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956 A Pentagonal Puzzle for Paula Beardell Krieg

/ ivasallay / 1 Comment

Paula Beardell Krieg likes mathematics. She is also an expert paper folder. Lately, she has been turning pentagons into five-point stars. Here are a couple of her recent twitter posts:

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

On Monday I wrote about a centered pentagonal number and included a graphic. Paula saw the post and tweeted:

https://platform.twitter.com/widgets.js

Paula followed through and made my graphic into a beautiful five-point star:

https://platform.twitter.com/widgets.js

There is nothing like trying to do origami for the first time to make me realize how irregular my pentagon graphic is. I would call my first attempt an epic fail. Sorry, I didn’t take any pictures.

However, before I started folding anything, I made this puzzle for Paula because she inspired me to make a puzzle with a pentagon in it. I needed the puzzle to be at least a 13 x 13 puzzle to get the large pentagon in it, but I decided to make it a 14 x 14 instead. It’s a level 5 so there will be some tricky parts, especially since most of the multiples of 7 in the puzzle are also multiples of 14. Don’t let that stop you from trying to solve it.

Print the puzzles or type the solution in this excel file: 10-factors-951-958

Anyways, after making the puzzle and making my epic fail star, I made this closer-to-regular pentagon on my computer:

 

I made a star using it. It looked pretty good so I decided to give my graphic of centered pentagonal number 951 a second try.  I cut it to make it more regular. Then I followed the directions on the video Paula recommended. My previous folds caused me some problems, but I was able to make something that looks like a star. It isn’t as good as Paula’s, especially on the back, but I’m okay with it. Here are pictures, front and back, of both stars I made (flaws and all):

Now since this is my 956 post, I will share some information about that number:

956 is a palindrome in two other bases:
4C4 BASE 14 (C is 12 base 10) because 4(14²) + 12(14¹) + 4(14⁰) = 956
2H2 BASE 18 (H is 17 base 10) because 2(18²) + 17(18¹) + 2(18⁰) = 956

  • 956 is a composite number.
  • Prime factorization: 956 = 2 × 2 × 239, which can be written 956 = 2² × 239
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 956 has exactly 6 factors.
  • Factors of 956: 1, 2, 4, 239, 478, 956
  • Factor pairs: 956 = 1 × 956, 2 × 478, or 4 × 239
  • Taking the factor pair with the largest square number factor, we get √956 = (√4)(√239) = 2√239 ≈ 30.9192

 

729 What I Did for Paula Krieg’s Star Project

/ ivasallay / Leave a comment

Let me tell you a little about Paula Krieg’s Star Project:

The 12-Fold Rosette has its origin in Islamic Geometry. Paula Krieg is asking that people from a variety of backgrounds color in a section of the rosette and send it back to her so that she can assemble the different pieces into one lovely unified rosette. We are all stars, and some of the pieces look like stars, so she calls it her star project. You are invited to become a part of the project as well. Just click on this invitation to learn more. You can get a tile by either leaving a comment or e-mailing her a request, but do it soon. You will need to return your tile to her by 10 January 2016 to be included in the project.

A partially filled 12-fold Rosette

Rosette from Paula Krieg’s blog showing some possible coloring schemes for the tiles.

The piece that Paula sent me looked like this:

Rosette tile 21

I thought I would color it black, gray, white, and red because I think they look good together.

Since I don’t have any crayons in my house right now, and all my coloring pencils need to be sharpened, I decided to color it on my computer using the program Paint.

I brought up the e-mail Paula, opened the attachment, hit “Print Screen”, copied and pasted it into Paint, then cropped it.

I soon discovered that I could color in my entire tile using only black and gray and stay true to a goal I had given myself. The very tiny triangles insisted on staying white, but if I had increased the view size before I hit “Print Screen”, I would have been able to color them, too.

Tile 21

I saved my coloring, but immediately felt like it really should have some red in it. I brought up my saved work, clicked on “save as”, named it “red tile”, and added some red. However, saving the file changed how it now received color. When I added some red, it looked like this, and I liked it and saved it again:

Tile 21 red

Then I thought I might want a little more white, so I brought it back up and did “save as” again before adding some. This time it seem to allow even less additional color in each shape. (In other words, we see much more gray than black.)

Tile 21 red and white

 

I had so much fun doing this project. I have never considered myself to be much of an artist, but I felt a little artistic doing this project. I like all three tiles, but I think I like the final product the best.

You should join the project, too! You’re a star! Go ahead and shine! Whatever colors you use or whatever method you use to color your tile, your efforts will be appreciated. Also, because everyone is unique, every tile will look at least a little different. No tile will be ordinary.

Likewise, you might think that 729 is just an ordinary number, but much depends on how you look at it, too. Here are 729 equally sized squares.

729 Equal Squares

There’s a variety of ways to count the squares. Some ways remind us that 729 is a perfect square, other ways reminds us 729 is a perfect cube, and perhaps you can even find a way to see that 729 is a perfect 6th power, too.

279 is a perfect square, cube, and 6th power

729 can be written as the sum of consecutive numbers several different ways:

  • 364 + 365 = 729; that’s 2 consecutive numbers.
  • 242 + 243 + 244 = 729; that’s 3 consecutive numbers.
  • 119 + 120 + 121 + 122 + 123 + 124 = 729; that’s 6 consecutive numbers.
  • 77 + 78 + 79 + 80 + 81 + 82 + 83 + 84 + 85 = 729; that’s 9 consecutive numbers.
  • 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 = 729; that’s 18 consecutive numbers.
  • 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 = 729; that’s 27 consecutive numbers.

729 is 1,000,000 in base 3, and 1000 in base 9, and 100 in base 27.

729 is also a palindrome in two bases:

  • 1331 BASE 8; note that 1(512) + 3(64) + 3(8) + 1(1) = 729.
  • 121 BASE 26; note that 1(26^2) + 2(26) + 1(1) = 729.

It is interesting that one less than 9 is 8, and one less than 27 is 26, and those are the two bases in which 729 is a palindrome!

Here is the factoring information for the number 729:

  • 729 is a composite number. 729 = 27^2; 729 = 9^3; and 729 = 3^6
  • Prime factorization: 729 = 3 x 3 x 3 x 3 x 3 x 3, which can be written 729 = (3^6)
  • The exponent in the prime factorization is 6. Adding one we get (6 + 1) = 7. Therefore 729 has exactly 7 factors.
  • Factors of 729: 1, 3, 9, 27, 81, 243, 729
  • Factor pairs: 729 = 1 x 729, 3 x 243, 9 x 81, or 27 x 27
  • 729 is a perfect square √729 = 27.

729-factor-pairs