Mathematics

Saying Goodbye to 2025

/ ivasallay / Leave a comment

You probably have an opinion on what kind of year 2025 was. Regardless of that, 2025 will always be a fabulous number. I wrote a post about 2025 that had so many number facts/puzzles from X and Bluesky that WordPress wouldn’t allow me to add or subtract even one word! That post is broken. You can read it, but I can’t edit it at all.

For my post welcoming 2026, I am only going to include math facts/puzzles I find on Bluesky, along with my own graphics. It will likely be a shorter post, but I’m confident I won’t break WordPress this time.

So, as we say goodbye to the complicated year 2025, I’d like you to know that 2025° simplifies nicely to 45π/4 in radians. Also remember that
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)² = 45² = 2025², and
1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ = 2025.

 

Have a lovely time saying Goodbye to 2025!

No Two Snowflakes are Alike

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I was inspired by this post from Paddy MacMahon that I saw on Bluesky.

#MathsToday #ALevelMaths #FurtherMaths

Find the exact area of the snowflake.

[image or embed]

— Paddy MacMahon (@paddymacmahon.com) December 23, 2025 at 8:19 AM

I loved that he produced this lovely snowflake from just one equation,
r = 16 + 6sin6θ + 4cos36θ. Wow!

Obviously, he knows a lot about polar coordinates, much more than I do, but that can’t keep me from playing around with his equation in Desmos and learning a little bit more in the process! All I did was replace the constant and the coefficients with sliders. My efforts produced these two snowflakes…

…and many more!

Then I thought, “What if I change the coefficients of theta?” I quickly learned that those coefficients need to be multiples of 6 to maintain the snowflake’s 6-sided shape, but yep, I’m learning from experimentation! I varied the theta coefficients over three equations to get this 3-D look:

It all makes me smile and think, “Let it snow! Let it snow! Let it snow!”

If you’re dreaming of a white Christmas, I hope you get the real thing, but if not, I hope these snowflakes will delight you at least a little bit.

Eight Desmos Ornaments

/ ivasallay / 4 Comments

I made some Christmas ornaments in Desmos that I hope you will enjoy. If you click on and off the circles on the left of the descriptions, you can see all eight ornaments in one Desmos graph, or you can find them all pictured below in this post. If you click the arrow next to each description in Desmos, you can also see the equations used to produce each ornament. However, the snowflake and Rudolf’s face required many ordered pairs, which I put into a separate folder.

1. Decorated half red and half green:

2. Decorated with diagonal stripes:

3. Decorated with sines and secants:

4. Decorated with a snowflake:

5. Decorated with a checkerboard design: (This was a pleasant surprise that required only one equation!)

6. Decorated with a spiral

7. Decorated with ellipses for a 3D look:

8. Decorated with Rudolf’s face:

Perhaps you will choose to make an ornament yourself in Desmos. If so, I’d love to see it.

I hope you all have a very merry Christmas!

1806 Is a Primary Pseudoperfect Number

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Today’s Puzzle:

OEIS.org informs us that the first five primary pseudoperfect numbers are 2, 6, 42, 1806, and 47058.

I noticed that
1⋅2 = 2,
2⋅3 = 6,
6⋅7 = 42, and
42⋅43 = 1806.

But that pattern stops there. 47058 = 2⋅3⋅11⋅23⋅31.

Look at the graphic from Desmos below. Can you figure out why those five numbers are primary pseudoperfect numbers?

Factors of 1806:

I made a couple of factor trees for the number 1806. Which do you like better?

  • 1806 is a composite number.
  • Prime factorization: 1806 = 2 × 3 × 7 × 43.
  • 1806 has no exponents greater than 1 in its prime factorization, so √1806 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore, 1806 has exactly 16 factors.
  • The factors of 1806 are outlined with their factor pair partners in the graphic below.

More About the Number 1806:

1806₁₀ is 248₁₉ because
2¹(19²) + 2²(19¹) + 2³(19º) = 1806.

1804 Desmos Christmas

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Today’s Puzzle:

Merry Christmas, everybody! Can you make a Christmas design in Desmos?

Here’s how I solved this Desmos Christmas puzzle: A few weeks ago, I saw this post on Bluesky and was inspired by the climbing sine curves on the featured Desmos Christmas tree:

#mathstoday I began thinking about a Desmos activity for my year 11 in which they could make a Christmas tree. Then I got carried away, thought about climbing sine curves (tinsel) and translating polar graphs. I’m not sure it’s suitable for year 11 anymore… Oops

[image or embed]

— over-drawn.bsky.social (@over-drawn.bsky.social) November 28, 2024 at 12:34 PM

What is a climbing sine curve, and could I use one to decorate the plain Desmos Christmas tree I made last year? I had to google “climbing sine” to proceed, but I learned that it is a function such as y = x + sin(x). That’s a familiar function; I just didn’t know it had a cutesy name.

I multiplied that function by a constant. Can you figure out what that constant was?

Later, I embellished the tree even more with lights and falling snow. I hope you enjoy it!

Here are some other delightful Christmas Desmos designs I saw on Bluesky. this first one rotates in 3-D.

Happy Holidays! 🎄
http://www.desmos.com/3d/p5t7m4kh4s
#iTeachMath

[image or embed]

— Raj Raizada (@rajraizada.bsky.social) December 10, 2024 at 10:46 AM

Enjoyed re-creating this visual in the @desmos.com Geometry tool: http://www.desmos.com/geometry/lx7… #mathsky

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— Tim Guindon (@tguindon.bsky.social) December 11, 2024 at 1:08 PM

More snowflake fun in @desmos.com
I don’t think it can show text mirror-flipped yet (?), so for this, you type your word, screenshot it, then load it as an image.
I’m hoping to have students load in pics of their names, then snowflake-ify them.
http://www.desmos.com/geometry/afo…
#iTeachMath #MathSky

[image or embed]

— Raj Raizada (@rajraizada.bsky.social) December 17, 2024 at 11:17 AM

This next one isn’t a Desmos design, but I enjoyed its playful nature just the same. Do you recognize the number pattern?

Inspired by @studymaths.bsky.social – #MathPlay 🧮 via Pascal’s Dice 🎲🔺

#ITeachMath #MTBoS #STEM #Maths #ElemMathChat #Math #MathSky #MathsToday #EduSky

[image or embed]

— Libo Valencia 🧮 MathPlay (@mrvalencia24.bsky.social) December 12, 2024 at 4:00 AM

Factors of 1804:

I know 1804 is divisible by four because the last two digits are divisible by 4.
1804 ÷ 4 = 451. Oh, and 4 + 1 = 5, so 451 is divisible by eleven and forty-one! Here’s a factor tree for 1804:

  • 1804 is a composite number.
  • Prime factorization: 1804 = 2 × 2 × 11 × 41, which can be written 1804 = 2² × 11 × 41.
  • 1804 has at least one exponent greater than 1 in its prime factorization so √1804 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1804 = (√4)(√451) = 2√451.
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying, we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore, 1804 has exactly 12 factors.
  • The factors of 1804 are outlined with their factor pair partners in the graphic below.

More About the number 1804:

1804 is the hypotenuse of one Pythagorean triple:
396-1760-1804, which is (9-40-41) times 44.

1804 looks interesting in some other bases:
It’s A8A in base 13 because 10(13²) + 8(13) + 10(1) = 1804.
It’s 4A4 in base 20 because 4(20²) +10(20) + 4(1) = 1804.

Desmos Thanksgiving Mystery Dot-to-Dot

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Today’s Puzzle:

I wanted to create a Dot-to-Dot in Desmos for my students that wouldn’t require them to type in many ordered pairs. I concluded that if most points could be reflected over the x or y-axis, I could eliminate the need to type in about half the points. With that in mind, I recently created this mystery dot-to-dot you can enjoy over the Thanksgiving weekend.

What will this unfinished dot-to-dot become when the dots are connected, and 90% of the image is reflected over the y-axis?

My sister guessed it was a cat. The image reminds me of a snowman. What did you think it might be?

You can discover what it is by clicking on this pdf and following the instructions: Desmos Mystery Ordered Pair Dot-to-Dot

The instruction will look like this:

Depending on your device, you may be able to click on the lower right-hand corner of the Desmos image below to see how much fun I had transforming it four different ways: I made the image slide along the x-axis,  rotated it 90 degrees, reflected it over the x-axis, and dilated it. (The location of the turkey’s wattle can help you determine if an image is a reflection, a rotation, or a combination of both.) If clicking the lower right-hand corner does not work on your device, click this link. These transformations are all essential concepts for students to learn, and Desmos can make the process quite enjoyable.

Did you guess right? Have a very happy Thanksgiving!

1787 The11-Digit Palindromes of Base 2

/ ivasallay / 2 Comments

Today’s Puzzle:

1787 is an 11-digit palindrome in base 2. I wondered how many 11-digit palindromes there are, what they are, and what numbers they represent in base ten. I decided to try to make you wonder about all that as well. Try it out yourself before you read how I solved this puzzle.

The only digits in base 2, are 0 and 1. The first digit of any number must be 1 or else the number will not have eleven digits. The last digit also must be one for the number to be a palindrome. In fact, all five last digits will be determined by the first five digits. Thus, we only need to find all possible combinations of 0 and 1 that can occur in the second through sixth positions. There are 2⁵ ways to write 0 and 1 in those 5 positions. That means we know right away that there are 32 different 11-digit palindromes in base 2. I opened Excel and wrote those 32 different 11-digit numbers beginning with 00000 and ending with 11111. I put a 1 in front of them and had Excel copy the appropriate numbers into the last 5 spots. That gave me all the 11-digit palindromes. Then I had Excel multiply the values in each cell with the powers of 2 that head up each column to give the base 10 representations. This chart was the final product.

Did you notice that the first base 10 number in the chart is the number just after 2¹º and the last number is the number right before 2¹¹?

Factors of 1787:

  • 1787 is a prime number.
  • Prime factorization: 1787 is prime.
  • 1787 has no exponents greater than 1 in its prime factorization, so √1787 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1787 has exactly 2 factors.
  • The factors of 1787 are outlined with their factor pair partners in the graphic below.

How do we know that 1787 is a prime number? If 1787 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1787. Since 1787 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1787 is a prime number.

More About the Number 1787:

1787 and 1789 are twin primes.
1783, 1787, and 1789 are a prime triplet.

1787 is a palindrome in some other bases, too!
It’s 919 in base 14 because 9(14²) + 1(14) + 9(1) = 1787,
595 in base 18 because 5(18²) + 9(18) + 5(1) = 1787, and
191 in base 38 because 1(38²) + 9(38) + 1(1) = 1787.

1786 is a Centered Triangular Number

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Today’s Puzzle:

A formula for the nth triangular number is n(n+1)/2. Centered triangular numbers are the sum of three consecutive triangular numbers. What would be a formula for finding centered triangular numbers? What value of n in your formula would produce the number 1786?

Factors of 1786:

  • 1786 is a composite number.
  • Prime factorization: 1786 = 2 × 19 × 47.
  • 1786 has no exponents greater than 1 in its prime factorization, so √1786 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1786 has exactly 8 factors.
  • The factors of 1786 are outlined with their factor pair partners in the graphic below.

More About the Number 1786:

From OEIS.org we learn that 1786³ = 5,696,975,656. Notice that all those digits are 5 or greater.

1786 is 1G1 in base35,
because 1(35²) + 16(35) + 1(1) = 1786.