### 956 A Pentagonal Puzzle for Paula Beardell Krieg

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:

thanks for help/interest yesterday @DavidKButlerUoA @mikeandallie @Simon_Gregg in deciphering those folds. This is what I was playing with pic.twitter.com/4ULLgpN6VB

— Paula Beardell Krieg (@PaulaKrieg) November 9, 2017

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From squares to pentagons to stars https://t.co/NsOL5PmIlK

— Paula Beardell Krieg (@PaulaKrieg) November 10, 2017

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

A pentagon with 951 points. I think this graphic needs to become an origami star. https://t.co/pwan9tGa2E

— Paula Beardell Krieg (@PaulaKrieg) November 14, 2017

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

Front & back of folded star made from @findthefactors graphic for pentagonal number 951 pic.twitter.com/TgKk62lWdO

— Paula Beardell Krieg (@PaulaKrieg) November 14, 2017

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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 = 2
**²**× 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

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