A Multiplication Based Logic Puzzle

Print the puzzles or type the solution on this excel file: 10-factors 807-814

One of my education professors taught that you can teach any concept with a picture book.

I recently read the book, Stick and Stone, to a class of 6th graders. Yes, 6th graders. You can get away with reading something way below grade level if you tell them before you start reading that you will use the book to introduce them to something that is definitely NOT below grade level. The first few pages of the book are shared by its publisher here:

As you can see, those first few pages equate stone as a zero and stick as a lonely number one.

The middle part of the book teaches about synergizing, working together to make life good and helping each other through tough times.

By the end of the book Stick and Stone know how to work very well together, “Stick, Stone. Together again. Stick, Stone. A perfect ten.”

The book pretty much ends there, but making a perfect ten is only the beginning of what these two characters can do together. I used this book to teach the class not only about getting along and working together, but also about base 2, or binary as it is also called. Every counting number we know can be represented by using just 1’s and 0’s. I wrote on the board the numbers from 1 to 16 and represented the first few of those numbers in base 2. Then I invited class members to come up with how to write the rest of the numbers in base 2. Some students caught on immediately while the others were able to learn how to do it by watching their classmates and listening to them. Eventually with at least 12 different student’s inputs, we came up with a chart that looked something like this:

Notice that the numbers from 9 to 15 are just 1000 plus the numbers directly across from them in the first column.

Some of the sixth grade students had already heard of binary, so I showed them a little more about base 2: I wrote a bunch of 1’s and 0’s “off the top of my head” onto the board and added the headings to show place values: 1’s place, 2’s place, etc.

Then I told them to sum up the place values that contained a one:

The sixth graders were delighted with the answer.

Stick and Stone are the main two characters, but the book has one other character, Pinecone. At first Pinecone bullied Stone, but after Stick stood up to him, the three of them were eventually able to become friends. You might enjoy finding out more about Pinecone by listening to Sean Anderson read the entire book to his children, one of which seems to really enjoy numbers.

If you used a unique symbol to represent Pinecone, it could look like a 2. Then you also could use the symbols 0, 1, and 2 to represent every counting number in base 3. That’s another concept the picture book Stick and Stone could be used to introduce!

To make a chart for base 3, start with these 3 columns of numbers with 3 numbers in each:

Since this is base 3, where should 10 and 100 go? The bottom of the first column and the bottom of the third column both MUST look like a power of 10. The rest of the chart is easy to fill out. Notice the 1 and 2 look exactly the same in base 10 and base 3. Also since 4 = 3 + 1, 5 = 3 + 2, and 6 = 3 + 3, we can easily fill in the 2nd column. Two more addition facts will finish the third column: 7 = 6 + 1, and 8 = 6 + 2.

Now add what you learned about 4, 5, 6, 7, 8, and 9 to column 1 and put the numbers 10 – 18 in the base 10 second column and numbers 19 – 27 in the base 10 third column. Again the bottom of the first column and the bottom of the third column both MUST look like a power of 10, so we now know where to put 1000.

To fill in the rest of the chart simply add 100 to the base 3 numbers in column 1 to get the the base 3 numbers in column 2. Then add 200 to the base 3 numbers in column 1 to get the remaining base 3 numbers in column 3.

You could do this process again to determine the first 81 counting numbers in base 3 with 81 being represented by 10000.

For base 4, you could do something similar with 4 columns. However, for counting in bases 4, 5, 6, 7, 8, and 9 I would suggest using the very versatile hundred chart. You can give instructions without even mentioning the concept of differing bases. For example, cross out every number on the hundred chart that has 7, 8, or 9 as one or more of its digits. Can you tell even before you get started how many numbers will get crossed out? (100 – 7²) What pattern do the cross-outs make? If you arrange the remaining numbers in order from smallest to largest, then you will have the first 49 numbers represented in base 7. With a minimal amount of cutting and taping you could have a “hundred” chart in base 7. Easy peasy.

This excel file not only has several puzzles, including today’s, but also a hundred chart and even a thousand chart because I know some of you might want to play with 3-digit numbers, too.

Now let me tell you a little bit about the number 810:

  • 810 is a composite number.
  • Prime factorization: 810 = 2 x 3 x 3 x 3 x 3 x 5, which can be written 810 = 2 x 3^4 x 5
  • The exponents in the prime factorization are 1, 4 and 1. Adding one to each and multiplying we get (1 + 1)(4 + 1)(1 + 1) = 2 x 5 x 2 = 20. Therefore 810 has exactly 20 factors.
  • Factors of 810: 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405, 810
  • Factor pairs: 810 = 1 x 810, 2 x 405, 3 x 270, 5 x 162, 6 x 135, 9 x 90, 10 x 81, 15 x 54, 18 x 45 or 27 x 30
  • Taking the factor pair with the largest square number factor, we get √810 = (√81)(√10) = 9√10 ≈ 28.4604989.

Since 810 has so many factors, it has MANY possible factor trees. If most people made a factor tree for 810, they would probably start with 81 × 10 or 9 x 90. NOT ME! Here are two less-often-used factor trees for 810:

Finally, here is an easy way to express 810 is in a different base:

  1. Make a cake in which you divide 810 by the base number repeatedly, keeping track of the remainders, including zero, as you go.
  2. Keep dividing until the number at the top of the cake is 0.
  3. List the remainders in order from top to bottom and indicate the base you used to do the division.
  4. This method is illustrated for BASE 2 and BASE 3 below:

That’s all pretty good work for a stone, a stick, and a pine cone!

By the way, using that method will also produce the following results:

  • 810 is 30222 BASE 4
  • 810 is 11220 BASE 5
  • 810 is 3430 BASE 6 and so forth.

 

 

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Print the puzzles or type the solution on this excel file: 10-factors 807-814

Normally I would tell you that 809 is a palindrome in two different bases:

  • 676 in BASE 11 because 6(121) + 7(11) + 6(1) = 809
  • 575 in BASE 12 because 5(144) + 7(12) +5(1) = 809

But whoop-de-doo, all that really means is that (x – 11) is a factor of 6x² + 7x – 803, and (x – 12) is a factor of 5x² + 7x – 804.

Isn’t it just as exciting that

  • (x – 2) is a factor of x^9 + x^8 + x^5 + x³ – 808 because 809 is 1100101001 in BASE 2?
  • (x – 3) is a factor of x^6 + 2x³ + 2x² +2x – 807 because 809 is 1002222 in BASE 3?
  • (x – 4) is a factor of 3x^4 + 2x² + 2x – 808 because 809 is 30221 in BASE 4?
  • (x – 5) is a factor of x^4 + x³ + 2x² + x – 805 because 809 is 11214 in BASE 5?

Notice that the last number in each of those polynomials is divisible by the BASE number.

Palindromes NEVER end in zero so the polynomials they produce will NEVER end in the original base 10 number.

So are palindromes really so special? Today I am much more excited that figuring out what a number is in another base can give us a factor of a corresponding polynomial!

How do I know what those polynomials are? Let me use 809 in BASE 6 as an example:

Since 809 is 3425 in BASE 6, I know that

  • 3(6³) + 4(6²) + 2(6¹) + 5(6º) = 809
  • 3(216) + 4(36) + 2(6) + 5(1) – 809 = 0
  • so 3(216) + 4(36) + 2(6) – 804 = 0
  • thus (x – 6) is a factor of 3x³ + 4x² + 2x – 804 because of the factor theorem.

If I told you what 809 is in Bases 7, 8, 9, and 10 would you be able to write the corresponding polynomials that are divisible by (x – 7), (x – 8), (x – 9), and (x – 10) respectively?

  • 2234 in BASE 7
  • 1451 in BASE 8
  • 1088 in BASE 9
  • 809 in BASE 10

Scroll down past 809’s factoring information to see if you found the correct polynomials.

—————–

  • 809 is a prime number.
  • Prime factorization: 809 is prime and cannot be factored.
  • The exponent of prime number 809 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 809 has exactly 2 factors.
  • Factors of 809: 1, 809
  • Factor pairs: 809 = 1 x 809
  • 809 has no square factors that allow its square root to be simplified. √809 ≈ 28.4429253066558.

How do we know that 809 is a prime number? If 809 were not a prime number, then it would be divisible by at least one prime number less than or equal to √809 ≈ 28.4. Since 809 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 809 is a prime number.

Here’s another way we know that 809 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 5² = 809 with 28 and 5 having no common prime factors, 809 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √809 ≈ 28.4. Since 809 is not divisible by 5, 13, or 17, we know that 809 is a prime number.

——————–

Were you able to find those polynomials from knowing what 809 is in other bases? Check your work with the answers below:

  • 2234 Base 7 tells us (x – 7) is a factor of 2x³ + 2x² + 3x – 805
  • 1451 Base 8 tells us (x – 8) is a factor of x³ + 4x² + 5x – 808
  • 1088 Base 9 tells us (x – 9) is a factor of x³ + 8x – 801
  • 809 Base 10 tells us (x – 10) is a factor of 8x² – 800

If you’ve made it this far, even if I’ve made you feel a little dizzy, you’ve done GREAT! Keep up the good work!

 

Happy birthday to my good friend, Justin! He seems to always remember the birthdays of everyone he knows, so this is how I am remembering his special day today. Justin is highly intelligent, thoughtful, and very friendly. I am confident he can solve this Level 6 puzzle that looks a little like a birthday cake.

Print the puzzles or type the solution on this excel file: 10-factors 807-814

This is my 808th post so I thought I would also make a factor cake for the number 808. It’s prime factor, 101, is at the top of the cake. Justin, I hope you live to be 101!

808 is a palindrome. That means it looks the same forwards and backwards. It is also a strobogrammatic number. That means it looks the same right side up or upside down.

ALL of the factors of 808 are also palindromes, and four of them are strobogrammatic numbers, too. Can you figure out which ones are both?

  • 808 is a composite number.
  • Prime factorization: 808 = 2 x 2 x 2 x 101, which can be written 808 = (2^3) x 101
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 808 has exactly 8 factors.
  • Factors of 808: 1, 2, 4, 8, 101, 202, 404, 808
  • Factor pairs: 808 = 1 x 808, 2 x 404, 4 x 202, or 8 x 101
  • Taking the factor pair with the largest square number factor, we get √808 = (√4)(√202) = 2√202 ≈ 428.425340807

Here are the factors that make puzzle #808 act like a multiplication table. It is followed by a table of logical steps to arrive at that solution.

807 and Level 1

What can I say about the number 807?

807 is palindrome 151 in BASE 26 because 1(26²) + 5(26) + 1(1) = 807.

Anything else? Well, I can figure out a few other things because 807’s has two prime factors, 3 and 269:

We can write ANY number (unless it’s a power of 2) as the sum of consecutive numbers in at least one way. 807 has three different ways to do that:

  • 403 + 404 = 807 because 807 isn’t divisible by 2.
  • 268 + 269 + 270 = 807 because it is divisible by 3.
  • 132 + 133 + 134 + 135 + 136 + 137 = 807 since it is divisible by 3 but not by 6.

I know that one of 807’s factors, 269, is a hypotenuse of a Pythagorean triple, so 807 is also. Thus. . .

  • (3·69)² + (3·260)² = (3·269)², or in other words, 207² + 780² = 807²

Since 807 has two odd sets of factor pairs, I know that 807 can be written as the difference of two squares two different ways:

  • 136² – 133² = 807
  • 404² – 403² = 807

I don’t usually do this, but today’s puzzle has something in common with 807. Can you tell what it is?

Print the puzzles or type the solution on this excel file: 10-factors 807-814

  • 807 is a composite number.
  • Prime factorization: 807 = 3 x 269
  • 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 807 has exactly 4 factors.
  • Factors of 807: 1, 3, 269, 807
  • Factor pairs: 807 = 1 x 807 or 3 x 269
  • 807 has no square factors that allow its square root to be simplified. √807 ≈ 28.4077454

 

 

 

 

 

When I put this post together I took a second look at today’s puzzle and thought, “That looks a little like a dunce cap.” That thought led me to two very interesting articles whose information surprised me greatly.

The first one titled “The Dunce Cap Wasn’t Always so Stupid” explains that long ago when the dunce cap was first introduced by the brilliant Scotsman John Duns Scotus, it became a symbol of exceptional intellect. In fact wizard hats were most likely modeled after them. Unfortunately, this positive perception of the caps remained for only a couple of centuries.

The second article is short but helped me visualize Topology’s Dunce Hat. I enjoyed watching the animation of this mathematical concept.

I hope you will enjoy trying to solve the puzzle. It is a Level 6, so it won’t be easy. If you succeed, you’ll deserve to feel that you have exceptional intellect.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Now here is some information about the number 806:

  • 806 is a composite number.
  • Prime factorization: 806 = 2 x 13 x 31
  • 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 x 2 x 2 = 8. Therefore 806 has exactly 8 factors.
  • Factors of 806: 1, 2, 13, 26, 31, 62, 403, 806
  • Factor pairs: 806 = 1 x 806, 2 x 403, 13 x 62, or 26 x 31
  • 806 has no square factors that allow its square root to be simplified. √806 ≈ 28.390139.

806 is a palindrome in three different bases:

  • 11211 BASE 5 because 1(625) + 1(125) + 2(25) + 1(5) + 1(1) = 806
  • 1C1 BASE 23 (C is 12 base 10) because 1(23²) + 12(23) + 1(1) = 806
  • QQ BASE 30 (Q is 26 base 10) because 26(30) + 26(1) = 806, which follows naturally from the fact that 26 × 31 = 806

806 is the hypotenuse of Pythagorean triple 310-744-806 which is 5-12-13 times 62.

And 806 can be written as the sum of three squares seven different ways:

  • 26² + 11² + 3² = 806
  • 26² + 9² + 7² = 806
  • 25² + 10² + 9² = 806
  • 23² + 14² + 9² = 806
  • 21² + 19² + 2² = 806
  • 21² + 14² + 13² = 806
  • 19² + 18² + 11² = 806

 

805 and Level 4

23 × 35 = 805 so we shouldn’t be surprised that 805 is palindrome NN in BASE 34. N is the same as 23 in base 10. Thus NN can be derived from 23(34) + 23(1) = 23(34 + 1) = 23 × 35 = 805. NN obviously is divisible by 11 like all 2 digit palindromes are.

Since 23 = 22 + 1, should we expect that 805 is a palindrome in BASE 22? No, and that is for the same reason that not all multiples of 11 are palindromes.

Finding the factors in today’s puzzle shouldn’t be very difficult, but the last few might be trickier than the rest:

Print the puzzles or type the solution on this excel file: 10-factors 801-806

  • 805 is a composite number.
  • Prime factorization: 805 = 5 x 7 x 23
  • 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 x 2 x 2 = 8. Therefore 805 has exactly 8 factors.
  • Factors of 805: 1, 5, 7, 23, 35, 115, 161, 805
  • Factor pairs: 805 = 1 x 805, 5 x 161, 7 x 115, or 23 x 35
  • 805 has no square factors that allow its square root to be simplified. √805 ≈ 28.37252

805 is the hypotenuse of a Pythagorean triple:

  • 483-644-805, which is 3-4-5 times 161

805 can be written as the sum of three squares four ways:

  • 25² + 12² + 6² = 805
  • 24² + 15² + 2² = 805
  • 20² + 18² + 9² = 805
  • 18² + 16² + 15² = 805

 

 

 

Factor Rainbows can be a wonderful way to display the factors of a number. Not only are all the factors listed in order from smallest to greatest, but the factor pairs are joined together with the same color band.

The number 804 has 12 factors so it makes a lovely rainbow with 6 different color bands.

Is there a pot of gold at the end of this factor rainbow? I’ll let you decide the answer to that question.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Finding golden nuggets of information about a number might be less difficult than finding pots of gold.

I always begin the painstaking mining process by looking at the factors of the number:

  • 804 is a composite number.
  • Prime factorization: 804 = 2 x 2 x 3 x 67, which can be written 804 = (2^2) x 3 x 67
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 804 has exactly 12 factors.
  • Factors of 804: 1, 2, 3, 4, 6, 12, 67, 134, 201, 268, 402, 804
  • Factor pairs: 804 = 1 x 804, 2 x 402, 3 x 268, 4 x 201, 6 x 134, or 12 x 67
  • Taking the factor pair with the largest square number factor, we get √804 = (√4)(√201) = 2√201 ≈ 28.3548937575

About half of everything there was already in the factor rainbow.

Finding nuggets of information about the number 804 has been a little difficult and disappointing:

  • None of 804’s prime factors can be written as 4N+1, so 804 is NOT the hypotenuse of any Pythagorean triples.
  • 804 is NOT a palindrome in base 36 or any base less than that.
  • 804 is NOT the sum of any consecutive prime numbers.

Even though I did not find any golden nuggets in those places, I kept looking and finally found a couple of gems about the number 804:

804 can be written as the sum of three squares four different ways, and all of those ways have some definition of double in them:

  • 28² + 4² + 2² = 804
  • 26² + 8² + 8² = 804
  • 22² + 16² + 8² = 804
  • 20² + 20² + 2² = 804

Stetson.edu also gives us a nugget about the number 804 that may be a bit too heavy for most people to handle: “804 is a value of n for which 2φ(n) = φ(n+1).” That basically means that there are exactly half as many numbers less than 804 that are NOT divisible by its prime factors (2, 3, or 67) as there are numbers less than 805 that are NOT divisible by its prime factors (5, 7, or 23).

I started looking for golden specs about 804 in places that I don’t usually look.

267 + 268 + 269 = 804 so 804 is the sum of 3 consecutive numbers.

As stated before 804 is never the hypotenuse of a Pythagorean triple. However to find all the times it is a leg in a triple will require a lot of labor especially since 804 has so many factors, including 4, and two of its factor pairs have factors where both factors are even.

  • 134 × 6 is an even factor pair, so (134 + 6)/2 = 70, and (134-6)/2 = 64. Thus 804 = 134·6 = (70 + 64)(70 – 64) = 70² – 64² .
  • 402 × 2 is another even factor pair, so (402 + 2)/2 = 202, and (402 – 2)/2 = 200. Thus 804 = 402·2 = (200 + 2)(200 – 2) = 202² – 200²
  • Likewise odd or even sets of factor pairs of any of 804’s factors can also be used to find Pythagorean triples.

So to find all Pythagorean triples that contain the number 804, we will have to find all the times 804 satisfies one of these FOUR conditions:

  1. 804 = 2k(a)(b) so that 804 is in the triple 2k(a)(b), k(a² – b²), k(a + b²) OR the triple k(a² – b²), 2k(a)(b), k(a + b²).
  2. 804 = 2(a)(b) so that 804 is in the triple 2(a)(b), a² – b², a + b² OR the triple a² – b², 2(a)(b), a + b².
  3. 804 = a² – b² so that 804 is in the triple a² – b², 2(a)(b), a + b² OR the triple 2(a)(b), a² – b², a + b².
  4. 804 = k(a² – b²) so that 804 is in the triple k(a² – b²), 2k(a)(b), k(a + b²) OR the triple 2k(a)(b), k(a² – b²), k(a + b²).

Let the mining process begin! I’ll list the triples with the shortest legs first and color code each triple according to the condition I used.

  • 335-804-871 which used 804 = 2·67(3)(2) to make a triple that is 5-12-13 times 67
  • 603-804-1005 which used 804 = 2·201(2)(1) to make a triple that is 3-4-5 times 201
  • 804-1072-1340 which used 804 = 268(2² – 1²) to make a triple that  is 3-4-5 times 268
  • 804-2345-2479 which used 804 = 2·67(6)(1) to make a triple that is 12-35-37 times 67
  • 804-4453-4525, which used 804 = 2(6)(67)
  • 804-8960-8996, which used 804 = 70² – 64² or 804 = 4(35² – 32²) to make a triple that is 201-2240-2249 times
  • 804-17947-17965, which used 804 = 2(134)(3)
  • 804-26928-26940 which used 804 = 12(34² – 33²) to make a triple that is 67-2244-2245 times 12
  • 804-40397-40405, which used 804 = 2(201)(2)
  • 804-53865-53871 which used 804 = 2·3(134)(1) to make a triple that is 268-17955-17957 times 3
  • 804-80800-80804 which used 804 = 202² – 200²  or 804 = 4(101² – 100²) to make a triple that is 201-20200-20201 times 4
  • 804-161603-161605, a primitive Pythagorean triple, that used 804 = 2(402)(1)

If you look for a pot of gold at the end of a rainbow, you’re bound to be disappointed. Science/How Stuff Works just had to crush dreams and dispel 10 Myths About Rainbows. Unfortunately a pot of gold being at the rainbow’s end is included on that list. Still I suppose we could still put every golden spec or nugget about 804 into a little pot and call it a pot of gold.

Or if you are as clever and quick as a leprechaun, perhaps you will consider finding Pythagorean triples to be like finding pots of gold.

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