Mathematics

2026 Math Facts and Factors

/ ivasallay / 1 Comment

In this blog post, I’m sharing as many math facts about the number 2026 as I can find on my own or shared by others on Bluesky. I’m publishing the post before the year ends, but I will add additional number facts after the new year begins as well.

Countdown to 2026:

Here’s a countdown you can use right before midnight on New Year’s Eve:

2026 Countdown

make misc GIFs like this at MakeaGif

Here’s another countdown I saw on Bluesky:

Here’s another nerdy & numerical New Year countdown I came across! This time, for the upcoming 2026. Enjoy using this for the upcoming year! :3 #Mathematics #MathSky #MathChat #MathsChat #NewYearCountdown #NewYear2026

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— Math Nerd 1729 (@mathnerd1729.bsky.social) December 18, 2025 at 4:50 PM

If you have any problem seeing the 2026 video, here is a screenshot of its final frame.

Factors of 2026:

  • 2026 is a composite number.
  • Prime factorization: 2026 = 2 × 1013.
  • 2026 has no exponents greater than 1 in its prime factorization, so √2026 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying, we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore, 2026 has exactly four factors.
  • The factors of 2026 are outlined with their factor pair partners in the graphic below.

What Kind of Shape is 2026 in?

2026 can’t claim a shape of its own, but it does have a relationship with some other numbers that can.

2026 is one more cube than the sum of the cubes from 1 to 9.
1³ + 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ = 2026.

2026 is the sum of three perfect squares in several different ways.

45² + 1² = 2026, so it’s shaped like this: 

(You can click on images to see them better. Each of the 2026 cells is numbered.)

2(23² + 22²) = 2026. Since it’s double the sum of consecutive squares, it’s also shaped like this:

Since 2026 = (2(22) + 1)² +1, it is also one more than the 22nd centered octagonal number.

2026 is also the sum of three triangular numbers in 31 different ways:

Changing from 2025 to 2026:

Happy New Year everyone!

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— Isokon Gallery (@isokongallery.bsky.social) December 31, 2025 at 6:30 PM

2026 = 2025 + 1.

Yes, 2026 is one more than 2025, or in the case of this video, 2026 is one more car than 2025.

2025 🔜 2026

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— 🍁🇨🇦Team Canada Forever🇨🇦🍁 (@teamcanadaforever.bsky.social) December 28, 2025 at 4:40 PM

And in a brilliant post on Bluesky, 2026 is one more billiard ball than 2025.

Some “Powerful” Facts About the Number 2026:

Since 2026 is the sum of an even number of consecutive numbers,
505 + 506 + 507 + 508 = 2026, we get this “powerful fact”:
508² – 507² + 506² – 505² = 2026.

45² + 1² = 2026, so
(3² + 6²)² + 1² = 2026.

2026 would be a palindrome in base 13 and in base 45:
11(13²)+12(13¹)+11(13º) = 2026.
1(45²) + 0(45¹) + 1(45°) = 2026.

2026 is in exactly two Pythagorean triples:
90-2024-2026, and
2026-1026168-1026170.

On October 24, 2026, we can celebrate the Pythagorean triple,
10-24-26.

I’ll finish off this topic with a “powerful” math joke about 2026:

The Revolutionary Discovery That 2026 Equals 2026

#Math #Exponents #Tautology #Obvious #2026

https://sciencehumor.io/math-memes/the-revolutionary-discovery-that-2026-equals-2026-terq

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— ScienceHumor.io (@sciencehumor-io.bsky.social) December 28, 2025 at 7:41 PM

2026 in Complex Numbers

2026 is the sum of two squares, 45² + 1², therefore it is the product of two complex numbers. If you are logged into Bluesky, you can see that complex product in the following post:

Here are two other complex number facts involving the number 2026:

2026 Games and Puzzles

Math=Love has mazes and other puzzles featuring the number 2026.

2026 is divisible by 2, but not by 4, so it is the magic sum of a magic square puzzle, specifically the one that uses the numbers from 499 to 514.

Happy 2026!

Here’s when 2026 starts around the world! It’s that time of year again, and we can play the 2026 NCTM Year Game in our January lessons. Use the digits in the year 2026 and the operations +, −, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), along with grouping symbols to…

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— Colleen Young (@colleenyoung.bsky.social) December 31, 2025 at 6:00 PM

Peter Rowlet reports on Bluesky that the 2026 game works for the numbers 1 to 10 but doesn’t spoil your fun.

MathGames has also shared some 2026 fun facts and games.

Among other mathematical facts about the number 2026, Scientific American’s reprint of Spektrum der Wissenschaft’s Why 2026 Is a Mathematically Special Number informs us that the 8-magnetic-disk version of the mathematical game, Tower of Hanoi, will always take at least 2026 moves to solve.

More About the Number 2026:

0 + 12 × 34 × 5 – 6 – 7 + 8 – 9 = 2026. You can find another count-up equation on the MathGames blog.

2026 degrees is 1013π/90 radians.

If you are logged into Bluesky, you can see The Maths Bazaar’s post informing us that
2026 = 2¹¹ – 2 × 11,
2026 = √(2²²) – 22, and
that each of the following is a prime number:
2026 + 1,
2 + 0 × 26 + 1,
20 × 26 + 1,
202 × 6 + 1.

2026 = √(2²²) – 22 inspired me to make a graphic for it and some other equations I had already found:

Mathometry has some suggestions for math activities involving the number 2026.

From this Bluesky post and its link to OEIS, I learned that there are “2026 hyperforests spanning 10 unlabeled nodes without isolated vertices.”

2026

https://mathr.co.uk/web/2026.html

#math #mathematics

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— The Mathematician (@math.blaze.email) January 1, 2026 at 11:10 AM

May 2026 be a delightful year for you and yours!

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.

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— 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

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— 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

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— 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

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— 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

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— 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.