Trick or Treat?

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I’m not much of an artist, but I still like playing around in Desmos. Here’s what I made this week. Have a Happy Halloween!

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

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

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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

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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!

1793 Are You Easily Distracted?

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Today’s Puzzle:

Don’t let the x’s in the puzzle distract you! This puzzle can actually be solved quite easily! Just follow the previous suggestion of putting a 12 in one of the last two boxes, fill in the rest of the boxes (don’t worry if any of the numbers are greater than 12), identify the largest number, and adjust all of the numbers so that that largest number becomes the new 12.

Factors of 1793:

Solve this problem: 1 – 7 + 9 – 3 =

If the answer is 0 or any other multiple of 11, then 1793 is a multiple of 11.

  • 1793 is a composite number.
  • Prime factorization: 1793 = 11 × 163.
  • 1793 has no exponents greater than 1 in its prime factorization, so √1793 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 1793 has exactly 4 factors.
  • The factors of 1793 are outlined with their factor pair partners in the graphic below.

More About the Number 1793:

1793 is a palindrome in base 32:
1O1 1(32²) + 24(32) + 1(1) = 1024 + 768 + 1 = 1793.
(O is the 15th letter of the alphabet, and 15 + 9 = 24, so O would be 24 if we all had 32 fingers.)

OEIS.org informs us that 1793 is a Fibonacci-inspired Pentanacci number.

 

Why Is 1792 a Friedman Number?

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Today’s Puzzle:

I’ve mentioned before that putting a 12 in one of the last two boxes will let you avoid negative numbers as you explore the relative relationship of the clues. For this puzzle, I would suggest that you put the 12 in the third from the last box. Why? Because the last triangle on the bottom has an 8 in it, and we will need to use either 12 – 8 = 4, and 4 – 2 = 2 for the last three boxes or 11 – 8 = 3, and 3 – 2 = 1.

.

After you make your way to the empty triangle on the left of the puzzle, you will notice that you are missing the numbers 1 and 8. There isn’t any way to get a 5 by subtracting those two numbers, but if you realize that 13 – 5 = 8, you should know what adjustments you need to make to solve the puzzle.

Factors of 1792:

If the last digit of a number is 2 or 6, and the next-to-the-last digit is odd, then the whole number is divisible by 4.

If the last digit of a number is 0, 4, or 8, and the next-to-the-last digit is even, then the whole number is also divisible by 4.

1792 will allow us to apply those two divisibility observations several times as we make this factor tree:

  • 1792 is a composite number.
  • Prime factorization: 1792 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7, which can be written 1792 = 2⁸ × 7.
  • 1792 has at least one exponent greater than 1 in its prime factorization so √1792 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1792 = (√256)(√7) = 16√7.
  • The exponents in the prime factorization are 8 and 1. Adding one to each exponent and multiplying we get (8 + 1)(1 + 1) = 9 × 2 = 18. Therefore 1792 has exactly 18 factors.
  • The factors of 1792 are outlined with their factor pair partners in the graphic below.

More About the Number 1792:

1792 is a Friedman number because 7·2⁹⁻¹ = 1792.

Notice that the digits 1, 7, 9, and 2 and only those digits are used on both sides of the equal sign, and they are used the same number of times. 1792 is only the 26th Friedman number.

1792 is the difference of two squares in SEVEN different ways:
449² – 447² = 1792,
226² – 222² = 1792,
116² – 108² = 1792,
71² – 57² = 1792,
64² – 48² = 1792,
46² – 18² = 1792, and
44² – 12² = 1792.

1791 What a Distraction This Puzzle Is!

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Today’s Puzzle:

If you followed my advice from other posts and put a 12 in one of the last two boxes, you’ll be able to place five other numbers before hitting the roadblock that is the empty triangle.

Now we see that the highest known value is 15. The following numbers less than 15 are missing 1, 2, 3, 4, 5, 6, 8, 10, and 13. Since we have a 15, and our largest number can’t be greater than 12, let’s eliminate the smallest (15 – 12 = 3) three numbers from the list. We now have 4, 5, 6, 8, 10, and 13.

What can you do now? I suggest that you put an x such that -11 < x < 11 in the empty triangle and continue writing in values for the squares.

Regardless if x is a positive number or a negative number, the smallest number in a box will be either 7 or else 5 + x.

Since there isn’t a 6 + x or an 8, we know that one of those circled positions must be 1 and the other must be 2. If we assume the 7 should have been 2, we can lower the six numbers on the right of the puzzle by 5.

Then assuming that 5 + x must be 1 and filling in the puzzle we would get:

Uh oh! We can’t have two 9’s, 6’s, or 10’s, so those were NOT good assumptions.

I assure you that if switch the positions of the 1 and the 2, you will be able to complete the puzzle and place each number up to 12 in a box:

Factors of 1791:

  • 1791 is a composite number.
  • Prime factorization: 1791 = 3 × 3 × 199, which can be written 1791 = 3² × 199.
  • 1791 has at least one exponent greater than 1 in its prime factorization so √1791 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1791 = (√9)(√199) = 3√199.
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1791 has exactly 6 factors.
  • The factors of 1791 are outlined with their factor pair partners in the graphic below.

More About the Number 1791:

1791 is the difference of two squares in three different ways:
896² – 895² = 1791,
300² – 297² = 1791, and
104² – 95² = 1791.

1791 is A7A in base 13 because 10(13²) + 7(13) + 10(1) = 1791, and
636 in base 17 because 6(17²) + 3(17) + 6(1) = 1791.

1790 How Can You Solve This Subtraction Distraction?

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Today’s Puzzle:

How would I solve this puzzle? I would want to find a set of 12 consecutive numbers that were all positive and relatively small. Since there is an empty triangle near the right side of the puzzle, I would begin with the triangle on the bottom with a 4 in it and write 12 in a box above it. Then I would think and write 12 – 4 = 8 for the other box above the 4. My thinking would look like this:

  • 12
  • 12 – 4 = 8 (going to the right of the 12)
  • 12 – 6 = 6 (going to the left of the 12)
  • 6 + 7 = 13
  • 13 – 6 = 7
  • 7 – 2 = 5
  • 5 + 5 = 10
  • 10 + 5 = 15
  • 15 – 6 = 9

So that the puzzle looks like this:

I would note that I’m missing the following numbers: 1, 2, 3, 4, 11, and 14, and would figure out which of those missing numbers fit in the last three squares. Because I have a 15, I would note that 15 – 12 = 3 and would subtract 3 from each square to get numbers from 1 to 12. Figuring out what belongs in the empty triangle won’t be difficult either.

Factors of 1790:

  • 1790 is a composite number.
  • Prime factorization: 1790 = 2 × 5 × 179.
  • 1790 has no exponents greater than 1 in its prime factorization, so √1790 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 1790 has exactly 8 factors.
  • The factors of 1790 are outlined with their factor pair partners in the graphic below.

More About the Number 1790:

1790 is the hypotenuse of a Pythagorean triple:
1074-1432-1790 which is (3-4-5) times 358.

1789 is 414 in base 21, but
1790 is 4I4 in base 19 because 4(19²) + 18(19) + 4(1) = 1790.