# 1447 Christmas Light Puzzle

If you’ve ever had a string of lights go out because ONE bulb went bad, it can be a very frustrating puzzle to figure out which light is causing the problem.

This is not that kind of puzzle. For this one, you just need to figure out where to put 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. There is only one solution, and if you always use logic, it will not be a frustrating puzzle to solve. I gave that puzzle the puzzle number 1447. That number won’t help you solve the puzzle, but here are some facts about it anyway:

• 1447 is a prime number.
• Prime factorization: 1447 is prime.
• 1447 has no exponents greater than 1 in its prime factorization, so √1447 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1447 has exactly 2 factors.
• The factors of 1447 are outlined with their factor pair partners in the graphic below. How do we know that 1447 is a prime number? If 1447 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1447. Since 1447 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1447 is a prime number.

1447 is also the difference of two consecutive squares:
724² – 723² = 1447

# 1438 and Level 5

You can solve this puzzle by using logic and multiplication/division facts. The unique solution requires all the numbers from 1 to 10 in both the first column and the top row. Can you solve it? Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

Today’s puzzle was given the number 1438. Here are a few facts about that number:

• 1438 is a composite number.
• Prime factorization: 1438 = 2 × 719
• 1438 has no exponents greater than 1 in its prime factorization, so √1438 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 1438 has exactly 4 factors.
• The factors of 1438 are outlined with their factor pair partners in the graphic below. 1438 is the sum of four consecutive numbers:
358  + 359 + 360 + 361 = 1438

# 1423 Lollipop

Lollipops are candies that easily delight children. Will today’s lollipop-wannabe puzzle be a delight? Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

The puzzle number is 1423. Here are a few facts about that number:

• 1423 is a prime number.
• Prime factorization: 1423 is prime.
• 1423 has no exponents greater than 1 in its prime factorization, so √1423 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1423 has exactly 2 factors.
• The factors of 1423 are outlined with their factor pair partners in the graphic below. How do we know that 1423 is a prime number? If 1423 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1423. Since 1423 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1423 is a prime number.

# 1414 Your Math Education Post Will Add So Much to This Month’s Carnival!

Have you written a blog post that would bring delight to a preschool, K-12 or homeschool mathematics teacher or student? Then submit it to this month’s Playful Math Education Blog Carnival or message me on Twitter by Friday, September 20th! I’m hosting the carnival this month, and I would love to read your post. So come join the fun!

Today’s puzzle looks a little like a wild, but fun? carnival ride. The numbers 36 and 12 went together on the ride. They managed to stay with each other but the ride went so fast, you can see 36 and 12 in two different places at the same time. There’s also poor number 40. You can see it in THREE places at the same time.

Oh my! Can you use logic to find where the numbers 1 to 10 need to go in both the first column and the top row so that this wild ride will behave like a multiplication table? It’s a level 5 so it won’t be easy to find its unique solution. Are you brave enough to try? Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

That puzzle’s number is 1414. Let me tell you a little about that number:

• 1414 is a composite number.
• Prime factorization: 1414 = 2 × 7 × 101
• 1414 has no exponents greater than 1 in its prime factorization, so √1414 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 1414 has exactly 8 factors.
• The factors of 1414 are outlined with their factor pair partners in the graphic below. 1414 is also the hypotenuse of a Pythagorean triple:
280-1386-1414 which is 14 times (20-99-101)

# 1398 and Level 5

You might find this puzzle to be a little tricky, but if you always use logic before you write any of the factors, you should succeed! Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here is some information about the number 1398:

• 1398 is a composite number.
• Prime factorization: 1398 = 2 × 3 × 233
• 1398 has no exponents greater than 1 in its prime factorization, so √1398 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 1398 has exactly 8 factors.
• The factors of 1398 are outlined with their factor pair partners in the graphic below. 1398 is the hypotenuse of a Pythagorean triple:
630-1248-1398 which is 6 times (105-208-233)

# 1383 and Level 5

Level 5 puzzles aren’t any harder than level 4 puzzles unless I trick you into starting with the common factor of a pair of clues that have more than one possibility. You won’t let me trick you, will you? Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here are a few facts about the number 1383:

• 1383 is a composite number.
• Prime factorization: 1383 = 3 × 461
• 1383 has no exponents greater than 1 in its prime factorization, so √1383 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 1383 has exactly 4 factors.
• The factors of 1383 are outlined with their factor pair partners in the graphic below. 1383 is the hypotenuse of a Pythagorean triple:
783-1140-1383 which is 3 times (261-380-461)

# 1361 and Level 5

If you carefully use logic instead of guessing and checking you can find the unique solution to this puzzle without tearing your hair out! Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are some facts about the number 1361:

• 1361 is a prime number.
• Prime factorization: 1361 is prime.
• 1361 has no exponents greater than 1 in its prime factorization, so √1361 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1361 has exactly 2 factors.
• The factors of 1361 are outlined with their factor pair partners in the graphic below.

How do we know that 1361 is a prime number? If 1361 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1361. Since 1361 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1361 is a prime number. 1361 is the first prime number after 1327. That was 34 numbers ago!

1361 is the sum of two squares:
31² + 20² = 1361

1361 is the hypotenuse of a Pythagorean triple:
561-1240-1361 calculated from 31² – 20², 2(31)(20), 31² + 20²

Here’s another way we know that 1361 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 20² = 1361 with 31 and 20 having no common prime factors, 1361 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1361. Since 1361 is not divisible by 5, 13, 17, or 29, we know that 1361 is a prime number.

# 1354 Solving a Level 5 Puzzle

What are the common factors of 16 and 4? Don’t guess which one to use. Use logic to figure it out as you find all the factors for this puzzle! Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

If you get stuck, you can watch this video:

Now I’ll share some information about the puzzle number, 1354:

• 1354 is a composite number.
• Prime factorization: 1354 = 2 × 677
• 1354 has no exponents greater than 1 in its prime factorization, so √1354 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 1354 has exactly 4 factors.
• The factors of 1354 are outlined with their factor pair partners in the graphic below. 1354 is the sum of two squares:
27² + 25² = 1354

1354 is the hypotenuse of a Pythagorean triple:
104-1350-1354 which is 2 times (52-675-677)
and can also be calculated from 27² – 25², 2(27)(25), 27² + 25²

# 1338 and Level 5

Finding the most logical place to start a level 5 puzzle is only a little bit trickier than for a level 4 puzzle. Will you figure it out and not be tricked? Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here is some information about the number 1338:

• 1338 is a composite number.
• Prime factorization: 1338 = 2 × 3 × 223
• 1338 has no exponents greater than 1 in its prime factorization, so √1338 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 1338 has exactly 8 factors.
• The factors of 1338 are outlined with their factor pairs in the graphic below. 1338 is in a couple of Pythagorean triples:
1338-447560-447562 calculated from 2(669)(1), 669² – 1², 669² + 1² and
1338-49720-49738 calculated from 2(223)(3), 223² – 3², 223² + 3²

# 1328 Christmas Tree

Can you figure out where to put the numbers from 1 to 10 in both the first column and the top row so that the lights on this Christmas tree work properly?

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

Now I’ll share some facts about the puzzle number, 1328:

• 1328 is a composite number.
• Prime factorization: 1328 = 2 × 2 × 2 × 2 × 83, which can be written 1328 = 2⁴ × 83
• The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1328 has exactly 10 factors.
• Factors of 1328: 1, 2, 4, 8, 16, 83, 166, 332, 664, 1328
• Factor pairs: 1328 = 1 × 1328, 2 × 664, 4 × 332, 8 × 166, or 16 × 83
• Taking the factor pair with the largest square number factor, we get √1328 = (√16)(√83) = 4√83 ≈ 36.44173

Because 28 is divisible by 4, but not by 8, and 3 (the digit before the 28) is an odd number, I know that 1328 is divisible by 8. I can use that fact to make this simple factor tree: 1328 is the difference of two squares three different ways:
333² – 331² = 1328
168² – 164² = 1328
87² – 79²  = 1328