1555 Two Turkeys Too Tough To Try?

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?

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.

1552 Look for Clues around the Corner

Today’s Puzzle:

This puzzle has four sets of clues that turn the corner. You will need to look around those corners to solve it. Use logic and have fun!

Factors of 1552:

  • 1552 is a composite number.
  • Prime factorization: 1552 = 2 × 2 × 2 × 2 × 97, which can be written 1552 = 2⁴ × 97
  • 1552 has at least one exponent greater than 1 in its prime factorization so √1552 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1552 = (√16)(√97) = 4√97
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1552 has exactly 10 factors.
  • The factors of 1552 are outlined with their factor pair partners in the graphic below.

More about the Number 1552:

1552 is the sum of two squares:
36² + 16²  = 1552

1552 is the hypotenuse of a Pythagorean triple:
1040-1152-1552, calculated from 36² – 16², 2(36)(16), 36² + 16².
It is also 16 times (65-72-97).

OEIS.org looked around the corner at the two numbers preceding 1552 to find something special about that number: The sum of its prime factors equals the sum of the prime factors of those previous two numbers! That’s a cool enough fact that I decided to make this graphic:

1551 Two Straight Lines

Today’s Puzzle:

The clues in this level 4 puzzle form two straight lines, and most of the logic needed to solve it is rather straightforward. Can you find the factors from 1 to 10 without getting twisted up?

Factors of 1551:

1+5 = 6, a multiple of 3, so 1551 is divisible by 3.
Since 1551 is a palindrome with an even number of digits, it is also divisible by 11.

  • 1551 is a composite number.
  • Prime factorization: 1551 = 3 × 11 × 47.
  • 1551 has no exponents greater than 1 in its prime factorization, so √1551 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 1551 has exactly 8 factors.
  • The factors of 1551 are outlined with their factor pair partners in the graphic below.

A Little More about the Number 1551:

Did you notice how many of 1551’s factors are palindromes?

1551 is the difference of two squares in four different ways:
776² – 775² = 1551,
260² – 257² = 1551,
76² – 65² = 1551, and
40² – 7² = 1551. (Yes, we are just 7² or 49 numbers away from 1600, the next perfect square!)

1550 Lucky for You: Solving This Puzzle Is as Easy as Climbing Down a Ladder

Today’s Puzzle:

It’s Friday the 13th, so don’t walk under any ladders! Still, there isn’t any reason to avoid them entirely. Because this ladder puzzle is a level 3, the clues are given in a logical order to help you find the solution. Start at the top of the ladder, find the common factor of 10 and 18, then work your way down the ladder rung by rung, writing all the numbers from 1 to 10 in both the first column and the top row until you reach the bottom of the ladder.  Good Luck!

Here’s the same puzzle with no colors to distract you.

Factors of 1550:

  • 1550 is a composite number.
  • Prime factorization: 1550 = 2 × 5 × 5 × 31, which can be written 1550 = 2 × 5² × 31
  • 1550 has at least one exponent greater than 1 in its prime factorization so √1550 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1550 = (√25)(√62) = 5√62
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1550 has exactly 12 factors.
  • The factors of 1550 are outlined with their factor pair partners in the graphic below.

1550 Factor Tree:

Here’s one of a few possible different factor trees for 1550:

More about the Number 1550:

1550 is the hypotenuse of TWO Pythagorean triples:
434-1488-1550, which is (7-24-25) times 62, and
930-1240-1550, which is (3-4-5) times 310.

1548 Puzzling Gerrymandering Questions

Today’s Puzzle:

If you had two types of candy and four kids, this is a good way to divide the candy. But if you had two types of voters and four congressional districts, is it a fair way to determine those congressional districts?

Gerrymandering happens when congressional district boundaries are drawn to give an advantage to one political party over another. The green party might be given an advantage in the drawing above, but does simply dividing the graphic into four quadrants give an advantage to the yellow party?

One article complains that one of the worst examples of gerrymandering is a congressional district shaped like a duck, but it is unclear if that duck is keeping like-minded people together or keeping them apart.

or

It is unclear to us if like-minded people were grouped together, but we can be certain that it was very clear to those who drew the boundaries.

Each of the graphics above had 12 yellow sections and 36 green sections. If you think it is only fair to let like-minded people elect someone who thinks like them,  how should congressional boundaries be drawn if the 12 yellow sections and the 36 green sections look like this?

No matter how you draw the boundaries, green will be in the majority in each congressional district, and most likely the majority will choose each district’s representative. But if you believe that gerrymandering is justified to benefit like-minded people, should those people NOT be represented by a like-minded representative simply because they don’t live next to each other?

These are good questions to puzzle over. Denise Gaskins has created the Gerrymandering Project to help you manipulate a 10 by 10 map of your creation in a variety of ways. This project will help every voter and future voter understand the mathematics and the politics of drawing boundaries on a larger scale. Check it out!

Factors of 1548:

  • 1548 is a composite number.
  • Prime factorization: 1548 = 2 × 2 × 3 × 3 × 43, which can be written 1548 = 2² × 3² × 43
  • 1548 has at least one exponent greater than 1 in its prime factorization so √1548 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1548 = (√36)(√43) = 6√43
  • The exponents in the prime factorization are 2, 2, and 1. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1548 has exactly 18 factors.
  • The factors of 1548 are outlined with their factor pair partners in the graphic below.

Factor Tree for 1548:

The last two digits of 1548 are a multiple of 4, so 1548 is divisible by 4.
1 + 5 + 4 + 8 = 18, a multiple of 9, so 1548 is divisible by 9.

Here’s how I used those two facts to make an autumn factor tree for 1548:

More about the Number 1548:

1548 is the difference of two squares in three different ways:
388² – 386² = 1548,
132² – 126² = 1548, and
52² – 34²  = 1548.

 

1547 is a Hexagonal Pyramidal Number

Today’s Puzzle:

1547 is the 13th hexagonal pyramidal number. Looking at the graphic that I made below of 1547 tiny squares, can you determine what the prime factorization of 1547 is?

Factors of 1547:

  • 1547 is a composite number.
  • Prime factorization: 1547 = 7 × 13 × 17.
  • 1547 has no exponents greater than 1 in its prime factorization, so √1547 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 1547 has exactly 8 factors.
  • The factors of 1547 are outlined with their factor pair partners in the graphic below.

More about the Number 1547:

1547 is the hypotenuse of FOUR Pythagorean triples:
147-1540-1547, which is 7 times (21-220-221),
595-1428-1547, which is (5-12-13) times 119,
728-1365-1547, which is (8-15-17) times 91, and
980-1197-1547, which is 7 times (140-171-221).

1546 Celebrating My 7th Blogiversary

Today’s Puzzle:

Seven years ago today at 2:11 in the morning, I published my first blog post. It featured a Find the Factors 1 – 10 puzzle with only perfect square clues. In honor of that first post, I publish this puzzle, also with only perfect square clues. This one looks like an exclamation point because I have enjoyed creating puzzles and writing these posts for you so much these seven years!

How do you solve the puzzle? Use logic to write the numbers from 1 to 10 in both the first column and the top row so that the puzzle functions like a multiplication table.

Factors of 1546:

  • 1546 is a composite number.
  • Prime factorization: 1546 = 2 × 773.
  • 1546 has no exponents greater than 1 in its prime factorization, so √1546 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 1546 has exactly 4 factors.
  • The factors of 1546 are outlined with their factor pair partners in the graphic below.

More about the Number 1546:

1546 is the sum of two squares:
39² + 5² = 1546.

1546 is the hypotenuse of a Pythagorean triple:
390-1496-1546, calculated from 2(39)(5), 39² – 5², 39² + 5².
It is also 2 times (195-748-773)

1544 Final Letter of the Message

Today’s Puzzle:

This is the sixth and final letter of the message that I made for you. It’s a mystery puzzle, but that doesn’t necessarily mean that’s it’s a difficult puzzle. Give it a try and think about why I sent the message.

Factors of 1544:

  • 1544 is a composite number.
  • Prime factorization: 1544 = 2 × 2 × 2 × 193, which can be written 1544 = 2³ × 193
  • 1544 has at least one exponent greater than 1 in its prime factorization so √1544 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1544 = (√4)(√386) = 2√386
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1544 has exactly 8 factors.
  • The factors of 1544 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1544:

1544 is the sum of two squares:
38² + 10² = 1544.

1544 is the hypotenuse of a Pythagorean triple:
760-1344-1544, calculated from 2(38)(10), 38² – 10², 38² + 10².
It is also 8 times (95-168-193).

1542 Fourth Letter of the Message

Today’s Puzzle:

This level one puzzle is the easiest of all the puzzles in my message to you. It might also be the letter that helps you know exactly what the message says.

Factors of 1542:

  • 1542 is a composite number.
  • Prime factorization: 1542 = 2 × 3 × 257.
  • 1542 has no exponents greater than 1 in its prime factorization, so √1542 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 1542 has exactly 8 factors.
  • The factors of 1542 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1542

1542 is the hypotenuse of a Pythagorean triple:
192-1530-1542, which is 6 times (32-255-257).