Month: December 2024

2025 Factors and Facts

/ ivasallay / Leave a comment

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

Countdown to 2025:

2025 Countdown

make science GIFs like this at MakeaGif

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

[image or embed]

— 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

[image or embed]

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

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

[image or embed]

— 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

[image or embed]

— 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...

[image or embed]

— 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...

[image or embed]

— 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

/ ivasallay / Leave a comment

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

[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

[image or embed]

— 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.