Mystery Level Puzzle

1030 Cupid’s Arrow

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Love can often be like a puzzle. When cupid’s arrow hits its mark, at first everything might seem to fall into place, but before long, love starts getting complicated and has to be figured out.

That’s the way this Cupid’s Arrow puzzle is, too. It’s easy to find the logic to start it, but then the logic will be more difficult to see. May you be able to figure out this puzzle as well as the important relationships in your life!

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

What can I tell you about the number 1030?

It’s the sum of two consecutive prime numbers:
509 + 521 = 1030

It’s the hypotenuse of a Pythagorean triple:
618-824-1030 which is (3-4-5) times 206

It’s palindrome 1102011 in BASE 3 because 3⁶ + 3⁵ + 2(3³) + 3¹ + 3⁰ = 1030

  • 1030 is a composite number.
  • Prime factorization: 1030 = 2 × 5 × 103
  • 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 1030 has exactly 8 factors.
  • Factors of 1030: 1, 2, 5, 10, 103, 206, 515, 1030
  • Factor pairs: 1030 = 1 × 1030, 2 × 515, 5 × 206, or 10 × 103
  • 1030 has no square factors that allow its square root to be simplified. √1030 ≈ 32.09361

 

1029 A Rose for Your Valentine

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A dozen roses can be pretty pricey around Valentine’s Day, but at least one website asserts that a single rose can make just as big a statement and just as big an impact. Today’s mystery level puzzle looks like a single rose. I hope you will enjoy its beauty even if its thorns are prickly.

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

Here are a few facts about the number 1029:

It’s easy to see that 1029 can be evenly divided by 3 because 1 + 0 + 2 + 9 = 12, a number divisible by 3.

It’s not quite as easy to tell that it can be evenly divided by 7:
It is because 102 – 2(9) = 102 – 18 = 84, a number divisible by 7.

I like the way 1029 looks when it is written in some other bases:
It’s 4433 in BASE 6 because 4(6³) + 4(6²) + 3(6¹) + 3(6⁰) = 4(216 + 36) + 3(6 + 1) = 1029,
3000 in BASE 7 because 3(7³) = 3(343) = 1029,
399 in BASE 17 because 3(17²) + 9(17) + 9(1) = 3(289 + 51 + 3) = 3(343) = 1029
333 in BASE 18 because 3(18² + 18 + 1) = 3(343) = 1029

  • 1029 is a composite number.
  • Prime factorization: 1029 = 3 × 7 × 7 × 7, which can be written 1029 = 3 × 7³
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (1 + 1)(3 + 1) = 2 × 4 = 8. Therefore 1029 has exactly 8 factors.
  • Factors of 1029: 1, 3, 7, 21, 49, 147, 343, 1029
  • Factor pairs: 1029 = 1 × 1029, 3 × 343, 7 × 147, or 21 × 49
  • Taking the factor pair with the largest square number factor, we get √1029 = (√49)(√21) = 7√21 ≈ 32.07803

 

1028 A Valentine Mystery

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Valentine’s Day is almost here so I’ve made three Valentine related puzzles this week. I’ve labeled all of three of them Mystery Level because you might find some of them to be difficult. Use logic and an ordinary 12 × 12 multiplication table. I promise that each one of them can be solved, and I hope that you LOVE working on them! There will be some easier puzzles later on in the week.

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

Let me tell you a little about the number 1028:

32² + 2² = 1028 so 1028 is the hypotenuse of a Pythagorean triple:
128-1020-1028 which is 4 times (32-255-257) and can be calculated from 2(32)(2), 32² – 2², 32² + 2²

1028 is a palindrome when it is written in a couple of different bases:
404 in BASE 16 because 4(16²) + 4(1) = 4(257) = 1028
2G2 in BASE 19 (G is 16 base 10) because 2(19²) + 16(19) + 2(1) = 1028

  • 1028 is a composite number.
  • Prime factorization: 1028 = 2 × 2 × 257, which can be written 1028 = 2² × 257
  • 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 1028 has exactly 6 factors.
  • Factors of 1028: 1, 2, 4, 257, 514, 1028
  • Factor pairs: 1028 = 1 × 1028, 2 × 514, or 4 × 257
  • Taking the factor pair with the largest square number factor, we get √1028 = (√4)(√257) = 2√257 ≈ 32.062439

 

1026 One Last Mystery

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I made a week’s worth of mystery level puzzles, and today’s puzzle is the last one in the set. Find the Factors of the clues in this puzzle by using logic and your knowledge of the multiplication table. It may not be easy for you, but give it a try anyway. If you find it too difficult, I will soon publish some more easier-level puzzles.

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

Now let me share a few facts about the number 1026:

1026 is the sum of the fourteen prime numbers from 43 to 103.

I like the way 1026 looks when it is written in these other bases:
It’s 2002 in BASE 8,
396 in BASE 17,
330 in BASE 18,
1G1 in BASE 25 (G is 16 base 10), and
123 in BASE 31

  • 1026 is a composite number.
  • Prime factorization: 1026 = 2 × 3 × 3 × 3 × 19, which can be written 1026 = 2 × 3³ × 19
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 4 × 2 = 16. Therefore 1026 has exactly 16 factors.
  • Factors of 1026: 1, 2, 3, 6, 9, 18, 19, 27, 38, 54, 57, 114, 171, 342, 513, 1026
  • Factor pairs: 1026 = 1 × 1026, 2 × 513, 3 × 342, 6 × 171, 9 × 114, 18 × 57, 19 × 54, or 27 × 38
  • Taking the factor pair with the largest square number factor, we get √1026 = (√9)(√114) = 3√114 ≈ 32.03123

1025 Mystery Date

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It is a mystery why we in the United States write our dates “month-day-year”. It makes about as much sense as saying larger, large, largest or better, good, best. Nevertheless, it is what it is.

So today in the United States it is 2-7-18, the e-day of the century. It’s not quite as exciting as 2-7-1828 might have been, but still pretty exciting. e is also known as Euler’s number, and like pi, it is an irrational number. A college professor of mine taught me how to remember its first few digits by remembering 2.7, the year 1828 twice, and 45-90-45 (that very important isosceles triangle). Thus, e ≈ 2.718281828459045.

The difficulty level of today’s puzzle is also a mystery. Nevertheless, you can still solve it by applying logic and facts from a simple 10×10 multiplication table:

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

Here are a few facts about the number 1025:

1025 can be written as the sum of consecutive prime numbers two different ways:
97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 2025; that’s nine consecutive prime numbers.
71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 2025; that’s eleven consecutive prime numbers.

1025 is the sum of two squares three different ways:
25² + 20² = 1025
32² + 1² = 1025
31² + 8² = 1025

That previous fact contributes to the fact that 1025 is the hypotenuse of SEVEN Pythagorean triples:
64-1023-1025 calculated from 2(32)(1), 32² – 1², 32² + 1²
225-1000-1025 which is 25 times (9-40-41) and can also be calculated from 25² – 20², 2(25)(20), 25² + 20²
287-984-1025 which is (7-24-25) times 41
420-935-1025 which is 5 times (84-187-205)
496-897-1025 calculated from 2(31)(8), 31² + 8², 31² + 8²
615-820-1025 which is (3-4-5) times 205
665-780-1025 which is 5 times (133-156-205)

1025 is also a wonderful palindrome in three different bases.
10000000001 in BASE 2
100001 in BASE 4
101 in BASE 32

  • 1025 is a composite number.
  • Prime factorization: 1025 = 5 × 5 × 41, which can be written 1025 = 5² × 41
  • 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 1025 has exactly 6 factors.
  • Factors of 1025: 1, 5, 25, 41, 205, 1025
  • Factor pairs: 1025 = 1 × 1025, 5 × 205, or 25 × 41
  • Taking the factor pair with the largest square number factor, we get √1025 = (√25)(√41) = 5√41 ≈ 32.01562

 

Mysterious 1023

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2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ + 2³ + 2² + 2¹ + 2⁰ = 1023. That makes 1023 a pretty cool and rather mysterious number.

This puzzle that I’ve numbered 1023 is pretty cool and mysterious, too. I’m sure you will enjoy solving it if you only use logic to find the solution.

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

Here are some other fascinating facts about the number 1023:

It is formed by using a zero and three other consecutive numbers, so it is divisible by 3.

1 – 0 + 2 – 3 = 0, so 1023 is divisible by eleven.

31 × 33 = 1023 so (32 – 1)(32 + 1) = 1023, AND it is 32² – 1, making it one away from the next square number!

It is the sum of five consecutive prime numbers:
193 + 197 + 199 + 211 + 223 = 1023

1023 looks quite interesting when it is written in several different bases:
First of all, it’s 1111111111 in BASE 2 because it is the sum of the all those powers of 2 from 0 to 9 that were included at the top of this post.

It’s also 33333 in BASE 4 because 3(4⁴ + 4³ + 4² + 4¹ + 4⁰) = 3(341) = 1023.
That also means that 3(2⁸ + 2⁶ + 2⁴ + 2² + 2⁰) = 1023

It’s 393 in BASE 17 because 3(17²) + 9(17) + 3(1) = 1023,
VV in BASE 32 (V is 31 base 10) because 31(32) + 31(1) = 31(33) = 1023, and
V0 in BASE 33 because 31(33) = 1023

  • 1023 is a composite number.
  • Prime factorization: 1023 = 3 × 11 × 31
  • 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 1023 has exactly 8 factors.
  • Factors of 1023: 1, 3, 11, 31, 33, 93, 341, 1023
  • Factor pairs: 1023 = 1 × 1023, 3 × 341, 11 × 93, or 31 × 33
  • 1023 has no square factors that allow its square root to be simplified. √1023 ≈ 31.98437

1022 Friedman Number Mystery

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1022 is the 15th Friedman number. “What is a Friedman number and why is 1022 one of them?” you may ask. I will solve that little mystery for you. 1022 is a Friedman number because
2¹⁰ – 2 = 1022. Notice that the expression 2¹⁰ – 2 uses the digits 1, 0, 2, and 2 in some order and a subtraction sign. A Friedman number can be written as an expression that uses all of its own digits the exact number of times that they occur in the number. The expression must include at least one operator (+, -, ×, ÷) or a power. Parenthesis are allowed as long as the other rules are followed.

Now I would like you to solve the mystery of this puzzle using logic and the multiplication facts. Can you do it?

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

1022 is the hypotenuse of a Pythagorean triple:
672-770-1022 which is 14 times (48-55-73)

  • 1022 is a composite number.
  • Prime factorization: 1022 = 2 × 7 × 73
  • 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 1022 has exactly 8 factors.
  • Factors of 1022: 1, 2, 7, 14, 73, 146, 511, 1022
  • Factor pairs: 1022 = 1 × 1022, 2 × 511, 7 × 146, or 14 × 73
  • 1022 has no square factors that allow its square root to be simplified. √1022 ≈ 31.96873

1021 Mystery Level

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Here is the second puzzle in a week’s worth of mystery level puzzles. Will it be very difficult or not so bad? That’s the mystery. You’ll have to try it to know for sure. You only need to use logic and your knowledge of the multiplication table to solve it. I wish you luck!

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

That puzzle may have been a mystery, but the number 1021 isn’t much of a mystery at all. It is the second number of twin primes, 1019 and 1021.

Since it is a prime number, and it has a remainder of one when it is divided by 4, it can be written as the sum of two squares:
30² + 11² = 1021.

Since it can be written as the sum of two squares, it is the hypotenuse of a Pythagorean triple:
660-779-1021 calculated from 2(30)(11), 30² – 11², 30² + 11²

It is also palindrome 141 in BASE 30 because 1(30²) + 4(30) + 1(1) = 1021

  • 1021 is a prime number.
  • Prime factorization: 1021 is prime.
  • The exponent of prime number 1021 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1021 has exactly 2 factors.
  • Factors of 1021: 1, 1021
  • Factor pairs: 1021 = 1 × 1021
  • 1021 has no square factors that allow its square root to be simplified. √1021 ≈ 31.95309

How do we know that 1021 is a prime number? If 1021 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1021 ≈ 31.95. Since 1021 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1021 is a prime number.

Here’s another way we know that 1021 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 11² = 1021 with 30 and 11 having no common prime factors, 1021 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1021 ≈ 31.95. Since 1021 is not divisible by 5, 13, 17, or 29, we know that 1021 is a prime number.

1020 A Week of Mystery

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Sometimes changing things up a little is good. I decided to make a week’s worth of mystery level puzzles. The actual difficulty level will vary from puzzle to puzzle so give each one of them a try. If you think one is too easy or too difficult, the next one might not be. Here’s the first one:

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

Here’s a little bit about the number 1020:

It is only 8 more than 1008, the previous number that also had 24 factors. Here are a few of its MANY possible factor trees.

1020 is the sum of six consecutive prime numbers:
157 + 163 + 167 + 173 + 179 + 181 = 1020

1020 is the hypotenuse of four Pythagorean triples:
156-1008-1020 which is 12 times (13-84-85)
432-924-1020 which is 12 times (36-77-85)
480-900-1020 which is (8-15-17) times 60
612-816-1020 which is (3-4-5) times 204

1020 looks interesting when it is written using some different bases:
It’s 33330 in BASE 4 because 3(4⁴ + 4³ + 4² + 4¹) = 3(340) = 1020,
848 in BASE 11 because 8(11²) + 4(11) + 8(1) = 1020,
606 in BASE 13 because 6(13²) + 6(1) = 6(170) = 1020,
480 in BASE 15 because 4(15²) + 8(15) = 4(225 + 30) = 4(255) = 1020,
390 in BASE 17 because 3(17²) + 9(17) = 3(289 + 51) = 3(340) = 1020,
UU in BASE 33 (U is 30 base 10) because 30(33) + 30(1) = 30(34) = 1020, and
U0 in BASE 34 because 30(34) = 1020

  • 1020 is a composite number.
  • Prime factorization: 1020 = 2 × 2 × 3 × 5 × 17, which can be written 1020 = 2² × 3 × 5 × 17
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1020 has exactly 24 factors.
  • Factors of 1020: 1, 2, 3, 4, 5, 6, 10, 12, 15, 17, 20, 30, 34, 51, 60, 68, 85, 102, 170, 204, 255, 340, 510, 1020
  • Factor pairs: 1020 = 1 × 1020, 2 × 510, 3 × 340, 4 × 255, 5 × 204, 6 × 170, 10 × 102, 12 × 85, 15 × 68, 17 × 60, 20 × 51, or 30 × 34
  • Taking the factor pair with the largest square number factor, we get √1020 = (√4)(√255) = 2√255 ≈ 31.9374

1013 Wolf Moon Mystery

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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.