My brother, Doug, recently visited me. He told me about his goal to get 11,000 steps every day. The American Heart Association recommends 10,000 steps a day. Is it worth trying to get a thousand steps more than the recommended number?

My brother shared the cool mathematics of an 11,000 daily step goal with me, and now I want to share it with you:

Now I think 11,000 steps a day is a very worthy goal! It helps me see the big picture of 1,000,000 steps each quarter and 4,000,000 steps each year and that will help me be more likely to meet the 11,000 step goal EVERY day.

It will take you far fewer than 11,000 steps to complete this multiplication table puzzle. It isn’t the most difficult puzzle I make, but it can still be a challenge:

Print the puzzles or type the solution on this excel file: 12 factors 815-820

Now here is a little bit about the number 818:

818 is a composite number.

Prime factorization: 818 = 2 x 409

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 818 has exactly 4 factors.

Factors of 818: 1, 2, 409, 818

Factor pairs: 818 = 1 x 818 or 2 x 409

818 has no square factors that allow its square root to be simplified. √818 ≈ 600699

818 looks the same right side up or upside down so we call it is a strobogrammatic number.

23² + 17² = 818

Finally, 818 can be found in these two Pythagorean triple equations:

Start at the top row of this level three puzzle and work down one cell at a time until you’ve written each number from 1 through 12 in both the top row and the first column. You will have solved the puzzle if all the clues given in the puzzle are the products of the numbers you wrote. You can do this!

Print the puzzles or type the solution on this excel file: 12 factors 815-820

817 is a composite number.

Prime factorization: 817 = 19 x 43

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 817 has exactly 4 factors.

Factors of 817: 1, 19, 43, 817

Factor pairs: 817 = 1 x 817 or 19 x 43

817 has no square factors that allow its square root to be simplified. √817 ≈ 28.58321

817 is the sum of three consecutive prime numbers:

269 + 271 + 277 = 817

Because it has two factor pairs in which the numbers in the pair are either both odd or both even, 817 can be written as the difference of two squares two different ways:

817 = 31² – 12², which I’ve illustrated below.

817 = 409² – 408²

Being able to be written as the difference of two squares means that 817 is a leg in two Pythagorean triples so:

744² + 817² = 1105²

817² + 333744² = 333745²

Last, but certainly not least, 817 is the difference of two consecutive cubes, namely 17³ – 16³ = 817. That means that 817 is the 17th centered hexagonal number as well! There are 17 squares on each side of the figure below and 817 tiny purple squares in all. The horizontal row in the middle has 2(17) – 1 = 33 tiny purple squares.

That’s from a³ – b³ = (a – b)(a² + ab + b²) and is illustrated below using 2D cubes instead of squares. In 3D, when the sides are folded up, the darker cubes can be claimed by two different sides so the figure will look like a 17 x 17 x 17 inside corner such as in a room where three perpendicular lines meet, but in 2D it’s just the flat net you see here:

Eight is half of sixteen, so 816 is divisible by 6. You probably weren’t expecting that divisibility rule, but it’s true.

816 can also be easily divided by 2, 4, and 8. How many factors does 816 have in all? Plenty! Scroll down past the puzzle and see!

Print the puzzles or type the solution on this excel file: 12 factors 815-820

816 is a composite number.

Prime factorization: 816 = 2 x 2 x 2 x 2 x 3 x 17, which can be written 816 = 2⁴ x 3 x 17

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 x 2 x 2 = 20. Therefore 816 has exactly 20 factors.

Since multi-digit 815 ends with 5, it is a composite number, and it is also the hypotenuse of a Pythagorean triple:

489-652-815 which is 163 times 3-4-5.

Can you write the numbers 1 – 12 in both the first column and the top row so that this puzzle functions like a multiplication table?

Print the puzzles or type the solution on this excel file: 12 factors 815-820

815 is a composite number.

Prime factorization: 815 = 5 x 163

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 815 has exactly 4 factors.

Factors of 815: 1, 5, 163, 815

Factor pairs: 815 = 1 x 815 or 5 x 163

815 has no square factors that allow its square root to be simplified. √815 ≈ 5482048.

814 is even, so it is divisible by 2. What else is it divisible by? Let’s apply 11’s divisibility rule:

8 – 1 + 4 = 11, so 814 can also be evenly divided by 11. (If the sum of every other digit minus the sum of the missed digits is divisible by 11, then the number is divisible by 11.)

This Level 6 Puzzle is a doozie, but I assure you that it CAN BE be solved using logic without guessing and checking!

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

Now here’s a little bit more about the number 814:

814 is a composite number.

Prime factorization: 814 = 2 x 11 x 37

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 814 has exactly 8 factors.

Factors of 814: 1, 2, 11, 22, 37, 74, 407, 814

Factor pairs: 814 = 1 x 814, 2 x 407, 11 x 74, or 22 x 37

814 has no square factors that allow its square root to be simplified. √814 ≈ 28.530685

Because 37 is one of its factors, 814 is the hypotenuse of a Pythagorean triple:

264-770-814 which is 22 times (12-35-37)

814 is also repddigit MM in BASE 36 (M is 22 base 10) because 22(36) + 22(1) = 22(36 + 1) = 22 × 37 = 814

Here are some initial logical notes to solve Find the Factors Puzzle #814:

10, 8, 4, and 6 must use all the 1’s and 2’s, so clue 4 cannot use both 2’s and must use 1 & 4.

The 8’s must be used by 72 and one of the 40’s. That 40 will also use a 5.

63 & 72 will use both 9’s so 18 must use a 3, as does 21.

That means 30 can’t be 3 x 10 and must use a 5.

Thus a 40 and 30 will use both 5’s so clue 10 can’t use a 5 and must use 1 & 10

In case there is still any confusion, here’s where the factors went:

Here she is sitting with my husband, me, and her big sister. My husband and I are pretty new to her so she’s probably thinking in Mandarin, “Who are these people?” On the other hand, we are delighted to be a part of her life now.

Here is today’s puzzle:

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

Here’s a little about the number 813:

813 is a composite number.

Prime factorization: 813 = 3 x 271

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 813 has exactly 4 factors.

Factors of 813: 1, 3, 271, 813

Factor pairs: 813 = 1 x 813 or 3 x 271

813 has no square factors that allow its square root to be simplified. √813 ≈ 51315.

813 is repdigit 111 in BASE 28 because 28² + 28¹ + 28º = 813

Stetson.edu and my calculator informed me that 813^e = 81,366,615.06223032 . . .

Here’s a puzzle for you. How many total triangles are there in the figure below?

Okay, if you guessed 812 because this is my 812th post, you would be right. But what if you were asked that question in some real world situation where accessing the internet to get the answer isn’t permitted. How would you know the answer then?

The total number of triangles = ⌊n(n+2)(2n+1)/8⌋ where the brackets mean round decimals DOWN to the closest integer.

Here n = 14, so the number of triangles is 14×16×29/8 = 812. Rounding down wasn’t necessary since the product of two consecutive even numbers is always divisible by 8.

Still, you probably wouldn’t remember that formula unless you had seen it VERY recently or you have a photographic memory.

You could actually COUNT all the triangles. In the post about 658 total triangles, I noted that the 13 rows of small triangles formed a total of 169 of the smallest triangles, but I suggested that it would be easier to ignore that nice square number and instead count the number of triangles pointing UP separately from the number pointing DOWN. You will add up a lot of triangular numbers as you sum up the number of them pointing up and again as you sum up the number pointing down.

Making a chart of the number of triangles pointing UP would be easy. It’s just a list of triangular numbers in order. However, the chart for the ones pointing down might be confusing because you don’t use all of the triangular numbers, and the ones you use will be different for an even number of rows than for an odd number of rows. For example,

the pointing DOWN portion of the chart for 14 rows of triangles below uses these seven triangular numbers: 1, 6, 15, 28, 45, 66, and 91,

while the pointing DOWN chart for 13 rows of triangles uses six different triangular numbers: 3, 10, 21, 36, 55, and 78.

Here is a chart listing the number of triangles of any size that are contained in a triangular figure made with 14 rows of small triangles. Interesting note: Because 14 + 2 = 16, a multiple of 8, the total number of triangles in this case will be divisible by 14, the number of line segments on each side.

Making such a chart works. However, remembering which triangular numbers to use and how many you should use, especially when counting the odd number of rows of triangles pointing DOWN, might be difficult.

Today I was thinking about triangular numbers and their relationship to square numbers:

The sum of two consecutive triangular numbers always makes a square number. On the chart above, I’ve paired up certain consecutive triangular numbers by coloring them the same color. The sizes of the triangles being paired together by like colors are not the same size (with one exception), but their sums are nevertheless square numbers every time.

Thus, we also can come up with a single list of numbers to add to arrive at the total number of triangles as illustrated below. This has made me changed my mind about using those square numbers to help count! Notice how every other number on each list below, including the last number to be added, is a square number.

Of the methods discussed in this post, this one that includes square numbers might be the easiest one to remember. Here are the steps I used to find the total number of triangles:

Count the rows and determine if that number is even or odd.

Write the triangular numbers in order until the amount of numbers written equals the number of rows.

If the number of rows is even, replace the 2nd, 4th, 6th, etc. entries with corresponding square numbers: 4, 16, 36, etc.

If the number of rows is odd, replace the 1st, 3rd, 5th, etc entries with corresponding square numbers: 1, 9, 25, etc.

Add all the numbers remaining in the list. The sum will be the total number of triangles for that many rows of small triangles.

Okay, that puzzle was rather difficult and took a while to explain. You may find this Level 4 puzzle easier:

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

Here’s a little more about the number 812:

812 is a composite number.

Prime factorization: 812 = 2 x 2 x 7 x 29, which can be written 812 = 2² x 7 x 29

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 812 has exactly 12 factors.

My youngest grandchild is 7 months old and is just beginning to crawl. I love the way he wipes drool off his own face in this video. It’s pretty funny.

Here’s today’s puzzle:

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

811 is a prime number.

Prime factorization: 811 is prime.

The exponent of prime number 811 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 811 has exactly 2 factors.

Factors of 811: 1, 811

Factor pairs: 811 = 1 x 811

811 has no square factors that allow its square root to be simplified. √811 ≈ 28.4780617

How do we know that 811 is a prime number? If 811 were not a prime number, then it would be divisible by at least one prime number less than or equal to √811 ≈ 28.5. Since 811 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 811 is a prime number.

811 is the second half of a twin prime, and it is also the sum of five consecutive prime numbers:

151 + 157 + 163 + 167 + 173 = 811.

Stetson.edu informs us that the smallest prime factor of 24!+1 is 811. Wow! Really?

If you type 24!+1 into your computer calculator you get 620,448,401,733,239,439,360,001.

If you divide that number by 811, you get 765,041,185,860,961,084,291.

If you type that into Number Empire’s Prime Number Checker, you will see that it is prime.

Thus 811 is the smaller of 24!+1’s two prime factors!

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 × 3⁴ × 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.

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:

Make a cake in which you divide 810 by the base number repeatedly, keeping track of the remainders, including zero, as you go.

Keep dividing until the number at the top of the cake is 0.

List the remainders in order from top to bottom and indicate the base you used to do the division.

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.

And just so you’ll know, 810 is the sum of consecutive primes 401 and 409.

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 ⁰¹²³⁴⁵⁶⁷⁸⁹

(x – 2) is a factor of x⁹+ x⁸ + x⁵ + x³ – 808 because 809 is 1100101001 in BASE 2?

(x – 3) is a factor of x⁶ + 2x³ + 2x² +2x – 807 because 809 is 1002222 in BASE 3?

(x – 4) is a factor of 3x⁴ + 2x² + 2x – 808 because 809 is 30221 in BASE 4?

(x – 5) is a factor of x⁴ + 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.

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!