Mystery Level Puzzle

1299 Is This Puzzle a Real Turkey?

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Happy Thanksgiving, everyone!

Turkeys run but they cannot hide. They all will eventually end up on somebodies’ table. There doesn’t seem to be much of a mystery about that, but I’ve created a mystery level puzzle for today anyway. I promise it can be solved using logic and the basic facts in a 12 × 12 multiplication table.

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Now I’ll share some facts about the number 1299:

  • 1299 is a composite number.
  • Prime factorization: 1299 = 3 × 433
  • 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 1299 has exactly 4 factors.
  • Factors of 1299: 1, 3, 433, 1299
  • Factor pairs: 1299 = 1 × 1299 or 3 × 433
  • 1299 has no square factors that allow its square root to be simplified. √1299 ≈ 36.04164

1299 is the hypotenuse of a Pythagorean triple:
435-1224-1299 which is 3 times (145-408-433)

OEIS.org informs us that 8¹²⁹⁹ ≈ 1299 × 10¹¹⁷⁰. You can see it for yourself on a computer calculator!

 

1298 Another Mystery

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Hint #1: Which two clues will use both of the nines?
Hint #2: Which two clues will use both of the sixes?

6, 7, 8, or 12, two of them will use the 1’s. One of those 1’s will be in the first column. They both can’t be. That fact was important when I worked to solve this puzzle.

It won’t be easy, but why don’t you give it a try?

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

It won’t help to solve the puzzle, but I’ll share some information now about our puzzle number, 1298:

  • 1298 is a composite number.
  • Prime factorization: 1298 = 2 × 11 × 59
  • 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 1298 has exactly 8 factors.
  • Factors of 1298: 1, 2, 11, 22, 59, 118, 649, 1298
  • Factor pairs: 1298 = 1 × 1298, 2 × 649, 11 × 118, or 22 × 59
  • 1298 has no square factors that allow its square root to be simplified. √1298 ≈ 36.02777

1298 is the sum of four consecutive prime numbers:
313 + 317 + 331 + 337 = 1298

1297 Mystery

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Today’s puzzle has 12 clues, but it still presents quite a mystery. Will you be able to figure out where the factors from 1 to 12 go in the 1st column and the top row of the puzzle, or will you let this mystery stump you?

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Now I’ll share a little bit of information about the number 1297:

  • 1297 is a prime number.
  • Prime factorization: 1297 is prime.
  • The exponent of prime number 1297 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1297 has exactly 2 factors.
  • Factors of 1297: 1, 1297
  • Factor pairs: 1297 = 1 × 1297
  • 1297 has no square factors that allow its square root to be simplified. √1297 ≈ 36.01389

How do we know that 1297 is a prime number? If 1297 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1297 ≈ 36.01. Since 1297 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1297 is a prime number.

1297 is the sum of two squares:
36² +  1² = 1297

1297 is the hypotenuse of a Pythagorean triple:
72-1295-1297 calculated from 2(36)(1), 36² –  1², 36² +  1²

Here’s another way we know that 1297 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 36² + 1² = 1297 with 36 and 1 having no common prime factors, 1297 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1297 ≈ 36.01. Since 1297 is not divisible by 5, 13, 17, or 29, we know that 1297 is a prime number.

1295 Mystery Level

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This pinwheel-shaped puzzle is no little kid’s toy. The logic needed to solve it might be a mystery to see, but it is still there!

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Here are some facts about the number 1295:

  • 1295 is a composite number.
  • Prime factorization: 1295 = 5 × 7 × 37
  • 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 1295 has exactly 8 factors.
  • Factors of 1295: 1, 5, 7, 35, 37, 185, 259, 1295
  • Factor pairs: 1295 = 1 × 1295, 5 × 259, 7 × 185, or 35 × 37
  • 1295 has no square factors that allow its square root to be simplified. √1295 ≈ 35.98611

Since 35 × 37 = 1295, we know that 1295 is one number less than 36².

1295 is the hypotenuse of FOUR Pythagorean triples:
399-1232-1295 which is 7 times (57-176-185)
420-1225-1295 which is 35 times (12-35-37)
728-1071-1295 which is 7 times (104-153-185)
777-1036-1295 which is (3-4-5) times 259

1288 Mystery Puzzle

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How difficult is this mystery level puzzle? That is part of the mystery, but I assure you that if you use logic and basic multiplication facts you can find the unique solution.

Print the puzzles or type the solution in this excel file: 10-factors-1281-1288

That was puzzle #1288. Now I’ll tell you some facts about the number 1288.

  • 1288 is a composite number.
  • Prime factorization: 1288 = 2 × 2 × 2 × 7 × 23, which can be written 1288 = 2³ × 7 × 23
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1288 has exactly 16 factors.
  • Factors of 1288: 1, 2, 4, 7, 8, 14, 23, 28, 46, 56, 92, 161, 184, 322, 644, 1288
  • Factor pairs: 1288 = 1 × 1288, 2 × 644, 4 × 322, 7 × 184, 8 × 161, 14 × 92, 23 × 56, or 28 × 46
  • Taking the factor pair with the largest square number factor, we get √1288 = (√4)(√322) = 2√322 ≈ 37.88872

1288 has four factor pairs that contain only even factors so 1288 can be written as the difference of two squares four different ways:
323² – 321² = 1288
163² – 159² = 1288
53² – 39² = 1288
37² – 9² = 1288

 

1287 Mystery Level

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I’ve put twenty clues in this mystery level puzzle. Some of the factors will be easy to find, but some of them won’t be quite as easy. Use logic the entire time, and you will be able to solve it!

Print the puzzles or type the solution in this excel file: 10-factors-1281-1288

Now I’ll share some facts about the number 1287:

  • 1287 is a composite number.
  • Prime factorization: 1287 = 3 × 3 × 11 × 13, which can be written 1287 = 3² × 11 × 13
  • 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 1287 has exactly 12 factors.
  • Factors of 1287: 1, 3, 9, 11, 13, 33, 39, 99, 117, 143, 429, 1287
  • Factor pairs: 1287 = 1 × 1287, 3 × 429, 9 × 143, 11 × 117, 13 × 99, or 33 × 39
  • Taking the factor pair with the largest square number factor, we get √1287 = (√9)(√143) = 3√143 ≈ 35.87478

1287 is the hypotenuse of a Pythagorean triple:
495-1188-1287 which is (5-12-13) times 99

1270 What’s Brewing on My 5-Year Blogiversary

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As Halloween approaches, I remember that five years ago today, I hit the publish button for the first time, and my puzzles became available for anyone with an internet connection to use.

Today’s puzzle looks a little bit like a cauldron. What’s brewing on my 5-year blogiversary?

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

I continue to be very grateful to WordPress and the WordPress community for making blogging and publishing easy and enjoyable. I am also very grateful to my readers who have done so much to make this blog grow.

I’m a lot busier now than I was five years ago. Besides blogging, I have a full-time job and a part-time job. I like both of these jobs because I like helping students understand mathematics better. Sometimes I don’t have the time I would like to work on my blog. Nevertheless, I still have blogging goals I want to reach so lately I find myself playing catch-up more often than not.

Now I’ll write a little about the number 1270:

  • 1270 is a composite number.
  • Prime factorization: 1270 = 2 × 5 × 127
  • 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 1270 has exactly 8 factors.
  • Factors of 1270: 1, 2, 5, 10, 127, 254, 635, 1270
  • Factor pairs: 1270 = 1 × 1270, 2 × 635, 5 × 254, or 10 × 127
  • 1270 has no square factors that allow its square root to be simplified. √1270 ≈ 35.63706

1270 is the hypotenuse of a Pythagorean triple:
762-1016-1270 which is (3-4-5) times 254

1268 Halloween Cat Mystery

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Cats can be quite mysterious. They are a favorite pet for many every day, even though suspicious stories abound about them on Halloween. Can you solve the mystery of this cat-like puzzle?

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

Now I’ll share a few facts about the number 1268:

  • 1268 is a composite number.
  • Prime factorization: 1268 = 2 × 2 × 317, which can be written 1268 = 2² × 317
  • 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 1268 has exactly 6 factors.
  • Factors of 1268: 1, 2, 4, 317, 634, 1268
  • Factor pairs: 1268 = 1 × 1268, 2 × 634, or 4 × 317
  • Taking the factor pair with the largest square number factor, we get √1268 = (√4)(√317) = 2√317 ≈ 35.60899

28² + 22² = 1268

1268 is the hypotenuse of a Pythagorean triple:
300-1232-1268 calculated from 28² – 22², 2(28)(22), 28² + 22².
It is also 4 times (75-308-317)

1267 Frankenstein Mystery

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There are legends of Dr. Frankenstein creating a monster years ago. Nowadays Frankenstein’s Monster can often be seen walking through neighborhoods on Halloween night. This puzzle looks a little bit like him.

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

But if you take all the color away, he looks completely different and quite harmless:

Now I’ll share some information about the number 1267:

  • 1267 is a composite number.
  • Prime factorization: 1267 = 7 × 181
  • 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 1267 has exactly 4 factors.
  • Factors of 1267: 1, 7, 181, 1267
  • Factor pairs: 1267 = 1 × 1267 or 7 × 181
  • 1267 has no square factors that allow its square root to be simplified. √1267 ≈ 35.59494

1267 is the sum of nine consecutive prime numbers:
113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 = 1267

1267 is the hypotenuse of a Pythagorean triple:
133-1260-1267 which is 7 times (19-180-181)

 

1265 More Candy Corn

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People don’t each candy corn every day and the only corn in it is corn syrup.  We usually only see it this time of year. Here’s a puzzle with some candy corn for you to enjoy:

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

  • 1265 is a composite number.
  • Prime factorization: 1265 = 5 × 11 × 23
  • 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 1265 has exactly 8 factors.
  • Factors of 1265: 1, 5, 11, 23, 55, 115, 253, 1265
  • Factor pairs: 1265 = 1 × 1265, 5 × 253, 11 × 115, or 23 × 55
  • 1265 has no square factors that allow its square root to be simplified. √1265 ≈ 35.56684

1265 is the sum of the fifteen prime numbers from 53 to 113.

1265 is the hypotenuse of a Pythagorean triple:
759-1012-1265 which is (3-4-5) times 253