A Multiplication Based Logic Puzzle

Archive for the ‘Level 5 Puzzle’ Category

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:

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

 

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948 and Level 5

All you need is a little bit of logic and the multiplication facts in a standard 12 x 12 multiplication table to solve this puzzle. This puzzle is not just for kids. It can be challenging even for adults. Go ahead, give it a try. You’ll soon see that it’s tougher than it looks, but stick with it, you can conquer it!

Print the puzzles or type the solution in this excel file: 12 factors 942-950

Here’s something cool about the number 948 that I learned from Stetson.edu:

948 and 949 make a Ruth-Aaron pair because they are consecutive numbers and the sum of 948’s prime factors equals the sum of 949’s prime factors:

  • 948 is a composite number.
  • Prime factorization: 948 = 2 × 2 × 3 × 79, which can be written 948 = 2² × 3 × 79
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 948 has exactly 12 factors.
  • Factors of 948: 1, 2, 3, 4, 6, 12, 79, 158, 237, 316, 474, 948
  • Factor pairs: 948 = 1 × 948, 2 × 474, 3 × 316, 4 × 237, 6 × 158, or 12 × 79,
  • Taking the factor pair with the largest square number factor, we get √948 = (√4)(√237) = 2√237 ≈ 30.7896

938 and Level 5

Can you figure out where the factors 1 – 10 go in the first column and top row so that this level 5 puzzle will function as a multiplication table?

Print the puzzles or type the solution on this excel file: 10-factors-932-941

Now I’ll share a few facts about the number 938.

938 is a palindrome in two consecutive bases:
It’s 343 in BASE 17 because 3(17²) + 4(17¹) + 3(17º) = 938
It’s 2G2 in BASE 18 (G is 16 base 10), because 2(18²) + 16(18¹) + 2(18º) = 938

  • 938 is a composite number.
  • Prime factorization: 938 = 2 × 7 × 67
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 938 has exactly 8 factors.
  • Factors of 938: 1, 2, 7, 14, 67, 134, 469, 938
  • Factor pairs: 938 = 1 × 938, 2 × 469, 7 × 134, or 14 × 67
  • 938 has no square factors that allow its square root to be simplified. √938 ≈ 30.62678566

929 Little Green Monster

Here’s a little green monster just in time for Halloween. It’s a level 5 so it might be a little scary. Just don’t write any of the factors in the first column or top row unless you know for sure that factor belongs where you are putting it. Use logic and not guessing, and you’ll handle this little green monster just fine.

Print the puzzles or type the solution on this excel file: 12 factors 923-931

929 is the sum of nine consecutive prime numbers:
83  + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 = 929

23² + 20² = 929, so 929 is the hypotenuse of a Pythagorean triple:
129-920-929 which is 23² – 20², 2(23)(20), 23² + 20²

Obviously 929 is a palindrome in base 10.

It is also palindrome 131 in BASE 29 because 1(29²) + 3(29) + 1(1) = 929.

  • 929 is a prime number.
  • Prime factorization: 929 is prime.
  • The exponent of prime number 929 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 929 has exactly 2 factors.
  • Factors of 929: 1, 929
  • Factor pairs: 929 = 1 × 929
  • 929 has no square factors that allow its square root to be simplified. √929 ≈ 30.4795013

How do we know that 929 is a prime number? If 929 were not a prime number, then it would be divisible by at least one prime number less than or equal to √929 ≈ 30.4. Since 929 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 929 is a prime number.

921 Is This Bug Cute or Creepy?

Some bugs make creepy Halloween decorations. Other bugs, like ladybugs, might make a very cute costume.

Today’s puzzle looks like a bug, but there is no reason to run and hide from this one. Yes, it’s a level 5, so some parts of it may be tricky.

This is what you need to do to solve it: stay calm; don’t guess and check. Figure out where to put each number from 1 to 10 in both the top row and the first column so that the clues make the puzzle work like a multiplication table. Don’t write a number down unless you are absolutely sure it belongs where you’re putting it. Use logic, step by step, and this puzzle will be a treat.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

When you put on a Halloween costume, you may look completely different.

When a number is written in a different base, it may look completely different. For example,
921 looks like repdigit 333 in BASE 17 because 3(17²) + 3(17¹) + 3(17º) = 3(289 + 17 + 1) = 3(307) = 921
(307 is 111 in BASE 17)

921 looks like palindrome 1H1 in BASE 23 (H is 17 base 10). As you might suspect, 1(23²) + 17(23¹) + 1(23º) = 529 + 391 + 1 = 921

When it’s not written in a different base, 921 looks pretty familiar. You can tell quite quickly that it is divisible by 3:

  • 921 is a composite number.
  • Prime factorization: 921 = 3 × 307
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 921 has exactly 4 factors.
  • Factors of 921: 1, 3, 307, 921
  • Factor pairs: 921 = 1 × 921 or 3 × 307
  • 921 has no square factors that allow its square root to be simplified. √921 ≈ 30.34798181

910 and Level 5

910 is the hypotenuse of four Pythagorean triples:

  • 224-882-910, which is 14 times (16-63-65).
  • 350-840-910, which is (5-12-13) times 70.
  • 462-784-910, which is 14 times (33-56-65)
  • 546-728-910, which is (3-4-5) times 182.

Print the puzzles or type the solution on this excel file: 12 factors 905-913

910 is 190 in BASE 26, and 910 looks interesting in some other bases, too:

  • 4114 in BASE 6, because 4(6³) + 1(6²) + 1(6¹) + 4(6º) = 910
  • 1221 in BASE 9, because 1(9³) + 2(9²) + 2(9¹) + 1(9º) = 910
  • QQ in BASE 34 (Q is 26 base 10), because 26(34¹) + 26(34º) = 26(34 + 1) = 26(35) = 910
  • 26 0 BASE 35, because 26(35) + 0(1) = 26(35) = 910

What are the factors of 910?

  • 910 is a composite number.
  • Prime factorization: 910 = 2 × 5 × 7 × 13
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 910 has exactly 16 factors.
  • Factors of 910: 1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910
  • Factor pairs: 910 = 1 × 910, 2 × 455, 5 × 182, 7 × 130, 10 × 91, 13 × 70, 14 × 65, or 26 × 35
  • 910 has no square factors that allow its square root to be simplified. √910 ≈ 30.166206

 

902 and Level 5

902 is the hypotenuse of a Pythagorean triple:

198-880-902 which is 22 times (9-40-41)

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 902 is a composite number.
  • Prime factorization: 902 = 2 × 11 × 41
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 902 has exactly 8 factors.
  • Factors of 902: 1, 2, 11, 22, 41, 82, 451, 902
  • Factor pairs: 902 = 1 × 902, 2 × 451, 11 × 82, or 22 × 41
  • 902 has no square factors that allow its square root to be simplified. √902 ≈ 30.0333.

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