A Multiplication Based Logic Puzzle

Archive for September, 2015

631 and Level 3

631 is a prime number that happens to also be the sum of the eleven prime numbers from 37 to 79.

Sometimes mathematicians talk about triangular numbers, centered triangular numbers, hexagonal numbers, and centered hexagonal numbers. Is there any relationship between those terms?

  • The sum of any three consecutive triangular numbers is a centered triangular number.
  • ALL hexagonal numbers are triangular numbers.
  • Most centered hexagonal numbers are NOT centered triangular numbers.

In my last post, I pointed out that 630 is a triangular number and a hexagonal number.

Well, amazingly the next number, 631, is a centered triangular number AND a centered hexagonal number.

How often does one more than a triangular number equal a centered triangular number? Also how often does one more than a hexagonal number equal a centered hexagonal number?

Let’s compare a couple of lists of these types of numbers up to 631. Red means a number is in both lists. Blue means the number in the 1st list is one less than the number in the 2nd list.

  • Triangular numbers (with hexagonal numbers in bold type):
  • 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630.
  • Centered triangular numbers (with numbers that are also centered hexagonal numbers in bold type):
  • 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631.

1 is the only number in the lists that is a triangular, centered triangular, hexagonal, and centered hexagonal number all at the same time. (1 is also a square number, a pentagonal number, etc.)

(630, 631) is the first combination of numbers that has BOTH a triangular number immediately followed by a centered triangular number AND a hexagonal number immediately followed by a centered hexagonal number.

631 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

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  • 631 is a prime number.
  • Prime factorization: 631 is prime and cannot be factored.
  • The exponent of prime number 631 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 631 has exactly 2 factors.
  • Factors of 631: 1, 631
  • Factor pairs: 631 = 1 x 631
  • 631 has no square factors that allow its square root to be simplified. √631 ≈ 25.11971.

How do we know that 631 is a prime number? If 631 were not a prime number, then it would be divisible by at least one prime number less than or equal to √631 ≈ 25.1. Since 631 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 631 is a prime number.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 10.)  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.

631 Factors

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630 Factor Trees and Level 2

630 is the 7th number with exactly 24 factors. So far, the seven numbers counting numbers with 24 factors are 360, 420, 480, 504, 540, 600, and 630. No counting number less than 630 has more than 24 factors.

Two of those seven numbers make up the Pythagorean triple 378-504-630. Which factor of 630 is the greatest common factor of those three numbers in the triple?

Here are a few of the MANY possible factor trees for 630.

630 Factor Trees

630 is the sum of the six prime numbers from 97 to 113.

630 is the 35th triangular number because (35 x 36)/2 = 630. It is also the 18th hexagonal number because 18(2 x 18 – 1) = 630.

630 is a triangular number that is a multiple of other triangular numbers in more ways than you probably want to know:

  • 630 is three times the 20th triangular number, 210, because 3(20 x 21)/2 = 630.
  • 630 is 6 times the 14th triangular number, 105, because 6(14 x 15)/2 = 630.
  • 630 is 14 times the 9th triangular number, 45, because 14(9 x 10)/2 = 630.
  • 630 is 30 times the 6th triangular number, 21, because 30(6 x 7)/2 = 630.
  • 630 is 42 times the 5th triangular number, 15, because 42(5 x 6)/2 = 630.
  • 630 is 63 times the 4th triangular number, 10, because 63(4 x 5)/2 = 630.
  • 630 is 105 times the 3rd triangular number, 6, because 105(3 x 4)/2 = 630.
  • 630 is 210 times the 2nd triangular number, 3, because 210(2 x 3)/2 = 630.
  • and finally, 630 is 630 times the 1st triangular number, 1, because 630(1 x 2)/2 = 630

630 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

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  • 630 is a composite number.
  • Prime factorization: 630 = 2 x 3 x 3 x 5 x 7, which can be written 630 = 2 x (3^2) x 5 x 7
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 x 3 x 2 x 2 = 24. Therefore 630 has exactly 24 factors.
  • Factors of 630: 1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 30, 35, 42, 45, 63, 70, 90, 105, 126, 210, 315, 630
  • Factor pairs: 630 = 1 x 630, 2 x 315, 3 x 210, 5 x 126, 6 x 105, 7 x 90, 9 x 70, 10 x 63, 14 x 45, 15 x 42, 18 x 35, or 21 x 30
  • Taking the factor pair with the largest square number factor, we get √630 = (√9)(√70) = 3√70 ≈ 25.09980.

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630 Factors

629 and Level 1

629 is the sum of the 17 prime numbers from 7 to 71. Both of its prime factors, 17 and 37, are included in that list.

17 and 37 are both 10 numbers away from their average, 27. That means that 629 + 10² = 729 or 27².

25² + 2² = 629 and 23² + 10² = 629. Notice that 629 plus or minus 100 is a square number.

Both of 629’s prime factors have a remainder of one when divided by four so 629 is the hypotenuse of four Pythagorean triples, two of which are primitives.

  • 100-621-629, a primitive that reminds me of another primitive, 20-21-29
  • 204-595-629, three numbers whose greatest common factor is 17
  • 296-555-629, three numbers whose greatest common factor is 37
  • 429-460-629, a primitive whose shorter leg is exactly 200 less than its hypotenuse.

629 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

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  • 629 is a composite number.
  • Prime factorization: 629 = 17 x 37
  • 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 629 has exactly 4 factors.
  • Factors of 629: 1, 17, 37, 629
  • Factor pairs: 629 = 1 x 629 or 17 x 37
  • 629 has no square factors that allow its square root to be simplified. √629 ≈ 25.079872.

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629 Factors

628, Tau, Pi, and Level 6

The circumference of a circle with a radius of one is approximately 6.28. That’s an important enough number that it has been given the symbol “τ ” which is pronounced “tau”. τ looks a little like half of the number π, but τ = 2π.

Some people think we should get rid of π and only use τ. Other people feel that π has been used for centuries, and there is no compelling reason to change now.

π is perfect for finding the area of a circle: Area = πr². Here’s the area of a circle using tau: Area = r²τ/2.

τ is very good for finding the circumference of a circle: Circumference = τr, but that looks strange compared to 2πr. In fact, it can be difficult to tell if τr is one character or two.

The Tau Manifesto shows angle measurements in degrees, π radians and τ radians. You might want to look at some videos, too. Some people think the τ radians are simpler because the radians correspond exactly to the fractional pieces of the circumference of a circle or, get this, to the fractional pieces of a pie. (τ does that, not π.) Other people think that π radians are just as good because we’re used to them, and they correspond exactly to the area of any wedge in a unit circle or the area of any slice of pie. (Which would you rather eat the circumference or the area of a pie?)

Until I wrote this post and read the link shared in the comments, I hadn’t heard anybody say that π is better for some situations while τ is better for others. (Actually it appears that π is better except in formulas that use 2π.) Diameters and radii have co-existed peacefully for centuries. I don’t understand why π and τ can’t do the same. Here’s a great video that shows both sides of the argument.

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22² + 12² = 628.

628 is the hypotenuse of the Pythagorean triple 340-528-628. The greatest common factor of those three numbers is the same as the greatest common factor of 22² and 12².

7² + 11² + 13² + 17² = 628. Thank you Stetson.edu for that fun fact about the squares of those four consecutive prime numbers.

628 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

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  • 628 is a composite number.
  • Prime factorization: 628 = 2 x 2 x 157, which can be written 628 = (2^2) x 157
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 628 has exactly 6 factors.
  • Factors of 628: 1, 2, 4, 157, 314, 628
  • Factor pairs: 628 = 1 x 628, 2 x 314, or 4 x 157
  • Taking the factor pair with the largest square number factor, we get √628 = (√4)(√157) = 2√157 ≈ 25.059928.

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628 Logic

627 and Level 5

Here are two divisibility rules applied to 627 that give a positive result:

  • 6 + 2 + 7 = 15, a multiple of 3, so 627 is divisible by 3.
  • 6 – 2 + 7 = 11, so 627 is divisible by 11.

627 can be expressed as the sum of consecutive counting numbers 4 different ways. The numbers in bold are in the middle of each sum:

  • 208 + 209 + 210 = 627
  • 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 = 627
  • 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 627
  • 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11+ 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 = 627

The logic on this Level 5 puzzle gets a bit complicated right when it’s almost done:

627 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

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  • 627 is a composite number.
  • Prime factorization: 627 = 3 x 11 x 19
  • 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 x 2 x 2 = 8. Therefore 627 has exactly 8 factors.
  • Factors of 627: 1, 3, 11, 19, 33, 57, 209, 627
  • Factor pairs: 627 = 1 x 627, 3 x 209, 11 x 57, or 19 x 33
  • 627 has no square factors that allow its square root to be simplified. √627 ≈ 25.039968.

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627 Logic

626 Semordnilaps, Palindromes and Level 4

626 is the hypotenuse of the Pythagorean triple 50-624-626. What is the greatest common factor of those three numbers?

Today my son posted on facebook, “A word that when spelled backwards spells a different word is called a Semordnilap.” That’s a word I hadn’t heard before, and it also applies to phrases and sentences that form different phrases or sentences when read backwards.

“Semordnilap” is a semordnilap for the word “palindromes” which are words, phrases, sentences, and numbers that read the same forward and backward.

Palindrome sentences are sometimes made with words that are semordnilaps: was, saw, live, evil, on, no, desserts, stressed, stop, pots, tops, spot, diaper, repaid.

626 is a number that is a palindrome in several different bases:

  • 10001 in base 5; note that (5^4) + 1 = 626.
  • 626 in base 10
  • 272 in base 16; note that 2(16^2) + 7(16) + 2 = 626
  • 1DI in base 19, if “1” and “I” look the same, and too often they do. Note that 1(19^2) + 13(19) + 18 = 626
  • 101 in base 25; note that (25^2) + 1 = 626
  • 11 in base 625; note that 626 + 1 = 626

I guess we could say that in all other bases 626 is a semordnilap.

All of 626’s factors are palindromes, too.

626 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

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  • 626 is a composite number.
  • Prime factorization: 626 = 2 x 313
  • 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 626 has exactly 4 factors.
  • Factors of 626: 1, 2, 313, 626
  • Factor pairs: 626 = 1 x 626 or 2 x 313
  • 626 has no square factors that allow its square root to be simplified. √626 ≈ 25.019992.

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626 Logic

625 Where did all those Pythagorean triples come from?

625 Pythagorean triple

625 is the hypotenuse of FOUR Pythagorean triples. Where did all those Pythagorean triples come from?

They come from 4 of the 5 factors of 625, all of which are powers of 5, a prime number of the form (4n + 1).

5 is the hypotenuse of the primitive Pythagorean triple 3-4-5 that was calculated from (2^2) – (1^2); 2(2)(1); (2^2) + (1^2).

25 is the hypotenuse of 5 times that triple plus it has a primitive of its own:

  • 15-20-25 which is 5 times 3-4-5
  • 7-24-25 calculated from (4^2) – (3^2); 2(4)(3); (4^2) + (3^2)

125 is the hypotenuse of 5 times 25’s two triples plus it has a primitive of its own:

  • 75-100-125 which is 5 times 15-20-25 or 25 times 3-4-5
  • 35-120-125 which is 5 times 7-24-25
  • 44-117-125 calculated from 2(11)(2); (11^2) – (2^2); (11^2) + (2^2)

625 is the hypotenuse of 5 times 125’s three triples plus it has a primitive of its own:

  • 375-500-625 which is 5 times 75-100-125, or 25 times 15-20-25, or 125 times 3-4-5
  • 175-600-625 which is 5 times 35-120-125, or 25 times 7-24-25
  • 220-585-625 which is 5 times 44-117-125
  • 336-527-625 calculated from 2(24)(7); (24^2) – (7^2); (24^2) + (7^2)

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625 Short Leg

625 is also the short leg of FOUR Pythagorean triples. Where did all those Pythagorean triples come from?

They come from 4 of the 5 factors of 625, all of which are odd. (Every odd number greater than one is the short leg of at least one Pythagorean triple.)

5 is the short leg of the primitive Pythagorean triple 5-12-13. Notice that 12 + 13 = 25 which is 5^2.

25 is the short leg of 5 times that triple plus it has a primitive of its own:

  • 25-60-65 which is 5 times 5-12-13
  • 25-312-313; Notice that 312 + 313 = 625 = (25^2)

125 is the short leg of 5 times 25’s two triples plus it has a primitive of its own:

  • 125-300-325 which is 5 times 25 -60-65 or 25 times 5-12-13
  • 125-1560-1565 which is 5 times 25-312-313
  • 125-7812-7813; Notice that 7812 + 7813 = (125^2)

625 is the short leg of 5 times 125’s three triples plus it has a primitive of its own:

  • 625-1500-1625 which is 5 times 125-300-325, or 25 times 25-60-65, or 125 times 5-12-13
  • 625-7800-7825 which is 5 times 125-1560-1565 or 25 times 25-312-313
  • 625-39060-39065 which is 5 times 125-7812-7813
  • 625-195312-195313; Notice that 195312 + 195313 = (625^2)

Thus 625 appears in 8 Pythagorean triples, and now you know where they all came from.

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Here are some other fun facts about the number 625:

Look at this pattern:

  • (5^2) = 25
  • (25^2) = 625
  • (625^2) = 390,625
  • but (390,625^2) = 152,587,890,625 so one digit discontinued the pattern!

625 is the sum of the seven prime numbers from 73 to 103.

What would happen if we ran the following prime number tests on 625?

(24^2) + (7^2) = 625, and 24 and 7 have no common prime factors. That means that 625’s only possible prime factors less than √625 are 5, 13, and 17. Obviously 625 is divisible by 5 so it isn’t a prime number.

Also note that (20^2) + (15^2) = 625, but 20 and 15 have a common prime factor, 5. The fact that they have a common prime factor means that 625 cannot be a prime number.

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  • 625 is a composite number.
  • Prime factorization: 625 = 5 x 5 x 5 x 5, which can be written 625 = 5^4
  • The exponent in the prime factorization is 4. Adding one, we get (4 + 1) = 5. Therefore 625 has exactly 5 factors.
  • Factors of 625: 1, 5, 25, 125, 625
  • Factor pairs: 625 = 1 x 625, 5 x 125 or 25 x 25
  • 625 is a perfect square and a perfect fourth power. √625 = (√25)(√25) = 25

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