1231 Mystery Level Puzzle

For almost all the sets of clues in this puzzle, there is more than one permissible common factor. That makes the puzzle a little tricky, but with care, you can still solve it using logic and your knowledge of the basic 10×10 multiplication table. Good luck!

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Now I’ll tell you a little bit about the number 1231:

  • 1231 is a prime number.
  • Prime factorization: 1231 is prime.
  • The exponent of prime number 1231 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1231 has exactly 2 factors.
  • Factors of 1231: 1, 1231
  • Factor pairs: 1231 = 1 × 1231
  • 1231 has no square factors that allow its square root to be simplified. √1231 ≈ 35.08561

How do we know that 1231 is a prime number? If 1231 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1231 ≈ 35.1. Since 1231 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1231 is a prime number.

1231 is a palindrome in a couple of bases:
It’s A1A in BASE 11 because 10(11²) + 1(11) + 10(1) = 1231, and
it’s 1B1 in BASE 30 because 1(30²) + 11(30) + 1(1) = 1231

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1230 Mystery

Is this mystery level puzzle easy or difficult? The only way to know for sure is to start filling in the factors. Don’t guess and check. Use logic to find its unique solution!

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Here is some information about the number 1230:

1230 ends with a zero so it is divisible by 2 and 5.
It is a number formed by three consecutive numbers and a zero so it is divisible by 3.

  • 1230 is a composite number.
  • Prime factorization: 1230 = 2 × 3 × 5 × 41
  • 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 1230 has exactly 16 factors.
  • Factors of 1230: 1, 2, 3, 5, 6, 10, 15, 30, 41, 82, 123, 205, 246, 410, 615, 1230
  • Factor pairs: 1230 = 1 × 1230, 2 × 615, 3 × 410, 5 × 246, 6 × 205, 10 × 123, 15 × 82, or 30 × 41
  • 1230 has no square factors that allow its square root to be simplified. √1230 ≈ 35.07136

1230 is the hypotenuse of four Pythagorean triples:
270-1200-1230 which is 30 times (9-40-41)
504-1122-1230 which is 6 times (84-187-205)
738-984-1230 which is (3-4-5) times 246
798-936-1230 which is 6 times (133-156-205)

 

1226 Happy Birthday to My Sister, Sue

I don’t make puzzles bigger than 12 × 12 very often, but I decided to make this one, a 17 × 17 Mystery Level for my sister’s birthday. I know she can solve smaller ones without any problems, so I wanted to give her a challenge. Happy birthday, Sue. I hope you have a great day and enjoy solving this one.

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Note that with a bigger table there are several more possible common factors:

Is 4, 8, or 16 the common factor needed for 64 and 32 or for 16 and 48?
Is 7 or 14 the common factor needed for 14 and 70?
Is 6, 10, or 15 the common factor needed for 60 and 90?

As always there is only one solution. The table below will help anyone not familiar with some of the lesser known multiplication facts needed to solve the puzzle.

Now I’ll share some information about the number 1226:

  • 1226 is a composite number.
  • Prime factorization: 1226 = 2 × 613
  • 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 1226 has exactly 4 factors.
  • Factors of 1226: 1, 2, 613, 1226
  • Factor pairs: 1226 = 1 × 1226 or 2 × 613
  • 1226 has no square factors that allow its square root to be simplified. √1226 ≈ 35.01428

35² + 1² = 1226

1226 is the hypotenuse of a Pythagorean triple:
70-1224-1226 calculated from 2(35)(1), 35² – 1², 35² + 1²

1218 Mystery Level

Working on this Mystery level puzzle may seem easy at first, but will it remain easy until its unique solution is found? Only those who solve it will know the answer to that question!

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Here are some facts about the number 1218:

  • 1218 is a composite number.
  • Prime factorization: 1218 = 2 × 3 × 7 × 29
  • 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 1218 has exactly 16 factors.
  • Factors of 1218: 1, 2, 3, 6, 7, 14, 21, 29, 42, 58, 87, 174, 203, 406, 609,
  • 1218
  • Factor pairs: 1218 = 1 × 1218, 2 × 609, 3 × 406, 6 × 203, 7 × 174, 14 × 87, 21 × 58, or 29 × 42
  • 1218 has no square factors that allow its square root to be simplified. √1218 ≈ 34.89986

1218 is the hypotenuse of a Pythagorean triple:
840-882-1218 which is (20-21-29) times 42

 

1217 Squeegee Mystery

The murder weapon for today’s mystery appears to be a squeegee! Study all the clues in the puzzle and see if you can find all the needed factors to solve this mystery. As in every whodunnit, there is only one true solution.

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Now here are a few facts about the number 1217:

  • 1217 is a prime number.
  • Prime factorization: 1217 is prime.
  • The exponent of prime number 1217 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1217 has exactly 2 factors.
  • Factors of 1217: 1, 1217
  • Factor pairs: 1217 = 1 × 1217
  • 1217 has no square factors that allow its square root to be simplified. √1217 ≈ 34.88553

How do we know that 1217 is a prime number? If 1217 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1217 ≈ 34.9. Since 1217 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1217 is a prime number.

31² + 16² = 1217

1217 is the hypotenuse of a Pythagorean triple:
705-992-1217 calculated from 31² – 16², 2(31)(16), 31² + 16²

Here’s another way we know that 1217 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 16² = 1217 with 31 and 16 having no common prime factors, 1217 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1217 ≈ 34.9. Since 1217 is not divisible by 5, 13, 17, or 29, we know that 1217 is a prime number.

1209 Mystery Level

How easy or difficult is this mystery level puzzle? That’s part of the mystery! Once you solve it, you will know, and you don’t have to tell let anybody else know.

Print the puzzles or type the solution in this excel file: 10-factors-1199-1210

  • 1209 is a composite number.
  • Prime factorization: 1209 = 3 × 13 × 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 1209 has exactly 8 factors.
  • Factors of 1209: 1, 3, 13, 31, 39, 93, 403, 1209
  • Factor pairs: 1209 = 1 × 1209, 3 × 403, 13 × 93, or 31 × 39
  • 1209 has no square factors that allow its square root to be simplified. √1209 ≈ 34.7707

Did you notice the pattern in the factors?  3×13×31 = 1209
39 and 93 are two of its factors, as well!

1209 is also the hypotenuse of a Pythagorean triple:
465-1116-1209 which is (5-12-13) times 93

1208 Mystery Level

The factors in the multiplication table puzzle below are not in the usual order. Can you figure out where each factor from 1 to 10 belongs in both the first column and the top row?

Print the puzzles or type the solution in this excel file: 10-factors-1199-1210

Here are a few facts about the number 1208:

  • 1208 is a composite number.
  • Prime factorization: 1208 = 2 × 2 × 2 × 151, which can be written 1208 = 2³ × 151
  • 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 1208 has exactly 8 factors.
  • Factors of 1208: 1, 2, 4, 8, 151, 302, 604, 1208
  • Factor pairs: 1208 = 1 × 1208, 2 × 604, 4 × 302, or 8 × 151
  • Taking the factor pair with the largest square number factor, we get √1208 = (√4)(√302) = 2√302 ≈ 34.75629

1208 is also the sum of consecutive prime numbers:
601 + 607 = 1208

1197 Mystery Level

The first few moves needed to solve this puzzle might not be too hard, but soon enough it might get a bit tougher. Nevertheless, its one solution can be found using logic and an ordinary 12×12 multiplication table.

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Here are facts about the number 1197:

  • 1197 is a composite number.
  • Prime factorization: 1197 = 3 × 3 × 7 × 19, which can be written 1197 = 3² × 7 × 19
  • 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 1197 has exactly 12 factors.
  • Factors of 1197: 1, 3, 7, 9, 19, 21, 57, 63, 133, 171, 399, 1197
  • Factor pairs: 1197 = 1 × 1197, 3 × 399, 7 × 171, 9 × 133, 19 × 63, or 21 × 57
  • Taking the factor pair with the largest square number factor, we get √1197 = (√9)(√133) = 3√133 ≈ 34.59769

1197 is the sum of these eleven consecutive prime numbers:
83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 1197

1197 looks interesting to me when it is written in some other bases:
It’s 3330 in BASE 7 because 3(7³ + 7² + 7¹) = 3(399) = 1197,
and it’s 2255 in BASE 8.
It’s 999 in BASE 11, because 9(11² + 11 + 1) = 9(133) = 1197,
and it’s 1K1 in BASE 26 (K is 20 base 10)

 

1191 Happy Birthday, Brent

Today I’m saying “Happy Birthday” to my son, Brent with a yummy but mysterious puzzle cake. I’ve made the puzzle just a little bit harder than normal.

Adding 13 to the puzzle actually makes it easier to solve, but adding 14 makes some multiples of 7 more difficult. For example, the allowable common factors of 70 and 35 are now 7 AND 5, and the allowable common factors of 28 and 56 are now 4, 7, and 14.

As always there is only one solution. I know my son can solve this puzzle, Can you?

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Now I’ll share some facts about the number 1191:

  • 1191 is a composite number.
  • Prime factorization: 1191 = 3 × 397
  • 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 1191 has exactly 4 factors.
  • Factors of 1191: 1, 3, 397, 1191
  • Factor pairs: 1191 = 1 × 1191 or 3 × 397
  • 1191 has no square factors that allow its square root to be simplified. √1191 ≈ 34.51087

1191 is the hypotenuse of a Pythagorean triple:
684-975-1191 which is 3 times (228-325-397)

1191 is repdigit 111 in BASE 34 because
34² + 34 + 1 = 35(34) + 1 = 1191

1185 Mystery Level

The difficulty level of a Mystery Level puzzle is somewhere between fairly easy to extra hard. You won’t know how easy or how difficult it is until you give it a try. Use logic, not guessing and checking, and have fun with it!

Print the puzzles or type the solution in this excel file: 10-factors-1174-1186

What can I tell you about the number 1185?

  • 1185 is a composite number.
  • Prime factorization: 1185 = 3 × 5 × 79
  • 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 1185 has exactly 8 factors.
  • Factors of 1185: 1, 3, 5, 15, 79, 237, 395, 1185
  • Factor pairs: 1185 = 1 × 1185, 3 × 395, 5 × 237, or 15 × 79
  • 1185 has no square factors that allow its square root to be simplified. √1185 ≈ 34.42383

1185 is the hypotenuse of a Pythagorean triple:
711-948-1185 which is (3-4-5) times 237

I think 1185 looks interesting when it is written in some other bases:
It’s palindrome 102201 in BASE 4,
357 in BASE 19, and
palindrome 151 in BASE 32