Author: ivasallay

1563 The Holly Wreath

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Today’s puzzle:

A holly wreath is yet another symbol that connects Christmas with Easter. It symbolizes eternity in its color and shape. It bears white flowers, red berries, and thorns reminding us of purity, blood, and a crown of thorns.

You might find some of the clues in this level 5 puzzle to be like thorns, but don’t give up. Use logic and perseverance and you will be able to find its unique solution.

Here’s the same puzzle without all the added color:

Factors of 1563:

  • 1563 is a composite number.
  • Prime factorization: 1563 = 3 × 521.
  • 1563 has no exponents greater than 1 in its prime factorization, so √1563 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1563 has exactly 4 factors.
  • The factors of 1563 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1563:

1563 is the hypotenuse of a Pythagorean triple:
837-1320-1563, which is 3 times (279-440-521).

1562 Evergreen Tree

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Today’s Puzzle:

An evergreen tree doesn’t drop its leaves in the fall or look dead in the winter. As it reminds us of everlasting life, it makes a lovely symbol of Christmas.

Write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues function like a multiplication table. Here is the same puzzle that won’t use as much ink to print:

Factors of 1562:

  • 1562 is a composite number.
  • Prime factorization: 1562 = 2 × 11 × 71.
  • 1562 has no exponents greater than 1 in its prime factorization, so √1562 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 1562 has exactly 8 factors.
  • The factors of 1562 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1562:

2(5⁴ + 5³ + 5² + 5¹ + 5⁰) = 1562

1561 Virgács for Boots and Stockings

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Today’s Puzzle:

Children living in Hungary put their nicely polished boots or stockings by a window for Mikulás (Saint Nicholas) to fill tonight. When they awake in the morning, they will find candies, and maybe nuts or fruit to reward them for the good they’ve done this past year. Because even the best children have been at least a little bit naughty sometime during the year, they will also find virgács, gold-painted twigs typically bound together with red ribbon. Now, if a child lives in a place where virgács is not available at the local market, Mikulás could copy today’s virgács puzzle and put it in any boot or stocking left out for him tonight.

Since this is a level 3 puzzle, the clues are listed in a logical order from the top of the puzzle to the bottom. After the factors of 12 and 40 are put in their respective cells, the rest of the factors can be found by working down the puzzle cell by cell until all the factors are written in.

Factors of 1561:

  • 1561 is a composite number.
  • Prime factorization: 1561 = 7 × 223.
  • 1561 has no exponents greater than 1 in its prime factorization, so √1561 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1561 has exactly 4 factors.
  • The factors of 1561 are outlined with their factor pair partners in the graphic below.

More about the Number 1561:

1561 is the sum of two consecutive numbers:
780 + 781 = 1561.

1561 is also the difference of two consecutive square numbers:
781² – 780² = 1561.

Did you notice a pattern in those two statements?

1561 is the sum of seven consecutive numbers:
220 + 221 + 222 + 223 + 224 + 225 + 226 = 1561.

1561 is the sum of the fourteen consecutive numbers from 105 to 118.

1561 is the difference of these two other square numbers:
115² – 108² = 1561.

Did you notice any other patterns? Does your pattern hold true for other multiples of 7?

Why Do Factor Pairs of 1560 Make Sum-Difference?

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Today’s Puzzle:

1560 has 16 different factor pairs. One of those pairs sum up to 89, and another pair subtracts to 89. It is only the 50th time that the sum of a factor pair of a number equals the difference of one of its other factor pairs.

You may have seen other Sum-Difference Puzzles where I’ve paired one puzzle with another puzzle and mentioned that the second puzzle was really the first puzzle in disguise. That is not the case for this puzzle because 89 is a prime number. This puzzle is a primitive; there is not a simpler puzzle that is its equivalent. Don’t let that worry you, however, everything you need to solve this puzzle can be found in this post.

Although I am making a big deal about our number 1560, it is the 89 which allows this puzzle to exist in the first place. You see, 89 is the hypotenuse of a Pythagorean triple, (39-80-89), and thus, 39² + 80² = 89². Since that triple is a primitive, the sum-difference is a primitive as well.

Note that (40)(39) = 1560.
We want to use the quadratic formula to solve 40x² + 89x + 39 = 0 and 40x² + 89x – 39 = 0. Let’s combine the left sides of those two equations into one expression:
40x² + 89x ± 39.
The discriminant would be 89² – 4(40)(±39)
= 89² ± 4(40)(39)
= 89² ± 2(80)(39)
= 39² + 80² ± 2(80)(39), ( That’s because 39² + 80² = 89².)
= 80² ± 2(80)(39) + 39²
= (80 ± 39)², a perfect square!

That perfect square makes 1560 one of those relatively rare numbers with factor pairs that make sum-difference.

Factors of 1560:

  • 1560 is a composite number.
  • Prime factorization: 1560 = 2 × 2 × 2 × 3 × 5 × 13 , which can be written 1560 = 2³ × 3 × 5 × 13.
  • 1560 has at least one exponent greater than 1 in its prime factorization so √1560 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1560 = (√4)(√390) = 2√390.
  • The exponents in the prime factorization are 2, 1, and 2. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 × 2 = 32. Therefore 1560 has exactly 32 factors.
  • The factors of 1560 are outlined with their factor pair partners in the graphic below.

That’s a lot of factor pairs for one number! Here’s a graphic showing those same factor pairs but with their sums and differences included:

Find the 89 in the Sum column and the 89 in the Difference column, and you will see the factor pairs by those 89’s that make the sum and difference needed to solve the puzzle.

More about the Number 1560:

1560 = 39 × 40, so 1560 is the sum of the first 39 even numbers:
2 + 4 + 6 + 8 + . . . + 74 + 76 + 78 = 1560.

1560 is the hypotenuse of FOUR Pythagorean triples:
384-1512-1560, which is 24 times (16-63-65),
600-1440-1560, which is (5-12-13) times 120,
792-1344-1560, which is 24 times (33-56-65), and
936-1248-1560, which is (3-4-5) times 312.

Since 1560 is a hypotenuse four different ways, could it be the bottom part of four different Sum-Difference puzzles? Yes!

You can find the top part of the puzzle by finding one half of the product of the first two numbers in each triple:
384 × 1512 ÷ 2 = ____________,
600 × 1440 ÷ 2 = ____________,
792 × 1344 ÷ 2 = ____________,
936 × 1248 ÷ 2 = ____________.

The numbers that go in the blanks are all between 100 thousand and one million, but each one of those numbers will have a factor pair that adds up to 1560 as well as another one that subtracts to 1560. If you find them, go ahead and brag about it! It will be quite an accomplishment!

A Factor Tree for 1560:

Here is one of MANY possible factor trees for 1560:

 

1559 We Need a Little Christmas Now

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Today’s Puzzle:

2020 reminds me of another difficult year, 2001. In December of that year, Angela Lansbury sang We Need a Little Christmas Now with the Tabernacle Choir at Temple Square. I was able to watch the concert on television, and I remember the feeling the music brought me. What a wonderful gift music is! Yes, in 2020, we need a little Christmas now!

This level 2 puzzle brings a little Christmas now. Write the numbers from 1 to 12 in both the first column and the top row so that the puzzle functions as a type of multiplication table. I’m pretty sure you can figure it out!

Factors of 1559:

  • 1559 is a prime number.
  • Prime factorization: 1559 is prime.
  • 1559 has no exponents greater than 1 in its prime factorization, so √1559 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1559 has exactly 2 factors.
  • The factors of 1559 are outlined with their factor pair partners in the graphic below.

How do we know that 1559 is a prime number? If 1559 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1559. Since 1559 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1559 is a prime number.

More about the number 1559:

1559 is the sum of two consecutive numbers:
780 + 779 = 1559.

1559 is also the difference of two consecutive square numbers:
780² – 779² = 1559.

(Yes, I know, any odd whole number can make a similar claim.)

1558 The Reason for the Season

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Today’s Puzzle:

What is the reason we have the Christmas season? The answer to that question actually begins with a cross…

Write the numbers from 1 to 12 in both the first column and the top row so that the given clues are the products of those numbers.

Factors of 1558:

  • 1558 is a composite number.
  • Prime factorization: 1558 = 2 × 19 × 41.
  • 1558 has no exponents greater than 1 in its prime factorization, so √1558 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 1558 has exactly 8 factors.
  • The factors of 1558 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1558:

1558 is the hypotenuse of a Pythagorean triple:
342-1520-1558, which is 38 times (9-40-41).

1557 Happy Birthday to My Brother, Andy!

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Today’s puzzle:

Today is my brother Andy’s birthday. He enjoys solving puzzles so I made this 18 × 18 puzzle hoping that he will find it challenging.

This 18 × 18 multiplication table will be invaluable as you work to solve it. When you look for a clue in the table, its color will let you know how many times it appears in the table.

Print the puzzles or type the solution in this excel file: 10 Factors 1546-1557.

If you need a hint to solve the puzzle: One of the first things you will want to do is identify the clues that are multiples of 5, 10, or 15. Then use logic to determine which clues will use the two 5’s, the two 10’s, and the two 15’s.

Factors of 1557:

  • 1557 is a composite number.
  • Prime factorization: 1557 = 3 × 3 × 173, which can be written 1557 = 3² × 173.
  • 1557 has at least one exponent greater than 1 in its prime factorization so √1557 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1557 = (√9)(√173) = 3√173.
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1557 has exactly 6 factors.
  • The factors of 1557 are outlined with their factor pair partners in the graphic below.

More about the number 1557:

From OEIS.org we learn that 1557 has a rather fun square:
1557² = 2424249.

1557 is the sum of two squares:
39² + 6² = 1557.

1557 is the hypotenuse of a Pythagorean triple:
468-1485-1557, which is 9 times (52-165-173).
It can also be calculated from 2(39)(6), 39² – 6², 39² + 6².

1557 is also the difference of two squares three different ways:
779² – 778² = 1557,
261² – 258² = 1557, and
91² – 82² = 1557.

1556 Stacks Up Nicely!

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What is special about the number 1556?

What makes 1556 stack up?

From OEIS.org we learn that
2² + 3² + 5² + 7² + 11² + 13² + 17² + 19² + 23² =  1556.
Yes, that’s the sum of the squares of the first nine prime numbers.
Those perfect squares can be stacked on top of each other as I illustrate in the graphic below:

Factors of 1556:

1556 (and every other whole number whose last two digits are 56) is divisible by 4:

  • 1556 is a composite number.
  • Prime factorization: 1556 = 2 × 2 × 389, which can be written 1556 = 2² × 389.
  • 1556 has at least one exponent greater than 1 in its prime factorization so √1556 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1556 = (√4)(√389) = 2√389.
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1556 has exactly 6 factors.
  • The factors of 1556 are outlined with their factor pair partners in the graphic below.

More about the Number 1556:

1556 is the sum of two squares:
34² + 20² = 1556.

1556 is the hypotenuse of a Pythagorean triple:
756 -1360-1556, which is 4 times (189-340-389).
It can also be calculated from 34² – 20², 2(34)(20), 34² + 20².

1556 is also the difference of two squares:
390²  – 388²  = 1556.

1555 Two Turkeys Too Tough To Try?

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Today’s Puzzle:

Two turkeys too tough to try? That’s a six-word title made with alliteration and three homophones! It also describes the mystery-level turkey puzzles below. Those turkeys might look like identical twins at first glance, but if you look closely, you will see they are not quite the same.

Here are some questions to help you find a logical way to start either puzzle: Which two clues MUST use the two 6’s as factors? Are there any other clues that are multiples of 6? If so, what factors would those clues use?

Factors of 1555:

  • 1555 is a composite number.
  • Prime factorization: 1555 = 5 × 311.
  • 1555 has no exponents greater than 1 in its prime factorization, so √1555 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1555 has exactly 4 factors.
  • The factors of 1555 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1555:

1555 is the hypotenuse of a Pythagorean triple:
933-1244-1555, which is (3-4-5) times 311.

1554 What Patterns Do You See?

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Today’s Puzzle:

I like multiples of 111, including 1554. What cool patterns do you notice if a 2-digit number is multiplied by 111 as shown in the graphic below:

A Factor Tree for 1554:

Here’s a factor tree for 1554 that begins with the factor pair 14 × 111:

Factors of 1554:

  • 1554 is a composite number.
  • Prime factorization: 1554 = 2 × 3 × 7 × 37.
  • 1554 has no exponents greater than 1 in its prime factorization, so √1554 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1554 has exactly 16 factors.
  • The factors of 1554 are outlined with their factor pair partners in the graphic below.

More about the Number 1554:

1554 is the hypotenuse of a Pythagorean triple:
504-1470-1554, which is (12-35-37) times 42.