1164 Mathematics at Corvin Castle

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In the forests of Transylvania in what is now Hunedoara, Romania, a large, beautiful, well-restored, and fascinating castle awaits. At Corvin Castle, you can see how enemies were tortured and royal friends were entertained in the days of kings and knights. However, the room that intrigued me the most was full of mathematics. I didn’t take a picture of every geometric shape that graced its walls; I only took a few. You will have to go there yourself to see all the wonderful mathematical artwork. I thoroughly enjoyed myself!

I knew that fellow mathematical puzzle maker, Simona Prilogan,  was from Romania, but I didn’t realize until after I returned home that this castle is in her hometown! She recently wrote about the castle and her experience growing up in Hunedoara. As a child, she was chosen to write a poem about Romania’s president and present the poem at this very castle.

Now I’ll tell you a little something about the number 1164:

  • 1164 is a composite number.
  • Prime factorization: 1164 = 2 × 2 × 3 × 97, which can be written 1164 = 2² × 3 × 97
  • 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 1164 has exactly 12 factors.
  • Factors of 1164: 1, 2, 3, 4, 6, 12, 97, 194, 291, 388, 582, 1164
  • Factor pairs: 1164 = 1 × 1164, 2 × 582, 3 × 388, 4 × 291, 6 × 194, or 12 × 97
  • Taking the factor pair with the largest square number factor, we get √1164 = (√4)(√291) = 2√291 ≈ 34.117444

1164 is the sum of consecutive prime numbers FOUR different ways:
It’s the sum of the eighteen prime numbers from 29 to 103.
97 + 101 + 103 + 107 + 109 + 113 + 127 + 131+ 137 + 139 = 1164; that’s ten consecutive primes,
281 + 283 + 293 + 307 = 1164; that’s four consecutive primes, and
577 + 587 = 1164; that’s two consecutive primes.

1164 is the hypotenuse of a Pythagorean triple:
780-864-1164 which is 12 times (65-72-97)

1164 is palindrome 969 in BASE 11 because 9(11²) + 6(11) + 9(1) = 1164,
and it’s 345 in BASE 19 because 3(19²) + 4(19) + 5(1) = 1164

 

1163 and Level 2

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Can you write the numbers 1 to 12 in both the first column and the top row of this puzzle so those numbers and the clues function like a multiplication table? Sure you can!

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

Here is some information about the number 1163:

  • 1163 is a prime number.
  • Prime factorization: 1163 is prime.
  • The exponent of prime number 1163 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1163 has exactly 2 factors.
  • Factors of 1163: 1, 1163
  • Factor pairs: 1163 = 1 × 1163
  • 1163 has no square factors that allow its square root to be simplified. √1163 ≈ 34.10279

How do we know that 1163 is a prime number? If 1163 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1163 ≈ 34.1. Since 1163 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1163 is a prime number.

Prime number 1163 is also the sum of nine consecutive primes:
107 + 109 + 113 + 127 + 131 +137 + 139 + 149 + 151 = 1163

1162 Is a Pentagonal Number

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1162 is the 28th pentagonal number because (3(28²) – 28)/2 = 1162. I created the graphic below so you can see all the pentagonal numbers up to 1162 and figure out why all of them are called pentagonal numbers.

Here’s more about the number 1162:

  • 1162 is a composite number.
  • Prime factorization: 1162 = 2 × 7 × 83
  • 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 1162 has exactly 8 factors.
  • Factors of 1162: 1, 2, 7, 14, 83, 166, 581, 1162
  • Factor pairs: 1162 = 1 × 1162, 2 × 581, 7 × 166, or 14 × 83
  • 1162 has no square factors that allow its square root to be simplified. √1162 ≈ 34.08812

1162 is a palindrome in three bases:
It’s 343 in BASE 19 because 3(19²) + 4(19) + 3(1) = 1162,
2I2 in BASE 20 (I is 18 base 10) because 2(20²) + 18(20) + 2(1) = 1162, and
1G1 in BASE 27 (G is 16 base 10) because 27² + 16(27) + 1 = 1162

1161 and Level 1

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Solving this puzzle will help you review the multiplication table. Knowing the multiplication table inside and out will be a big PLUS in your life. It will save you so much time in all your mathematics classes!

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

Here is some information about the number 1161:

  • 1161 is a composite number.
  • Prime factorization: 1161 = 3 × 3 × 3 × 43, which can be written 1161 = 3³ × 43
  • 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 1161 has exactly 8 factors.
  • Factors of 1161: 1, 3, 9, 27, 43, 129, 387, 1161
  • Factor pairs: 1161 = 1 × 1161, 3 × 387, 9 × 129, or 27 × 43
  • Taking the factor pair with the largest square number factor, we get √1161 = (√9)(√129) = 3√129 ≈ 34.07345

1161 is the sum of the first twenty-six prime numbers. That’s all the primes from 2 to 101.

1161 is a palindrome in a couple of bases:
It’s 10010001001 in BASE 2 because 2¹⁰ + 2⁷ + 2³ + 2⁰ = 1161 and
1B1 in BASE 29 (B is 11 base 10) because 29² + 11(29) + 1 = 1161

1160 Find the Factors Challenge

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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

1159 Mystery Level

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This mysterious multiplication table does not have its factors in the usual places. Can you use logic to figure out where the factors from 1 to 10 go so that the given clues belong where they are?

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

What can I tell you about the number 1159?

  • 1159 is a composite number.
  • Prime factorization: 1159 = 19 × 61
  • 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 1159 has exactly 4 factors.
  • Factors of 1159: 1, 19, 61, 1159
  • Factor pairs: 1159 = 1 × 1159 or 19 × 61
  • 1159 has no square factors that allow its square root to be simplified. √1159 ≈ 34.04409

1159 is the sum of the twenty-five prime numbers from 3 to 101.

1159 is the hypotenuse of a Pythagorean triple:
209-1140-1159 which is 19 times (11-60-61)

1158 Mystery Level

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The twelve clues in this puzzle are easy to see, but solving the mystery of the puzzle might not be so easy. The clues work together to point to particular factors. Can you figure out where they all go?

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

Now I’ll write a little bit about the number 1158:

  • 1158 is a composite number.
  • Prime factorization: 1158 = 2 × 3 × 193
  • 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 1158 has exactly 8 factors.
  • Factors of 1158: 1, 2, 3, 6, 193, 386, 579, 1158
  • Factor pairs: 1158 = 1 × 1158, 2 × 579, 3 × 386, or 6 × 193
  • 1158 has no square factors that allow its square root to be simplified. √1158 ≈ 34.0294

1158 is the hypotenuse of a Pythagorean triple:
570-1008-1158 which is 6 times (95-168-193)

1158 is 123 in BASE 33 because 33² + 2(33) + 3(1) = 1158

1157 and Level 6

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The more you solve these puzzles, the easier most of them become. This one is no exception. Can you figure out the logic needed to make the first move?

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

Now I’ll share some facts about the number 1157:

  • 1157 is a composite number.
  • Prime factorization: 1157 = 13 × 89
  • 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 1157 has exactly 4 factors.
  • Factors of 1157: 1, 13, 89, 1157
  • Factor pairs: 1157 = 1 × 1157 or 13 × 89
  • 1157 has no square factors that allow its square root to be simplified. √1157 ≈ 34.0147

34² + 1² = 1157
31² + 14² = 1157

1157 is the hypotenuse of FOUR Pythagorean triples:
68-1155-1157 calculated from 2(34)(1), 34² – 1², 34² + 1²
445-1068-1157 which is (5-12-13) times 89
507-1040-1157 which is 13 times (39-80-89)
765-868-1157 calculated from 31² – 14², 2(31)(14), 31² + 14²

1157 is a palindrome in two different bases:
It’s 2D2 in BASE 21 (D is 13 base 10) because 2(21²) + 13(21) + 2(1) = 1157
101 in BASE 34 because 34² + 1 = 1157

What Kind of Shape is 1156 In?

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One of the shapes that 1156 is in is a pentagon. Below you see 1156 tiny squares arranged into a pentagon.

If you study the graphic above you will also see that 1156 is 1 more than five times the 21st triangular number.

The first few centered pentagonal numbers are 1, 6, 16, 31, 51, 76, 106. Double any centered pentagonal number 106 or greater, and you will get a number that can be represented as 552 in some base even if it’s a base nobody uses. Let’s see it at work here:
1156 × 2 = 2312, and 2312 is 552 in BASE 21

Why is that true? Because of the formula that calculates centered pentagonal numbers. For example, we know 1156 is the 22nd centered pentagonal number from
(5(21²) + 5(21) + 2(1))/2 = 1156

Written another way it looks like 1 more than 5 times the 21st triangular number:
5(21)(22)/2 + 1 = 1156

1156 is also a perfect square!

Not only is it a perfect square but it is the square of a composite number. The following two squares show how it can be made with multiple squares of its prime factors.

  • 1156 is a composite number.
  • Prime factorization: 1156 = 2 × 2 × 17 × 17, which can be written 1156 = 2²× 17²
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1156 has exactly 9 factors.
  • Factors of 1156: 1, 2, 4, 17, 34, 68, 289, 578, 1156
  • Factor pairs: 1156 = 1 × 1156, 2 × 578, 4 × 289, 17 × 68, or 34 × 34
  • 1156 is a perfect square. √1156 = 34

Another shape where you can find 1156 are these two Pythagorean triple triangles:
644-960-1156 and 544-1020-1156

1156 is also the sum of the twenty-four prime numbers from 5 to 101.

1156 is palindrome 1120211 in BASE 3 because 3⁶+3⁵+2(3⁴)+2(3²)+3+1=1156

Square number 1156 also looks like some other square numbers when it is written in these other bases:
It’s 961 in BASE 11 because 9(11²) + 6(11) + 1(1) = 1156,
484 in BASE 16 because 4(16²) + 8(16) + 4(1) = 1156,
400 in BASE 17 because 4(17²) = 1156,
169 in BASE 31 because 31² + 6(31) + 9(1) = 1156,
144 in BASE 32 because 32² + 4(32) + 4(1) = 1156,
121 in BASE 33 because 33² + 2(33) + 1(1) = 1156,
and 100 in BASE 34 because 34² = 1156

 

Reasons to Celebrate 1155

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1155 is the product of the four smallest odd prime numbers. That alone is a reason to celebrate. Here, I’ve made a factor cake with 1155 for you to enjoy!

1155 Factor Cake

 

make science GIFs like this at MakeaGif

Besides 1155 being the smallest number divisible by the first 4 odd prime numbers, what else can I say about it?

  • 1155 is a composite number.
  • Prime factorization: 1155 = 3 × 5 × 7 × 11
  • 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 1155 has exactly 16 factors.
  • Factors of 1155: 1, 3, 5, 7, 11, 15, 21, 33, 35, 55, 77, 105, 165, 231, 385, 1155
  • Factor pairs: 1155 = 1 × 1155, 3 × 385, 5 × 231, 7 × 165, 11 × 105, 15 × 77, 21 × 55, or 33 × 35
  • 1155 has no square factors that allow its square root to be simplified. √1155 ≈ 33.98529

Since it’s number of prime factors is a power of 2, it can make some very well-balanced and attractive factor trees:

1155 is the hypotenuse of a Pythagorean triple:
693-924-1155 which is (3-4-5) times 231

1155 is one number away from the next perfect square, AND it is the difference of two squares EIGHT different ways.
34² – 1² = 1155
38² – 17² = 1155
46² – 31² = 1155
58² – 47² = 1155
86² – 79² = 1155
118² – 113² = 1155
194² – 191² = 1155
578² – 577² = 1155

Those eight ways correspond to its eight factor pairs. Notice that both of the numbers in all of its factor pairs are odd numbers.

I like the way 1155 looks in some other bases:
It’s 3A3 in BASE 18 (A is 10 base 10) because 3(18²) + 10(18) + 3(1) = 1155
XX in BASE 34 (X is 33 base 10) because 33(34) + 33(1) = 33(35) = 1155
X0 in BASE 35 because 33(35) = 1155

XX X0 . . . Hugs and Kisses. You gotta love that!