Month: December 2019

1444 Christmas Wrapping Paper

/ ivasallay / Leave a comment

A fourth of the products in this multiplication table puzzle are already there just because I wanted the puzzle to have a wrapping paper pattern. Can you figure out what the factors are supposed to be and what all the other products are?

  • 1444 is a composite number and a perfect square.
  • Prime factorization: 1444 = 2 × 2 × 19 × 19, which can be written 1444 = 2²× 19²
  • 1444 has at least one exponent greater than 1 in its prime factorization so √1444 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1444 = (√4)(√361) = (2)(19) = 38
  • The exponents in the prime factorization are 2 and 2. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1444 has exactly 9 factors.
  • The factors of 1444 are outlined with their factor pair partners in the graphic below.

Square number 1444 looks a lot like another square number, 144.
If we keep adding 4’s to the end, will we continue to get square numbers?
No.

However, in different bases, 1444 looks like several other square numbers:
It’s 484 in BASE 18,
400 in BASE 19,
169 in BASE 35,
144 in BASE 36,
121 in BASE 37, and
100 in BASE 38.

1443 A Simple Gift for You

/ ivasallay / Leave a comment

This level one puzzle is my simple gift to you. Yes, you can solve it, and you don’t even have to wait until December 25th to discover all the factors and products to be found inside!

That was puzzle number 1443. Here are some facts about that number.

  • 1443 is a composite number.
  • Prime factorization: 1443 = 3 × 13 × 37
  • 1443 has no exponents greater than 1 in its prime factorization, so √1443 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1443 has exactly 8 factors.
  • The factors of 1443 are outlined with their factor pair partners in the graphic below.

(38-1)(38+1) = 1443 so 1443 is one number away from 38² = 1444.

Actually, 1443 is the difference of two squares in four different ways:
38² – 1² = 1443
62²- 49² = 1443
242² – 239² = 1443
722² – 721² = 1443

1443 is the hypotenuse of FOUR Pythagorean triples:
93-1440-1443 which is 3 times (31-480-481)
468-1365-1443 which is (12-35-37) times 39
555-1332-1443 which is (5-12-13) times 111
957-1080-1443 which is 3 times(319-360-481)

1442 A Birthday Present for My Brother, Andy

/ ivasallay / Leave a comment

My brother, Andy, has a birthday today. He’s very good at solving puzzles, so I made this Challenge puzzle as a  present for him.  Have a very happy birthday, Andy!

You can try to solve it too. If the box and ribbon are too distracting, here’s a copy of the puzzle without the added color. Click on it to see it better.

Print the puzzles or type the solution in this excel file: 10 Factors 1432-1442

That was puzzle number 1442. Here are some facts about that number:

  • 1442 is a composite number.
  • Prime factorization: 1442 = 2 × 7 × 103
  • 1442 has no exponents greater than 1 in its prime factorization, so √1442 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1442 has exactly 8 factors.
  • The factors of 1442 are outlined with their factor pair partners in the graphic below.

2(103)(7) = 1442, making it a leg in this Pythagorean triple:
1442-10560-10658 calculated from 2(103)(7), 103² – 7², 103² + 7²