1350 Logic is at the Heart of This Puzzle

By simply changing two clues of that recently published puzzle that I rejected, I was able to create a love-ly puzzle that can be solved entirely by logic. Can you figure out where to put the numbers from 1 to 12 in each of the four outlined areas that divide the puzzle into four equal sections? If you can, my heart might just skip a beat!

Now I’ll tell you a few things about the number 1350:

  • 1350 is a composite number.
  • Prime factorization: 1350 = 2 × 3 × 3 × 3 × 5 × 5, which can be written 1350 = 2 × 3³ × 5²
  • The exponents in the prime factorization are 1, 3 and 2. Adding one to each and multiplying we get (1 + 1)(3 + 1)(2 + 1) = 2 × 4 × 3 = 24. Therefore 1350 has exactly 24 factors.
  • Factors of 1350: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 27, 30, 45, 50, 54, 75, 90, 135, 150, 225, 270, 450, 675, 1350
  • Factor pairs: 1350 = 1 × 1350, 2 × 675, 3 × 450, 5 × 270, 6 × 225, 9 × 150, 10 × 135, 15 × 90, 18 × 75, 25 × 54, 27 × 50 or 30 × 45
  • Taking the factor pair with the largest square number factor, we get √1350 = (√225)(√6) = 15√6 ≈ 36.74235

1350 is the sum of consecutive prime numbers two ways:
It is the sum of the fourteen prime numbers from 67 to 131, and
673 + 677 = 1350

1350 is the hypotenuse of two Pythagorean triples:
810-1080-1350 which is (3-4-5) times 270
378-1296-1350 which is (7-24-25) times 54

1350 is also the 20th nonagonal number because 20(7 · 20 – 5)/2 = 1350

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1349 A Rejected Puzzle

I was in the mood to make a Find the Factors Challenge Puzzle that used the numbers from 1 to 12 as the factors. I’ve never made such a large puzzle before, but after I made it, I rejected it. All the puzzles I make must meet certain standards: they must have a unique solution, and that solution must be obtainable by using logic. Although the “puzzle” below has a unique solution, and you can fill in a few of the cells using logic, you would have to use guess and check to finish it. Besides that, you wouldn’t be able to know if you guessed right until almost the entire puzzle was completed. Thus, it doesn’t meet my standards.

Even though the puzzle was rejected, there were still some things about it that I really liked. In my next post, I’ll publish a slightly different puzzle that uses some of the same necessary logic that I appreciated but doesn’t rely on guess and check at all. This is NOT the first time I have tweaked a puzzle that didn’t initially meet my standards to make it acceptable. I just thought I would share the process this time. If you try to solve it, you will be able to see the problem with the puzzle yourself.

Now I’ll share some information about the number 1349:

1349 is the sum of 13 consecutive primes, and it is also the sum of three consecutive primes:
73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 1349
443 + 449 + 457 = 1349

1220 Challenge Puzzle

The last challenge puzzle was particularly difficult. This one won’t be nearly as bad. Try it and see if you can figure it out!

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

Here are a few facts about the number 1220:

  • 1220 is a composite number.
  • Prime factorization: 1220 = 2 × 2 × 5 × 61, which can be written 1220 = 2² × 5 × 61
  • 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 1220 has exactly 12 factors.
  • Factors of 1220: 1, 2, 4, 5, 10, 20, 61, 122, 244, 305, 610, 1220
  • Factor pairs: 1220 = 1 × 1220, 2 × 610, 4 × 305, 5 × 244, 10 × 122, or 20 × 61
  • Taking the factor pair with the largest square number factor, we get √1220 = (√4)(√305) = 2√305 ≈ 34.9285

1220 is the sum of consecutive prime numbers: 607 + 613 = 1220

1220 is the sum of two squares two different ways:
32² + 14² = 1220
34² + 8² = 1220

1220 is the hypotenuse of four Pythagorean triples:
220-1200-1220 which is 20 times (11-60-61)
544-1092-1220 calculated from 2(34)( 8), 34² – 8², 34² + 8²
and is also 4 times (136-273-305)
828-896-1220 calculated from 32² – 14², 2(32)(14), 32² + 14²
and is also 4 times (207-224-305)
732-976-1220 which is (3-4-5) times 244

1210 Challenge Puzzle

Some of these Find the Factors Challenge puzzles are easier than others. This one is probably not one of the easier ones. Have fun with it anyway!

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

Now I’ll share a few facts about the number 1210:

  • 1210 is a composite number.
  • Prime factorization: 1210 = 2 × 5 × 11 × 11, which can be written 1210 = 2 × 5 × 11²
  • 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 1210 has exactly 12 factors.
  • Factors of 1210: 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605, 1210
  • Factor pairs: 1210 = 1 × 1210, 2 × 605, 5 × 242, 10 × 121, 11 × 110, or 22 × 55,
  • Taking the factor pair with the largest square number factor, we get √1210 = (√121)(√10) = 11√10 ≈ 34.78505

1210 is the hypotenuse of a Pythagorean triple:
726-968-1210 which is (3-4-5) times 242

1210 is
1122211 in BASE 3
626 in BASE 14
181 in BASE 31

1198 Challenge Puzzle

You can solve this Find the Factors 1 – 10 puzzle if you use logic. Guessing and checking will likely only frustrate you. Go ahead and give logic a try!

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

Now I’ll share some facts about the number 1198:

  • 1198 is a composite number.
  • Prime factorization: 1198 = 2 × 599
  • 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 1198 has exactly 4 factors.
  • Factors of 1198: 1, 2, 599, 1198
  • Factor pairs: 1198 = 1 × 1198 or 2 × 599
  • 1198 has no square factors that allow its square root to be simplified. √1198 ≈ 34.61214

1198 is also palindrome 262 in BASE 23

1186 Challenge Puzzle

It shouldn’t be too hard to make your first move in this puzzle. After that, I don’t make any guarantees. You just need to write each number from 1 to 10 in each of the four boldly outlined areas so that the given clues are the products of the factors you wrote. Use logic to find all the factors and have fun doing it!

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

What have I found out about the number 1186?

  • 1186 is a composite number.
  • Prime factorization: 1186 = 2 × 593
  • 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 1186 has exactly 4 factors.
  • Factors of 1186: 1, 2, 593, 1186
  • Factor pairs: 1186 = 1 × 1186 or 2 × 593
  • 1186 has no square factors that allow its square root to be simplified. √1186 ≈ 34.43835

31² + 15² = 1186

1186 is the hypotenuse of a Pythagorean triple:
736-930-1186 calculated from 31² – 15², 2(31)(15), 31² + 15²

1186 is palindrome 989 in BASE 11

1173 Challenge Puzzle

Getting started on this Challenge Puzzle will take some thinking, but solving it is worth all the effort. Remember use logic, not guess and check, and you will eventually be successful!

Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Here’s some information about the number 1173:

  • 1173 is a composite number.
  • Prime factorization: 1173 = 3 × 17 × 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 1173 has exactly 8 factors.
  • Factors of 1173: 1, 3, 17, 23, 51, 69, 391, 1173
  • Factor pairs: 1173 = 1 × 1173, 3 × 391, 17 × 69, or 23 × 51
  • 1173 has no square factors that allow its square root to be simplified. √1173 ≈ 34.24909

1173 is the hypotenuse of a Pythagorean triple:
552-1035-1173 which is (8-15-17) times 69

1173 is palindrome 3B3 in BASE 18 (B is 11 base 10)
because 3(18²) + 11(18) + 3(1) = 1173

1160 Find the Factors Challenge

I love this particular puzzle. I had so much fun figuring out the logic. It’s a little bit complicated, but once you figure out the first move, it shouldn’t take too long to figure out most of the rest of the puzzle. Do give it a try!

Print the puzzles or type the solution in this excel file: 10-factors-1148-1160

Here are some facts about the number 1160:

  • 1160 is a composite number.
  • Prime factorization: 1160 = 2 × 2 × 2 × 5 × 29, which can be written 1160 = 2³ × 5 × 29
  • 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 1160 has exactly 16 factors.
  • Factors of 1160: 1, 2, 4, 5, 8, 10, 20, 29, 40, 58, 116, 145, 232, 290, 580, 1160
  • Factor pairs: 1160 = 1 × 1160, 2 × 580, 4 × 290, 5 × 232, 8 × 145, 10 × 116, 20 × 58, or 29 × 40
  • Taking the factor pair with the largest square number factor, we get √1160 = (√4)(√290) = 2√290 ≈ 34.05877

26² + 22² = 1160
34² +  2² = 1160

1160 is the hypotenuse of FOUR Pythagorean triples:
136-1152-1160 which is 8 times (17-144-145)
(It can also be calculated from 2(34)(2), 34² –  2², 34² +  2²)
192-1144-1160 which is 8 times (24-143-145)
(It can also be calculated from 26² – 22², 2(26)(22), 26² + 22²)
696-928-1160 which is (3-4-5) times 232
800-840-1160 which is (20-21-29) times 40

1160 is a palindrome when it is written in these three bases:
It’s 808 in BASE 12 because 8(12²) + 8(1) = 8(145) = 1160,
525 in BASE 15 because 5(15²) + 2(15) + 5(1) = 1160, and
404 in BASE 17 because 4(17²) + 4(1) = 4(290) = 1160

1147 Dragonfly Challenge

This “dragonfly” challenges you to solve the puzzle. It won’t be easy, but if you stick to using logic the entire time, you will be able to do it!

Print the puzzles or type the solution in this excel file: 12 factors 1134-1147

Here are a few facts about the number 1147:

  • 1147 is a composite number.
  • Prime factorization: 1147 = 31 × 37
  • 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 1147 has exactly 4 factors.
  • Factors of 1147: 1, 31, 37, 1147
  • Factor pairs: 1147 = 1 × 1147 or 31 × 37
  • 1147 has no square factors that allow its square root to be simplified. √1147 ≈ 33.86739

1147 is the hypotenuse of a Pythagorean triple:
372-1085-1147 which is 31 times (12-35-37)

34² – 3² = 1147, so we are only 9 numbers away from the next perfect square.

1147 is a leg in a few Pythagorean triples including this primitive:
204-1147-1165 calculated from 2(34)(3), 34² – 3², 34² + 3²

I like the way 1147 looks when it is written in some other bases:
It’s 5151 in BASE 6 because 5(6³) + 1(6²) + 5(6) + 1(1) = 1147,
7B7 in BASE 12 (B is 11 base 10) because 7(12²) + 11(12) + 7(1) = 1147,
and VV in BASE 36 (V is 31 base 10) because 31(36) + 31(1) = 31(37) = 1147