

The algebra students learned about adding and subtracting positive and negative integers without any problems, and they LOVED it.

Pumpkin Mix m&m’s have been replaced with other varieties. The colors don’t matter. You could have the sides with the “m” be positive and the side without the “m” be negative.

That summer I enrolled in a Teaching Secondary Mathematics class at the university. I needed to do some volunteer work in a school, reflect on the experience, and write a paper about it. I share that slightly edited paper with you today:

I worked with Mark’s classes. Shon and Serena volunteered there as well. Mark’s students are adults many of which are learning English as a second language as they prepare for the GED test.  Every student I observed was motivated to learn.  One of the students struggled with basic addition facts.  Another understood the concepts but wrote the symbols for algebraic sentences in a different order than we use.  Most of the students are learning Pre-Algebra concepts and getting individualized instruction from the computer program “Classworks.”  If students don’t pass a pretest, they can read a brief lesson on the computer, use some virtual manipulatives to learn the concept, and demonstrate what they have learned.  Several students worked on a lesson that required them to solve for x by balancing equations.  Some of the students seemed confused.  I thought it might be helpful if they had physical manipulatives that required them to do the balancing rather than the computer.  Mark allowed me to plan and prepare a lesson for the twelve students in his first-period class.

I typed and printed a worksheet that consisted of four equations and two large rectangles.  I purchased twelve 1.5 oz packages of Reece’s Pieces to use as Algebra Tiles.  The empty packages represented the variable “x,” each orange candy represented “+ 1” and each brown (or yellow) candy represented “-1.”  Shon and Serena also helped the students understand how to use the manipulatives.  I thought the lesson would only last about five minutes, but it lasted the remainder of the class period.  The students did well with the activity, but it would not have gone so well if my fellow students were not there assisting mostly because of English language issues.

Mark asked me to teach the lesson again to his third-period class.  He even bought more candy so I wouldn’t have to.  His third-period class had six or seven students in attendance.  Because Mark bought M&M’s which come in many different colors, I labeled the diagram I drew on the board with +’s and –‘s instead of O’s and B’s when I explained how their mats should look as we did each step.   Doing that made my explanation to third period clearer than my explanation was to the first period.

When we were almost finished, Mark asked me to write more problems so the students could continue practicing balancing equations.  Instead, I asked a student to write a problem for the class.  She quickly wrote one on paper and then on the whiteboard.  After most of the class members had solved her problem, I had her explain the steps to the class.  She did a terrific job and we all clapped.  I asked another student to write a problem.  She shared a problem, and we cheered for her after she explained the steps.  Eventually, every class member wrote a problem for the class to solve, and we cheered after they explained the steps to solve it with their newly acquired English skills.  Mark also wrote a problem, one that I had thought to be too simple to put on the board:
x – 2 = -2.  It turned out not to be too trivial.  Some students needed to manipulate what happens in that case as well.  Shon and Serena assisted some of the students, but clearly before the class period was over most of the students did not need much help.

When class was finished, Mark met with the three of us.  He told us he really liked the activity and that any time you mix candy and learning together, it’s going to be a hit.  He said when I introduce an activity, I need to slow down.  I need to make sure everyone understands what they are supposed to do.  He thanked me and started his next class.  Shon and Serena both enjoyed helping students with the activity.  Serena said slowing down when giving directions seems to be one of the most common suggestions she hears given to pre-service teachers.  She said it might be helpful to have an equal sign between the two rectangles on the mat so students would know that the two sides are supposed to be equal.  Shon mentioned that when I explained what to do, I didn’t stress that we were solving for x so students might not understand what they need to do when they have similar problems to solve but no candy to use as a manipulative.  All of these are good suggestions that will improve my presentation to help students learn better.

As this paper illustrated, reflection when teaching is very important.

Now for a little about the number 933:

All of its digits are divisible by 3, so 933 is divisible by 3.

  • 933 is a composite number.
  • Prime factorization: 933 = 3 × 311
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 933 has exactly 4 factors.
  • Factors of 933: 1, 3, 311, 933
  • Factor pairs: 933 = 1 × 933 or 3 × 311
  • 933 has no square factors that allow its square root to be simplified. √933 ≈ 30.545049

What Kind of Shape is 925 In?

925 is the 22nd Centered Square number because 22² + 21² = 925. I made this graphic to show this fact through the use of color. Look at the center of the centered square. Can you see how the single yellow square and the four small green squares in the center correspond to the same colored squares in the smaller squares? The pattern continues from the inside of the centered square to the outside.

925 is the sum of two squares these THREE ways:

 

  • 22² + 21² = 925
  • 27² + 14² = 925
  • 30² + 5² = 925

925 is the hypotenuse of SEVEN Pythagorean triples:

  • 43-924-925
  • 259-888-925
  • 285-880-925
  • 300-875-925
  • 520-765-925
  • 533-756-925
  • 555-740-925

925 is the 25th pentagonal number because 3((25²) – 25)/2 = 925. The shape in the graphic below of 925 tiny squares may look more like a house, but it is still very much a pentagon.

925 looks interesting in a few other bases:

4141 in BASE 6
1K1 BASE 22 (K is 20 in base 10)
151 BASE 28
PP in BASE 36 (P is 25 in base 10)

  • 925 is a composite number.
  • Prime factorization: 925 = 5 × 5 × 37, which can be written 925 = 2² × 37
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 925 has exactly 6 factors.
  • Factors of 925: 1, 5, 25, 37, 185, 925
  • Factor pairs: 925 = 1 × 925, 5 × 185, or 25 × 37
  • Taking the factor pair with the largest square number factor, we get √925 = (√25)(√37) = 5√37 ≈ 30.41381

 

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