A Multiplication Based Logic Puzzle

Archive for the ‘Level 5 Puzzle’ Category

1017 and Level 5

You might find today’s puzzle to be a little trickier than most level 5 puzzles, but don’t let that deter you from giving it your best effort. For example, it’s true that 6 and 12 are both common factors of 60 and 36, but some of the other clues will eliminate either the 6 or the 12. Can you figure out which one gets eliminated?

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

Let me share some reasons 1017 is an interesting number.

24² + 21² = 1007

1017 is the hypotenuse of a Pythagorean triple:
135-1008-1017 which is 9 times (15-112-113). It can also be calculated
from 24² – 21², 2(24)(21), 24² + 21²

1017 is also palindrome 1771 in BASE 8 because 1(8³) + 7(8²) + 7(8¹) + 1(8⁰) = 1017

  • 1017 is a composite number.
  • Prime factorization: 1017 = 3 × 3 × 113, which can be written 1017 = 3² × 113
  • 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 1017 has exactly 6 factors.
  • Factors of 1017: 1, 3, 9, 113, 339, 1017
  • Factor pairs: 1017 = 1 × 1017, 3 × 339, or 9 × 113
  • Taking the factor pair with the largest square number factor, we get √1017 = (√9)(√113) = 3√113 ≈ 31.890437

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999 and Level 5

What are the common factors of 6 and 12? There are many: 1, 2, 3, and 6 are all common factors and possibilities for this puzzle. Only one of those numbers will work with the other clues to make this puzzle a multiplication table. That why 6 and 12 is NOT a good place to start when solving this puzzle. Look at all the other rows and columns with two or more clues. ONE of them will be the perfect place to start. Good luck!

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

999 is the smallest number whose digits add up to 27.

999 is 666 upside-down.

Did you know that 999² = 998001? That’s cool because 998 + 001 = 999. Numbers whose squares can be separated and then added together to get the original number are so unusual, they’ve been given a name, Kaprekar numbers. There are only seven Kaprekar numbers less than 999. Stetson.edu was my source for this fun 999 fact.

999 is the hypotenuse of a Pythagorean triple:
324-945-999 which is 27 times (12-35-37)

999 is a repdigit in base 10 and it looks interesting in some other bases, too:
It’s 4343 in BASE 6 because 4(6³) + 3(6²) + 4(6¹) + 3(6⁰) = 999,
It’s palindrome 515 in BASE 14 because 5(14²) + 1(14) + 5(1) = 999,
and it’s repdigit RR in BASE 36 (R is 27 base 10) because 27(36) + 27(1) = 27(37) = 999

  • 999 is a composite number.
  • Prime factorization: 999 = 3 × 3 × 3 × 37, which can be written 999 = 3³ × 37
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 999 has exactly 8 factors.
  • Factors of 999: 1, 3, 9, 27, 37, 111, 333, 999
  • Factor pairs: 999 = 1 × 999, 3 × 333, 9 × 111, or 27 × 37
  • Taking the factor pair with the largest square number factor, we get √999 = (√9)(√111) = 3√111 ≈ 31.60696

990 Christmas Factor Trees

Today’s puzzle has a couple of small Christmas trees in it. Don’t let their smallness fool you into thinking this is an easy puzzle. Can you solve it? Remember to use logic and not guess and check to find the solution.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

There are many interesting facts about the number 990:

9 × 10 × 11 = 990

Because 44 × 45/2 = 990, it is the 44th triangular number. That means that the sum of all the numbers from 1 to 44 is 990.

990 is the sum of the twelve prime numbers from 59 to 107.
It is also the sum of six consecutive prime numbers:
151 + 157 + 163 + 167 + 173 + 179 = 990,
and the sum of two consecutive primes:
491 + 499 = 990

990 is the hypotenuse of a Pythagorean triple:
594-792-990 which is (3-4-5) times 198

990 looks interesting in some other bases:
It is 6A6 in BASE 12 (A is 10 base 10) because 6(144) + 10(12) + 6(1) = 990,
2E2 in BASE 19 (E is 14 base 10) because 2(19²) + 14(19) + 2(1) = 990
1K1 in BASE 23 (K is 20 base 10) because 1(23²) + 20(23) + 1(1) = 990
UU in BASE 32 (U is 30 base 10) because 30(32) + 30(1) = 30(33) = 990
U0 in BASE 33 because 30(33) = 990

  • 990 is a composite number.
  • Prime factorization: 990 = 2 × 3 × 3 × 5 × 11, which can be written 990 = 2 × 3² × 5 × 11
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 990 has exactly 24 factors.
  • Factors of 990: 1, 2, 3, 5, 6, 9, 10, 11, 15, 18, 22, 30, 33, 45, 55, 66, 90, 99, 110, 165, 198, 330, 495, 990
  • Factor pairs: 990 = 1 × 990, 2 × 495, 3 × 330, 5 × 198, 6 × 165, 9 × 110, 10 × 99, 11 × 90, 15 × 66, 18 × 55, 22 × 45, or 30 × 33
  • Taking the factor pair with the largest square number factor, we get √990 = (√9)(√110) = 3√110 ≈ 31.464265

982 Red-Hot Cinnamon Candy

When I was a child I remember eating a red-hot cinnamon ball around the holidays. I really like cinnamon, but I wasn’t sure I liked how hot the candy was. I hope you enjoy today’s red-hot cinnamon candy puzzle.

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Now here’s something interesting about the number 982:

It is palindrome 292 in BASE 20 because 2(20²) + 9(20) + 2(1) = 982.

  • 982 is a composite number.
  • Prime factorization: 982 = 2 × 491
  • 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 982 has exactly 4 factors.
  • Factors of 982: 1, 2, 491, 982
  • Factor pairs: 982 = 1 × 982 or 2 × 491
  • 982 has no square factors that allow its square root to be simplified. √982 ≈ 31.336879

 

975 and Level 5

This puzzle has eleven clues, some of which are designed to trick you possibly. Will you be tricked, or will you use logic to figure out where to put the factors from 1 to 10 in the first column and the top row? The finished puzzle looks like a multiplication table but with the factors out of numerical order.

Print the puzzles or type the solution in this excel file: 10-factors-968-977

Here are a few facts about the number 975:

975 is the hypotenuse of SEVEN Pythagorean triples. Some factor pairs of 975 are highlighted in red next to those triples.
108-969-975 which is 3 times (36-323-325)
240-945-975 which is 15 times (16-63-65)
273-936-975 which is (7-24-25) times 39
375-900-975 which is (5-12-13) times 75
495-840-975 which is 15 times (33-56-65)
585-780-975 which is (3-4-5) times 195
612-759-975 which is 3 times (204-253-325)

1(5) + 2(5³) + 3(5²) + 4(5¹) + 5(5) = 975

You might think that last fact means that 975 is 12345 in base 5, but it isn’t. The only digits used in base 5 are 0, 1, 2, 3, and 4.

Here is 975 written in some different bases:
1111001111 in BASE 2 because 2⁹ + 2⁸ + 2⁷+ 2⁶ + 2³ + 2² + 2¹ + 2⁰ = 975
33033 in BASE 4 because 3(4⁴) + 3(4³) + 0(4²) + 3(4¹) + 3(4⁰) = 3(256 + 64 + 4 + 1) = 3(325) = 975
1717 in BASE 8 because 1(8³) + 7(8²) + 1(8¹) + 7(8⁰) = 975
303 in BASE 18 because 3(18²) + 0(18¹) + 3(18⁰) = 3(324 + 1) = 3(325) = 975

  • 975 is a composite number.
  • Prime factorization: 975 = 3 × 5 × 5 × 13, which can be written 975 = 3 × 5² × 13
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 975 has exactly 12 factors.
  • Factors of 975: 1, 3, 5, 13, 15, 25, 39, 65, 75, 195, 325, 975
  • Factor pairs: 975 = 1 × 975, 3 × 325, 5 × 195, 13 × 75, 15 × 65, or 25 × 39,
  • Taking the factor pair with the largest square number factor, we get √975 = (√25)(√39) = 5√39 ≈ 31.22499

 

 

964 Bow and Arrow

You can google lots of images of bows and arrows related to Thanksgiving. Perhaps that is because Indians from the Wampanoag Nation joined the Pilgrims in what we call the first Thanksgiving. I’m sure you know the story very well told from the white man’s point of view. It will be well worth your time to read The REAL Story of Thanksgiving as well. Everyone’s History Matters is also an excellent read.

Print the puzzles or type the solution in this excel file: 12 factors 959-967

 

Now I’ll tell you a little about the number 964:

964 is the sum of the fourteen prime numbers from 41 to 101.

It is also the sum of four consecutive prime numbers:
233 + 239 + 241 + 251 = 964

30² + 8²  = 964 making it the hypotenuse of a Pythagorean triple:
480-836-964 which is 4 times (120-209-241)

964 is a palindrome in 2 bases and a repdigit in another.
1022201 in BASE 3
7A7 in BASE 11 (A is 10 in base 10)
444 in BASE 15

  • 964 is a composite number.
  • Prime factorization: 964 = 2 × 2 × 241, which can be written 964 = 2² × 241
  • 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 964 has exactly 6 factors.
  • Factors of 964: 1, 2, 4, 241, 482, 964
  • Factor pairs: 964 = 1 × 964, 2 × 482, or 4 × 241
  • Taking the factor pair with the largest square number factor, we get √964 = (√4)(√241) = 2√241 ≈ 31.048349

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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