A Multiplication Based Logic Puzzle

Posts tagged ‘Puzzle’

1018 and Level 6

Level 6 puzzles can be tricky to solve, but I promise that you can still solve this one using logic and knowledge of the basic multiplication table. Just write the numbers from 1 to 12 in both the first column and the top row so that the puzzle is like a partially filled out multiplication table with the factors in a different order. Like always, there is only one solution. Can you find it?

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

Look at these interesting facts about the number 1018:

27² + 17² = 1018
That means that 1018 is the hypotenuse of a Pythagorean triple:
440-918-1018 calculated from 27² – 17², 2(27)(17), 27² + 17²

It also means that (44² – 10²)/2 = 1018
Note that 27 + 17 = 44 and 27 – 17 = 10

1018 is full house 33322 in BASE 4 because 3(4⁴) + 3(4³) + 3(4²) + 2(4¹) + 2(4⁰) = 3(256 + 64 + 16) + 2(4 + 1) = 1018

Since 1018 is the sum of odd squares, it is divisible by 2. Since those odd squares have no common prime factors, you only have to check to see if 1018 is divisible by any Pythagorean triple hypotenuses less than or equal to (√1018)/2 ≈ 15.953. It is not divisible by 5 or 13, therefore 1018 only has two prime factors: 2 and 1018/2.

  • 1018 is a composite number.
  • Prime factorization: 1018 = 2 × 509
  • 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 1018 has exactly 4 factors.
  • Factors of 1018: 1, 2, 509, 1018
  • Factor pairs: 1018 = 1 × 1018 or 2 × 509
  • 1018 has no square factors that allow its square root to be simplified. √1018 ≈ 31.90611

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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

1016 and Level 4

If you’ve never tried a level 4 puzzle before, this is a great one to try. Of its twelve clues, eight have only one factor pair in which both factors are from 1 to 12. You should easily be able to place the factors for those eight clues. Since each factor from 1 to 12 must appear exactly one time in the first column and the top row, the factors from those eight clues will eliminate some of the possible factors of the other four clues. Don’t be afraid to give this puzzle a try!

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

Here are some reasons why 1016 is an interesting number:

1016 is the sum of seven consecutive powers of two:
2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ + 2³ = 1016
I know that’s true because 1016 is 1111111000 in BASE 2

1016 is a palindrome in a couple of bases as well:
It’s 13031 in BASE 5 because 1(5⁴) + 3(5³) + 0(5²) + 3(5¹) + 1(5⁰) = 1016
161 in BASE 29 because 1(29²) + 6(29¹) + 1(29⁰) = 1016

1016 is divisible by 2 because 6 is even.
1016 is divisible by 4 because 16 is divisible by 4. (And also because 6 is divisible by 2 but NOT by 4 and 1 is odd.)
1016 is divisible by 8 because 016 is divisible by 8. (And because 16 is divisible by 8 and 0 is an even number.
1016 is NOT divisible by 16 because 016 is divisible by 16 and 1 is an odd number.

  • 1016 is a composite number.
  • Prime factorization: 1016 = 2 × 2 × 2 × 127, which can be written 1016 = 2³ × 127
  • 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 1016 has exactly 8 factors.
  • Factors of 1016: 1, 2, 4, 8, 127, 254, 508, 1016
  • Factor pairs: 1016 = 1 × 1016, 2 × 508, 4 × 254, or 8 × 127
  • Taking the factor pair with the largest square number factor, we get √1016 = (√4)(√254) = 2√254 ≈ 31.87475

1015 and Level 3

If you glance at this puzzle for a few seconds, you may think there are three places in the top row where the number 1 will work, but in actuality, only one of those places will work with all the other clues in the puzzle. This is a level 3 puzzle, so start with the top cell in the first column and work down cell by cell placing factors in both the first column and the top row until the puzzle resembles a multiplication table.

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

Here are a few facts about the number 1015:

1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² + 14² = 1015. That makes 1015 the 14th square pyramidal number.

331 + 337 + 347 = 1015. That’s the sum of 3 consecutive prime numbers.

1015 is the hypotenuse of four Pythagorean triples:
119-1008-1015
168-1001-1015
609-812-1015
700-735-1015

1015 looks interesting when it is written is some other bases:
707 in BASE 12
1D1 in BASE 26 (D is 13 base 10)
TT in BASE 34 (T is 29 base 10)
T0 in BASE 35

  • 1015 is a composite number.
  • Prime factorization: 1015 = 5 × 7 × 29
  • 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 1015 has exactly 8 factors.
  • Factors of 1015: 1, 5, 7, 29, 35, 145, 203, 1015
  • Factor pairs: 1015 = 1 × 1015, 5 × 203, 7 × 145, or 29 × 35
  • 1015 has no square factors that allow its square root to be simplified. √1015 ≈ 31.85906

Level 2 and Simplifying √1014

Have you memorized a basic multiplication table? If you have, then you can solve this puzzle. The numbers being multiplied together aren’t where they are in a regular multiplication table, but you can still easily figure out where they need to go. There is only one solution. I bet you can find it!

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

If I wanted to find √1014, I would first check to see if it were divisible by 4 or by 9 because most numbers whose square roots can be simplified are divisible by 4 or by 9 or both.
1014 isn’t divisible by 4 because 14 isn’t divisible by 4.
It isn’t divisible by 9 because 1 + 0 + 1 + 4 = 6, and 6 is not divisible by 9.
However, it is divisible by both 2 and 3 and thus also by 6. Since most people are less likely to make a mistake dividing by 6 in ONE step instead of two, I would make a little division cake and do that division first:

Recognizing that 169 is a perfect square, I would then take the square root of everything on the outside of my little cake. (√6)(√169) = 13√6

***********************

Here are some facts about the number 1014:

Because 13² is one of its factors, 1014 is the hypotenuse of two Pythagorean triples:
714-720-1014 which is 6 times (119-120-169),
390-936-1014 which is (5-12-13) times 78

1014 looks interesting when written in some other bases:
It’s 600 in BASE 13 because 6(13²) = 6 (169) = 1014,
and 222 in BASE 22 because 2(22²) + 2(22) + 2(1) = 2(484 + 22 + 1) = 2(507) = 1014

  • 1014 is a composite number.
  • Prime factorization: 1014 = 2 × 3 × 13 × 13, which can be written 1014 = 2 × 3 × 13²
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1014 has exactly 12 factors.
  • Factors of 1014: 1, 2, 3, 6, 13, 26, 39, 78, 169, 338, 507, 1014
  • Factor pairs: 1014 = 1 × 1014, 2 × 507, 3 × 338, 6 × 169, 13 × 78, or 26 × 39,
  • Taking the factor pair with the largest square number factor, we get √1014 = (√169)(√6) = 13√6 ≈ 31.84337

1013 Wolf Moon Mystery

There will be a full moon tonight. Since it is the first full moon of the year, it is called a wolf moon. I learned that fact from reading a short post called Supermoon by the Lycanthropic author, Steve Morris. Wolves love howling at the moon. It has been said that some men have even howled at the moon…

It all sounds pretty mysterious to me, so I made this mystery level puzzle look a little like a full moon. The number 12 appears three times in it on a nice little arc of the circle.  You can work on solving this puzzle under the light of tonight’s full wolf moon … if you dare.

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

I will be having company for the next week. Just to make it easier for me I’m taking a week off from blogging. I hope you’ll rejoin me then.

1011 A Toast to My Readers

Thank you, dear readers, for stopping by today and/or any other time you have visited! My blog had the best year ever in 2017, and it is all because of you. I sincerely hope that I have been of service to you.

Here’s to you, dear friend. May lifting each other help us both to climb even higher this coming year!

May you be able to find the factors in this puzzle and, more importantly, the factors in life that will bring you happiness and success!

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

Let me tell you some things about the number 1011 that you probably didn’t know before:

The only nonzero digits in 1011 are three 1’s so 1011 is divisible by 3, and 1011 is included in this interesting pattern:

1011² = 1022121; Notice that the digits in bold are 1011 and all the other digits are 2’s. Thank you Stetson.edu for that fun fact.

1011 looks interesting in a few other bases:
It’s 33303 in BASE 4 because 3(4⁴ + 4³ + 4² + 4⁰) = 3(256 + 64 + 16 + 1) = 3(337) = 1011,
323 in BASE 18 because 3(18²) + 2(18) + 3(1) = 1011

1011 is the hypotenuse of a Pythagorean triple:
525-864-1011 which is 3 times (175-288-337)

  • 1011 is a composite number.
  • Prime factorization: 1011 = 3 × 337
  • 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 1011 has exactly 4 factors.
  • Factors of 1011: 1, 3, 337, 1011
  • Factor pairs: 1011 = 1 × 1011 or 3 × 337
  • 1011 has no square factors that allow its square root to be simplified. √1011 ≈ 31.7962

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