Mathematics

What can you see in the number 585?

/ ivasallay / 1 Comment

This week I watched an excellent video titled 5 x 9 is more than 45. Indeed 45 is so much more than simply 5 x 9. Every multiplication fact is much more than that mere fact, but Steve Wyborney used 5 x 9 = 45 in his video… Guess what! 585 is a multiple of 45.

As I thought about the number 585, I marveled at some of the hidden mysteries this number holds.

Since 585 is divisible by two different centered square numbers, 5 and 13, I saw that 585 could be represented by this lovely array that has 45 larger squares made up of 13 smaller colorful squares. When you look at the array, do you just see 585 squares or can you see even more multiplication and division facts? If you rotate the array 90 degrees, do the facts change?

585 Squares-1

What do you see in this array of 117 medium sized squares made up of 5 smaller squares:

585 Squares-2

Or this more simple looking array of sixty-five 3 x 3 squares.

585 Squares-3

All of these arrays are in just two dimensions. A 5 x 9 x 13 rectangular prism is ONE way to represent 585 in three dimensions.

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Since factors 5, 13, and 65 are hypotenuses of primitive Pythagorean triples, 585 is the hypotenuse of four Pythagorean triples. Each triple has a different greatest common factor. Can you figure out what each one is?

  • 144-567-585
  • 225-540-585
  • 297-504-585
  • 351-468-585

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And if all of that wasn’t enough, OEIS.org informs us that 585 is a palindrome in 3 different bases!

  • 585 = 1001001001 in base 2
  • 585 = 1111 in base 8
  • 585 = 585 in base 10

Here are two different methods of determining 585 in base 8. In both methods the base 8 representation is in blue.

585 base 8

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  • 585 is a composite number.
  • Prime factorization: 585 = 3 x 3 x 5 x 13, which can be written 585 = (3^2) x 5 x 13
  • 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 585 has exactly 12 factors.
  • Factors of 585: 1, 3, 5, 9, 13, 15, 39, 45, 65, 117, 195, 585
  • Factor pairs: 585 = 1 x 585, 3 x 195, 5 x 117, 9 x 65, 13 x 45, or 15 x 39
  • Taking the factor pair with the largest square number factor, we get √585 = (√9)(√65) = 3√65 ≈ 24.18677

578 When I stopped teaching about thousands, it made a BIG difference

/ ivasallay / 6 Comments

Several weeks ago I helped a young girl with a place value page in her workbook. The largest number she needed to write out in words was less than 900 million.

The young girl was very confused so I explained it exactly the same way the book explained it – which was exactly the same way it was explained to me when I was a kid.

She easily understood the ones place, the tens place, the hundreds place, and the thousands place.  The trouble began when we started working with the ten thousands place and became even worse when I started talking about the hundred thousands place, the millions place, the ten millions place, and the hundred millions place.

The young girl could easily and correctly separate the digits of any multi-digit number into groups of three, but reading or writing that number using words baffled her.

Also translating number words back into digits and putting those digits into the right places was equally challenging for her.

After explaining what to do on EVERY single problem, I knew she really didn’t get it….even though the assignment was finally finished.

Then last week on twitter I saw a retweet of this:

I wondered what crazy, radical thing was meant by that statement so I clicked on the link and read a very clear and easy-to-understand explanation of how to teach place value. I was very impressed.

This last Friday I saw that young girl again, and I said, “I want to show you something.”

I took my red pen and wrote in her workbook,

place value places

and this time I actually taught her the concept of place value. This time she got it!

Triumphantly she wrote down a 14-digit number made from some “random” numbers that popped into her head and then read it to me perfectly. It was as thrilling for me as it was for her!

An hour before she couldn’t correctly read 90% of the whole numbers less than a million, but now she had MASTERED thousands, millions, billions, and even trillions.

I really like that blogger/tweeter Michael Tidd chose “units” as the last category because that is the natural place to say miles, meters, dollars or whatever the unit happens to be. The unit this young girl chose for her 14-digit number was CATS.

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578 is the hypotenuse of two Pythagorean triples: 322-480-578 and 272-510-578. Each triple has a greatest common factor. Which factors of 578 could they be?

  • 578 is a composite number.
  • Prime factorization: 578 = 2 x 17 x 17, which can be written 578 = 2 x (17^2)
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3  = 6. Therefore 578 has exactly 6 factors.
  • Factors of 578: 1, 2, 17, 34, 289, 578
  • Factor pairs: 578 = 1 x 578, 2 x 289, or 17 x 34
  • Taking the factor pair with the largest square number factor, we get √578 = (√289)(√2) = 17√2 ≈ 24.04163

578 cake

2 x 17 x 17 = 578

561 gives a false positive to these 102 prime number tests

/ ivasallay / 2 Comments

We can use some easy divisibility tests to find two of the factors of 561.

  • 5 + 6 + 1 = 12, a multiple of 3 so 561 is divisible by 3.
  • 5 – 6 + 1 = 0, which is divisible by 11 so 561 can be evenly divided by 11.

But if we only do those divisibility tests, we will miss something very significant about the number 561:

If you divide 2^561 by 561, the remainder will be 2. When we are more interested in the remainder than the quotient, we can simply type “2, x^y, 561, Mod, 561, =” into the computer’s scientific calculator:

561 Mod Calculator

This is only a picture of a calculator.

2^561 (mod 561) = 2 means that 561 is VERY LIKELY a prime number, but this is one time when VERY LIKELY does not mean ACTUALLY!

561 has something in common with the number 341. Yes, both of them pass this quick prime number test, and both of them are composite numbers divisible by 11. Both numbers are called pseudo-prime numbers. (341 and 561 are the two smallest composite numbers to give a false positive to this particular test.)

561 is even more remarkable than 341:

  • 2^561 (mod 561) = 2, and 2^341 (mod 341) = 2 (Both numbers pass.)
  • 3^561 (mod 561) = 3, while 3^341 (mod 341) = 168 (561 passes; 341 fails.)
  • 5^561 (mod 561) = 5
  • 7^561 (mod 561) = 7
  • 11^561 (mod 561) = 11
  • 13^561 (mod 561) = 13
  • 17^561 (mod 561) = 17
  • etc.

There are 102 prime numbers less than 561, and p^561 (mod 561) = p for every single one of them! 561 acts like a prime number in those 102 ways.

In 1910 R. D. Carmichael discovered that 561 is the first COMPOSITE number that passes ALL those modular (remainder) prime number tests, so 561 is the first Carmichael number. Yes, there will be more – in fact, infinitely more.

R. D. Carmichael actually found that 561 passes ALL 559 prime number tests using each whole number between 1 and 561, for example 33^561 (mod 561) = 33. All prime numbers can make a similar claim, but 561 is the smallest composite number with that property.

(Note: I did not use the standard mathematical notation for this property, but what I used is equivalent to it and doesn’t require parenthesis when typing it into the computer’s scientific calculator. Also I think “=” is less intimidating looking for some of my readers than “≡”.)

There are other reasons why the number 561 is an interesting number:

Because 33 x 34/2 = 561, we know that 561 is the 33rd triangular number and is equal to 1 + 2 + 3 + . . . + 31 + 32 + 33, the sum of the first 33 whole numbers.

Because 17 x (2 x 17 – 1) = 561, we know that 561 is the 17th hexagonal number. (All hexagonal numbers are also triangular numbers.)

561 is also the hypotenuse of the Pythagorean triple 264-495-561. What is the greatest common factor of those three numbers? Hint: it is one of the factors of 561 listed below:

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  • 561 is a composite number.
  • Prime factorization: 561 = 3 x 11 x 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 x 2 x 2 = 8. Therefore 561 has exactly 8 factors.
  • Factors of 561: 1, 3, 11, 17, 33, 51, 187, 561
  • Factor pairs: 561 = 1 x 561, 3 x 187, 11 x 51, or 17 x 33
  • 561 has no square factors that allow its square root to be simplified. √561 ≈ 23.6854

Is 529 an Interesting Number?

/ ivasallay / 4 Comments
  • 529 is a composite number.
  • Prime factorization: 529 = 23^2
  • The exponent in the prime factorization is 2. Adding one we get (2 + 1) = 3. Therefore 529 has exactly 3 factors.
  • Factors of 529: 1, 23, 529
  • Factor pairs: 529 = 1 x 529 or 23 x 23
  • 529 is a perfect square. √529 = 23

In the United States you can save money for college in a tax-free 529 College Savings Plan. If you use a 529 Plan, when you are 23, you could be making a good, square income. That is how I remember that 529 = 23².

529 dots can be made into a square, but is 529 an interesting number?

23 is the 9th prime number, and 529 is the 9th number with exactly 3 factors. Is that interesting?

What about the fact that 529 is palindrome 121 in BASE 22 because 1(22²) + 2(22) + 1(1) = 529?

Or that 529 can be written as the sum of three squares three ways and each of those ways contain exactly TWO elements of the set {18², 6², 3²} plus one other square number?

  • 22² + 6² + 3² = 529
  • 18² + 14² + 3² = 529
  • 18² + 13² + 6² = 529

Wikipedia informs us that 529 is the 12th centered octagonal number because  4(12²) – 4(12) + 1 = 529.

Do any of those reasons make 529 an interesting number? Is ANY number interesting?

Recently when I wrote that 526 is a centered pentagonal number, Steve Morris of Blog, Blogger, Bloggest commented:

It’s intriguing how so many numbers have interesting or special properties. I used to think that there was something magical about this, that these patterns were somehow telling us something deep about the universe. Like, why is 3.14159… the value it is?

Nowadays I think that these special properties are things we invented. For instance, a centred pentagonal number is interesting if you think it is, and isn’t if you don’t think so. Some numbers have particularly important or curious properties (prime numbers for instance), but they are only interesting because we think they are!

What do you think?

I think that much of mathematics was discovered by astronomers, physicists, and even philosophers, and the patterns in mathematics do tell us something deep about the universe, but all of those mathematical properties existed before they were discovered. They are eternal principles that we each understand to the best of our abilities.

Are numbers with certain properties interesting only because some people think they are?

If the world never knew that pi is approximately equal to 3.14159…, we would still be as advanced as we are now. Pi seems like a very important number, but what if the world had never heard of pi? What if for thousands of years the world had instead used tau (τ ) which is approximately equal to 6.28318? Every important, magical discovery related to pi would still be known. Since τ = 2π, some people think we should all celebrate on June 28 by eating twice as much pie. Those people are really saying that tau is an interesting number only because pi is interesting. If we had never heard of pi, we wouldn’t eat pie to celebrate either day. As it is, most people have never heard of τ and would consider it to be a rather boring number.

Are any numbers inherently interesting?

Ancient mathematicians were fascinated that a series of dots could be made into different shapes. Sometimes a certain number of dots could be made into a triangle, or a square, or even a pentagon. A string of dots that couldn’t be made into a 2-dimensional rectangle represented a prime number. The fact that the ancients were interested in the shape of numbers makes them even more interesting to me.

It doesn’t matter what language you use or even what number base you use, 529 ♦’s can be arranged into a perfectly formed square just as ♦ ♦ ♦ ♦ can be. Most people can understand square numbers or even cubed numbers. However, for some people triangular numbers and pentagonal numbers are just mind-boggling, and they won’t consider such numbers to be interesting at all.

I don’t know if the ancients were aware of CENTERED pentagonal numbers, but I am fascinated by the fact that this is a shape that CAN consist of a prime number of dots (31, 181, 331, for example.)

My husband often says that nothing is boring. Two different people can listen to the same talk. One of the listeners might be moved to tears or inspired to action while the other person is bored out of his mind. It isn’t the subject matter that decides how people will react, it’s the people themselves.

Mathematician G. H. Hardy is an inspiration to me. He said, “Nothing I have ever done is of the slightest practical use.” He studied mathematical topics that interested him without regard to their usefulness. How could he potentially waste Ramanujan’s great mind on something as useless as partition theory. How did he convince Cambridge University to pay him while he obsessively explored this unimportant topic especially when he boasted that it didn’t have the slightest practical use? I don’t know how he did that, but it turns out that partition theory was not simply recreational mathematics; it does have a practical purpose! From G. H. Hardy I have learned that it is okay to explore topics that interest me even if they NEVER have a practical purpose. I haven’t figured out how to get someone to pay me to explore any unimportant topics, but I can still explore them to my heart’s content:

I often notice if the square root of a number can be reduced or not. I did not invent the idea of reducing square roots, but I’ve decided it is an interesting topic. I don’t know if anybody else looks for consecutive reducible square roots or calculates what percentage of numbers have reducible square roots, but I find it interesting, and I explore it. If you google reducible square roots, chances are that much of what you see will have been created by me.

Some mathematicians spend time trying to prove the Riemann hypothesis, but that topic doesn’t interest me as much as other topics because that is what I’ve decided.

What do you think? What makes a number interesting or not? What makes a mathematical topic interesting or not?

Square Roots Up to √513 That Can Be Simplified

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5 + 1 + 3 = 9 so 513 can be evenly divided by 9, and thus its square root can be simplified.

513 is the 200th counting number whose square root can be reduced. 200/513 ≈ .38986, which means, so far, 38.97% of the counting numbers have reducible square roots.

Here are the first 100 reducible square roots followed by the second hundred:

1st 100 reducible square roots

I highlighted the ones that are part of three or more consecutive reducible square roots.

2nd 100 reducible square roots

  • 513 is a composite number.
  • Prime factorization: 513 = 3 x 3 x 3 x 19, which can be written 513 = (3^3) x 19
  • 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 513 has exactly 8 factors.
  • Factors of 513: 1, 3, 9, 19, 27, 57, 171, 513
  • Factor pairs: 513 = 1 x 513, 3 x 171, 9 x 57, or 19 x 27
  • Taking the factor pair with the largest square number factor, we get √513 = (√9)(√57) = 3√57 ≈ 22.6495033

500 Pick Your Pony! Who’ll Win This Number of Factors Horse Race?

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Today I factor the number 500. How many factors does it have? Each number between 401 and 500 has at least 2 factors, but no more than 24 factors.

What if we had a horse race between the number of factors? Click on the graphic below to see a gif of the numbers racing against each other. Before you click, pick your pony. Will 2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, or 24 be the number of factors of more integers between 401 and 500 than any other number? Click on the graphic to find out!

500 Horse Race

Did you see the lead change a couple of times? How did your pony do? Which pony will you choose in the 501 to 600 race?

Remarkably, only 37 of these one hundred numbers have reducible square roots. That’s only 37%, which is significantly lower than in the 40% or 39% of previous hundreds as this graphic illustrates:

500 Reducible vs. Non-Reducible

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  • 500 is a composite number.
  • Prime factorization: 500 = 2 x 2 x 5 x 5 x 5, which can be written 500 = (2^2) x (5^3)
  • The exponents in the prime factorization are 2 and 3. Adding one to each and multiplying we get (2 + 1)(3 + 1) = 3 x 4 = 12. Therefore 500 has exactly 12 factors.
  • Factors of 500: 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500
  • Factor pairs: 500 = 1 x 500, 2 x 250, 4 x 125, 5 x 100, 10 x 50, or 20 x 25
  • Taking the factor pair with the largest square number factor, we get √500 = (√100)(√5) = 10√5 ≈ 22.36067977

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If you didn’t want to click, you can still see the horse race below, but the numbers from 401 to 500 will be much clearer if you click.

Factors of Numbers from 401 to 500 Horse Race

make animated gifs like this at MakeAGif

470 Greatest Common Factors of Pythagorean Triples.

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  • 470 is a composite number.
  • Prime factorization: 470 = 2 x 5 x 47
  • 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 470 has exactly 8 factors.
  • Factors of 470: 1, 2, 5, 10, 47, 94, 235, 470
  • Factor pairs: 470 = 1 x 470, 2 x 235, 5 x 94, or 10 x 47
  • 470 has no square factors that allow its square root to be simplified. √470 ≈ 21.67948

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470 is the hypotenuse of the non-primitive Pythagorean triple 282-376-470. What is the greatest common factor of those three numbers?

The greatest common factor will always be a factor of the smallest of the three numbers, but it will also be a factor of the smallest difference between the three numbers as well. Let’s find those differences. Note: the difference between the 282 and 470 will not be the smallest difference so there is no need to find that one. We only need to find these two differences:

470 difference

In the case of this Pythagorean triple the differences are equal to each other which means that the difference, 94*, is also the greatest common factor of the three numbers! Go ahead and try dividing each number in the triple by 94. You will discover that this Pythagorean triple is just 3-4-5 multiplied by 94.

*This statement is only true of Pythagorean triples. For example the following numbers also have differences of 94, but the greatest common factor is not 94, but a factor of 94:

  1. The greatest common factor of 283-377-471 is 1.
  2. The greatest common factor of 284-378-472 is 2
  3. The greatest common factor of 329-423-517 is 47

Mathchat has written an excellent post on finding the greatest common factor of three or more numbers that can be used for all integers in general.

But as far as Pythagorean triples are concerned, anytime the corresponding differences of a Pythagorean triple are equal to each other, then that Pythagorean triple is just 3-4-5 multiplied by the difference. There are an infinite number of such triples, and 282-376-470 is just one of them.

3-4-5 Pythagorean Triple Sequence

Now remember there is an infinite number of primitive Pythagorean triples, and every one of those triples can be multiplied by each of the infinitely many counting numbers. A graphic like the one above could be made for every primitive triple followed by each of its multiples. For example 5-12-13, 10-24-26, 15- 36-39, etc. would be another infinite series of Pythagorean triples.

You could say the total number of Pythagorean triples equals infinity times infinity!

457 A Pythagorean Triple Logic Puzzle

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457 = 4² + 21², and it is the hypotenuse of the primitive Pythagorean triple 168-425-457. Also, 457 is the sum of some consecutive prime numbers. One of my readers posted those primes in the comments.

A long time ago I decided that Pythagorean triples could make a great logic puzzle, so I created one. You can see it directly underneath the following directions:

This puzzle is NOT drawn to scale. Angles that are marked as right angles are 90 degrees, but any angle that looks like a 45 degree angle, isn’t 45 degrees. Lines that look parallel are NOT parallel. Shorter looking line segments may actually be longer than longer looking line segments. Most rules of geometry do not apply here: in fact non-adjacent triangles in the drawing might actually overlap.

No geometry is needed to solve this puzzle. All that is needed is the table of Pythagorean triples under the puzzle. The puzzle only uses triples in which each leg and each hypotenuse is less than 100 units long. The puzzle has only one solution.

If any of these directions are not clear, let me know in the comments. I will NOT be publishing the solution to this puzzle, but I will allow anyone who desires to put any or all of the missing values in the comments. Also, the comments will help me determine if I should publish another puzzle like this one.

Good Luck!

457 Puzzle

Sorted Triples

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

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  • 457 is a prime number.
  • Prime factorization: 457 is prime.
  • The exponent of prime number 457 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 457 has exactly 2 factors.
  • Factors of 457: 1, 457
  • Factor pairs: 457 = 1 x 457
  • 457 has no square factors that allow its square root to be simplified. √457 ≈ 21.3776

How do we know that 457 is a prime number? If 457 were not a prime number, then it would be divisible by at least one prime number less than or equal to √457 ≈ 21.3776. Since 457 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 457 is a prime number.

456 An Inchworm Measuring Marigolds

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456 is the sum of consecutive prime numbers in two different ways. One of my readers listed those ways in the comments. The factors of 456 are at the end of the post.

Inchworm, inchworm,
Measuring the marigolds
You and your arithmetic will probably go far.

Two plus two is four
Four plus four is eight
Eight and eight is sixteen
Sixteen and sixteen is thirty-two.

Inchworm, inchworm,
Measuring the marigolds
Seems to me you’d stop and see
How beautiful they are.

Today I taught a class of three year olds about being thankful for birds, insects, and creeping things. To keep their attention, I used a variety of stories, riddles, books, and games. I also sang a few songs including this one about an inchworm who is very good at arithmetic. I think preschool children can still enjoy songs like this even if they don’t understand everything the song is about or even if they are wiggling as much as an inchworm while they listen to it. Here is the song sung by Danny Kaye from the movie Hans Christian Andersen:

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Now for the number 456. The last two digits can be evenly divided by four, so the entire number is divisible by four. Also since it is formed from three consecutive numbers, it is divisible by 3. However since the number in the middle of those consecutive numbers is not 3, 6, 9 or another multiple of 3, we know that 456 is NOT divisible by 9.

Because it is divisible by four, we will use that fact first to determine how to reduce its square root.

456 divided by 4

456 ÷ 4 = 114. Notice that 114 is even, but 14 can’t be evenly divided by 4, so 114 cannot be either. Also notice that 114 is still divisible by 3. If we’re not sure whether or not 114 has any square factors, we are less likely to make a mistake if we divide it by 6 once, instead of by 2 and then by 3.

114 divided by 6

114 ÷ 6 = 19, a prime number, and we are certain there were no other square factors. Since we know 19 x 6 = 114, let’s backtrack a little and go back to that original one layer cake:

456 divided by 4

Take the square root of everything on the outside of the cake and get √456 = (√4)(√114) = 2√114

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  • 456 is a composite number.
  • Prime factorization: 456 = 2 x 2 x 2 x 3 x 19, which can be written 456 = (2^3) x 3 x 19
  • 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 x 2 x 2 = 16. Therefore 456 has exactly 16 factors.
  • Factors of 456: 1, 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, 228, 456
  • Factor pairs: 456 = 1 x 456, 2 x 228, 3 x 152, 4 x 114, 6 x 76, 8 x 57, 12 x 38, or 19 x 24
  • Taking the factor pair with the largest square number factor, we get √456 = (√4)(√114) = 2√114 ≈ 21.3542

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Picture credits: Inchworm and ruler: http://www.kindergartenkindergarten.com/2012/06/problem-solving-measurement.html;

441 Consecutive Numbers

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The first 6 triangular numbers are

  • 1 = 1
  • 1 + 2 = 3
  • 1 + 2 + 3 = 6
  • 1 + 2 + 3 + 4 = 10
  • 1 + 2 + 3 + 4 + 5 = 15
  • 1 + 2 + 3 + 4 + 5 + 6 = 21

For millennia mathematicians have thought triangular numbers were quite interesting. 21 is a triangular number and 21 x 21 = 441 which makes 441 interesting, too. But wait, there’s something else that is very interesting about triangular numbers:

Sum of consecutive cubes

It is amazing that when we begin with 1 cube, the sum of n consecutive cubes equals the nth triangular number squared every time!

What  follows is less amazing, but very practical. We can add consecutive numbers several ways to get 441. To find those ways we need to know the factors of 441.

  • 441 is a composite number.
  • Prime factorization: 441 = 3 x 3 x 7 x 7, which can be written 441 = (3^2) x (7^2)
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 x 3 = 9. Therefore 441 has exactly 9 factors.
  • Factors of 441: 1, 3, 7, 9, 21, 49, 63, 147, 441
  • Factor pairs: 441 = 1 x 441, 3 x 147, 7 x 63, 9 x 49, or 21 x 21
  • 441 is a perfect square. √441 = 21

Because all of the factors for 441 are odd numbers, it is so easy to find consecutive numbers whose sum equal 441. Check out all of these:

consecutive numbers equals 441If we allowed negative numbers in the list of consecutive numbers we could also see that 441 equals the sum of 147 consecutive numbers that are centered around the number 3, and 49 consecutive numbers that are centered around the number 7, and 63 consecutive numbers that are centered around the number 9. All of those sums would be quite long.

Here are some more reasonable-length sums using only consecutive ODD numbers.

consecutive odd numbers equal 441

That last sum reminds us that we always get n squared when we begin with one and add n consecutive odd numbers together.