2026 Math Facts and Factors

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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.

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.

If you are logged into Bluesky, you can see The Maths Bazaar’s post informing us that
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.

May 2026 be a delightful year for you and yours!

Saying Goodbye to 2025

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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!

2025 Factors and Facts

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Check back every now and then. As I find new facts about the number 2025, I’ll add them.

Countdown to 2025:

2025 Countdown

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Fun with the Digits of 2025:

The countdown to 2025 is on! I’ve been having so much fun making 2025-themed puzzles the last few weeks. It’s time to start sharing them!

2025 Square Edge Matching Puzzle

mathequalslove.net/yearly-squar…

#mtbos #iteachmath #edusky #mathsky #puzzlingclassroom

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— Sarah Carter (@mathequalslove.bsky.social) December 26, 2024 at 8:37 AM

Start the New Year with a fun #math challenge! 🎉🧊 Break the ice with your students using the 2025 Year Game.

How many expressions can you and your students create from the numbers 1 to 100—using only the digits in 2025? Try it now: nctm.link/LvN1e

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— NCTM (@nctm.org) December 26, 2024 at 6:23 AM

2024 was the year of the dragon, next year -is- the dragon

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— Odendo (@odendo.bsky.social) December 22, 2024 at 1:18 PM

Factors of 2025:

Unless you were born before the end of 1936,  2025 will be the only year in your lifetime with exactly 15 factors.

  • 2025 is a composite number.
  • Prime factorization: 2025 = 3 × 3 × 3 × 3 × 5 × 5, which can be written 1458 = 3⁴ × 5².
  • Since its prime factorization only contains even powers, 2025 is a perfect square. √2025 =
  • The exponents in the prime factorization are 4 and 2. Adding one to each exponent and multiplying, we get (4 + 1)(2 + 1) = 5 × 3 = 15. Therefore, 2025 has exactly 15 factors.
  • The factors of 2025 are outlined with their factor pair partners in the graphic below.

Here’s another way to display the factor pairs of 2025, although several are outside the visible gridlines.

Let’s continue the countdown to 2025.

Today’s 2025 Factor Tree Puzzle was inspired by Dr. Harold Reiter who presented on factor tree puzzles at a math teachers’ circle workshop I attended.

mathequalslove.net/2025-factor-…

#mtbos #iteachmath #puzzlingclassroom #mathsky #edusky

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— Sarah Carter (@mathequalslove.bsky.social) December 27, 2024 at 8:14 AM

Square Facts About 2025:

2025 is going to be a really square year, and also be on the lookout for this specific square date!

#math #ITeachMath #EduSky

youtube.com/shorts/CEZ6B...

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— Howie Hua (@howiehua.bsky.social) December 29, 2024 at 8:02 AM

This morning I learned that 2025 will be the first 'perfect square' year (45x45) since 1936 (44x44) and there won't be another until 2116 (46x46). Expressed in month/day/year format, there will be a total of 8 'perfect square dates' in 2025 (ex. 1/09/2025 ➡️ 1092025 ➡️ 1045x1045).

— Andrew in New Jersey (@aannddrreeww.bsky.social) November 19, 2024 at 9:38 AM

2025 Pythagorean Triples:

Then have a bit more of that info:

* 2025 is also Pythagorean: 27²+36²=45²=2025.
* Also, when it is 20:25 (does not work in am/pm notation) then exactly 35²=1225 minutes have passed, another square.
* Finally, 20 and 25 are letters T and Y.

Bottom line, 2025 will be a good and a polite one.

— Christophe Smet (@christophesmet.bsky.social) December 23, 2024 at 6:13 AM

2025 is the hypotenuse of two Pythagorean triples:

1215-1620-2025, which is (3-4-5) times 405, and
567-1944-2025, which is (7-24-25) times 81.

2025 is a leg in a bunch of Pythagorean triples:

2025-2700-3375, which is (3-4-5) times 675,
2025-4860-5265, which is (5-12-13) times 405,
2025-9000-9225, which is (9-40-41) times 225,
2025-15120-15255, which is (15-112-113) times 135,
1080-2025-2295, which is (8-15-17) times 135,
2025-25272-25353, which is (25-312-313) times 81,
2025-27300-27375, which is (27-364-365) times 75,
1260-2025-2385, which is (28-45-53) times 45,
2025-45540-45585, which is (45-1012-1013) times 45,
2025-8316-8559, which is (75-308-317) times 27,
2025-75924-75951, which is (75-2812-2813) times 27,
2025-82000-82025, which is (81-3280-3281) times 25,
2025-5280-5655, which is (135-352-377) times 15,
2025-136680-136695, which is (135-9112-9113) times 15,
2025-2448-3177, which is (225-272-353) times 9,
2025-227808-227817, which is (225-25312-25313) times 9,
2025-410060-410065, which is (405-82012-82013) times 5
156-2025-2031, which is (52-675-677) times 3,
2025-683436-683439, which is (675-227812-227813) times 3, and finally, this primitive,
2025-2050312-2050313 calculated from 2025, (2025²-1²)/2, (2025²+1²)/2.

Powerful Facts about 2025:

Pour les amoureux des maths :

2025 est le carré de la somme des chiffres de 1 à 9 :
(1+2+3+4+5+6+7+8+9)² = 2025

2025 est aussi la somme des cubes des chiffres de 1 à 9 :
1³+2³+3³+4³+5³+6³+7³+8³+9³= 2025.

C'est le théorème de Nicomaque :
fr.wikipedia.org/wiki/Somme_d...

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— Pierre Col (@pierrecol.bsky.social) December 23, 2024 at 9:41 AM

2025 Shapes:

2025 is a perfect 45×45 square, but what other shapes can it be in?

Maybe 2025 will be the only square number in your lifetime, but it definitely will be the only centered octagonal number you will live to see.

2025 Magic Square:

2025 is the sum of consecutive numbers:

Other Interesting 2025 Sums:

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

/ ivasallay / Leave a comment

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!