A Multiplication Based Logic Puzzle

Posts tagged ‘square number’

You’ll Be Impressed By 841’s Number Facts

841 is a very cool square number. 29 × 29 = 841. You can write it more compactly: 29² = 841. Here are some facts about this square number:

841 is the sum of the 29th and the 28th triangular numbers because 841 = 435 + 406.

What’s that you say? Every other square number can make a similar claim? Oh. … Wait a minute…Not EVERY square number can do that. One is a square number, and it’s NOT the sum of two triangular numbers. So there…

How about this… 841 is the sum of the first 29 odd numbers. That makes it the sum of all the odd numbers from 1 to 57. That’s impressive!

What? All square numbers do that? Huh??? n² is always the sum of the first n odd numbers? Why’d you have to tell me that fact isn’t particularly unique either!

Okay…841 is an interesting number in a few other bases. Pay attention….  841 is

  • 100 in BASE 29
  • 121 in BASE 28
  • 144 in BASE 27
  • 169 in BASE 26
  • 441 in BASE 14

Ha! You didn’t know that one! … Now don’t go spoiling my glee by telling me that those same square numbers show up for a few other square numbers, too. Don’t tell me that!

Give me one more chance to impress you….This square number, 841, is the sum of two consecutive square numbers, so 20² + 21² = 841 = 29². That hasn’t happened to a square number since 3² + 4² = 25 = 5². That makes 841 the 21st Centered Square Number but only the SECOND square number that is both kinds of squares! There may be an infinite number of squares that do the same thing, but it is still a fairly unique characteristic.

There are 441 blue squares in that graphic. There are 400 squares that are orange, red, green, purple, or black. 441 + 400 = 841.

Here are a few other tidbits about 841:

841 is the sum of the nine prime numbers from 73 to 109. It is also the sum of three consecutive primes: 277 +  281 + 283 = 841

841 is the hypotenuse of TWO Pythagorean triples:

  • 580-609-841 which is 29 times (20-21-29)
  • 41-840-841 calculated from 21² – 20², 2(21)(20), 21² + 20²

841 Pythagorean triple Recursion: Hmm…How Quickly Can I Give You a Headache? Warning! Reading this set of bullet points might overload your brain:

  • 5² + 2² = 29, so (2∙5∙2)² + (5² – 2²)² = (5² + 2²)² which means 20² + 21² = 29² = 841.
  • 21² + 20² = 841, so (2∙21∙20)² + (21² – 20²)² = (21² + 20²)² which means 840² + 41² = 841² = (29²)² = 707281.
  • 840² + 41² = 707281, so (2∙840∙41)² + (840² – 41²)² = (840² + 41²)² which means 68880² + 703919² = 707281² = ((29²)²)² = 500,246,412,961
  • We could go on forever with even bigger powers of 29 …

We’ll finish with just some simple, easy-on-the-brain facts about 841:

29 is the tenth prime number. Its square, 841, is only the tenth number to have exactly three factors.

  • 841 is a composite number.
  • Prime factorization: 841 = 29²
  • The exponent in the prime factorization is 2. Adding one we get (2 + 1) = 3. Therefore 841 has exactly 3 factors.
  • Factors of 841: 1, 29, 841
  • Factor pairs: 841 = 1 × 841 or 29 × 29
  • 841 is a perfect square. √841 = 29


I’m impressed by all this, even if you aren’t.

 

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441 Consecutive Numbers

The first 6 triangular numbers are

  • 1 = 1
  • 1 + 2 = 3
  • 1 + 2 + 3 = 6
  • 1 + 2 + 3 + 4 = 10
  • 1 + 2 + 3 + 4 + 5 = 15
  • 1 + 2 + 3 + 4 + 5 + 6 = 21

For millennia mathematicians have thought triangular numbers were quite interesting. 21 is a triangular number and 21 x 21 = 441 which makes 441 interesting, too. But wait, there’s something else that is very interesting about triangular numbers:

Sum of consecutive cubes

It is amazing that when we begin with 1 cube, the sum of n consecutive cubes equals the nth triangular number squared every time!

What  follows is less amazing, but very practical. We can add consecutive numbers several ways to get 441. To find those ways we need to know the factors of 441.

  • 441 is a composite number.
  • Prime factorization: 441 = 3 x 3 x 7 x 7, which can be written 441 = (3^2) x (7^2)
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 x 3 = 9. Therefore 441 has exactly 9 factors.
  • Factors of 441: 1, 3, 7, 9, 21, 49, 63, 147, 441
  • Factor pairs: 441 = 1 x 441, 3 x 147, 7 x 63, 9 x 49, or 21 x 21
  • 441 is a perfect square. √441 = 21

Because all of the factors for 441 are odd numbers, it is so easy to find consecutive numbers whose sum equal 441. Check out all of these:

consecutive numbers equals 441If we allowed negative numbers in the list of consecutive numbers we could also see that 441 equals the sum of 147 consecutive numbers that are centered around the number 3, and 49 consecutive numbers that are centered around the number 7, and 63 consecutive numbers that are centered around the number 9. All of those sums would be quite long.

Here are some more reasonable-length sums using only consecutive ODD numbers.

consecutive odd numbers equal 441

That last sum reminds us that we always get n squared when we begin with one and add n consecutive odd numbers together.

306 and Level 4

  • 306 is a composite number.
  • Prime factorization: 306 = 2 x 3 x 3 x 17, which can be written 306 = 2 x (3^2) x 17
  • 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 x 3 x 2 = 12. Therefore 306 has exactly 12 factors.
  • Factors of 306: 1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306
  • Factor pairs: 306 = 1 x 306, 2 x 153, 3 x 102, 6 x 51, 9 x 34, or 17 x 18
  • Taking the factor pair with the largest square number factor, we get √306 = (√9)(√34) = 3√34 ≈ 17.493

Happy Thanksgiving!

2014-47 Level 4

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-24

Here’s a little more about the number 306:

306 = 17 × 18, which means it is the sum of the first 17 even numbers.

  • Thus, 2 + 4 + 6 + 8 + . . .  + 30 + 32 + 34 = 306

It also means that we are halfway between 17² and 18², or halfway between 289 and 324. The average of those two numbers is 306.5.

AND it means that 17² + 18² – 1 = 2(306) = 2(17 × 18)

2014-47 Level 4 Logic

305 and Level 3

  • 305 is a composite number.
  • Prime factorization: 305 = 5 x 61
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 305 has exactly 4 factors.
  • Factors of 305: 1, 5, 61, 305
  • Factor pairs: 305 = 1 x 305 or 5 x 61
  • 305 has no square factors that allow its square root to be simplified. √305 ≈ 17.464

2014-47 Level 3

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-24

A Logical Approach to FIND THE FACTORS: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

2014-47 Level 3 Factors

304 and Level 2

  • 304 is a composite number.
  • Prime factorization: 304 = 2 x 2 x 2 x 2 x 19, which can be written 304 = (2^4) x 19
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 304 has exactly 10 factors.
  • Factors of 304: 1, 2, 4, 8, 16, 19, 38, 76, 152, 304
  • Factor pairs: 304 = 1 x 304, 2 x 152, 4 x 76, 8 x 38, or 16 x 19
  • Taking the factor pair with the largest square number factor, we get √304 = (√16)(√19) = 4√19 ≈ 17.436

This 1 – 10 multiplication table has only twelve clues. Can you still complete the entire table?

2014-47 Level 2

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-24

2014-47 Level 2 Factors

303 and Level 1

  • 303 is a composite number.
  • Prime factorization: 303 = 3 x 101
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 303 has exactly 4 factors.
  • Factors of 303: 1, 3, 101, 303
  • Factor pairs: 303 = 1 x 303 or 3 x 101
  • 303 has no square factors that allow its square root to be simplified. √303 ≈ 17.407

2014-47 Level 1

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-24

2014-47 Level 1 Factors

302 and Level 6

  • 302 is a composite number.
  • Prime factorization: 302 = 2 x 151
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 302 has exactly 4 factors.
  • Factors of 302: 1, 2, 151, 302
  • Factor pairs: 302 = 1 x 302 or 2 x 151
  • 302 has no square factors that allow its square root to be simplified. √302 ≈ 17.378

Here’s a very tricky level 6 puzzle to try.

2014-46 Level 6

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-17

The logic for this one is complicated. Here is an explanation that may be helpful:

9 is a clue two times in this puzzle. At most one of those 9’s can be 3 x 3. Clue 99 and the other nine will use both 9’s so 72 cannot be 8 x 9 in this puzzle. Here is an explanation of why 72 must be 6 x 12 with the 6 in the first column and the 12 in the top row.

Assuming 6 goes over 72 in top row

Now we can find the rest of the factors easily using logic:

2014-46 Level 6 Logic

 

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