# 1416 A Birthday Mystery

Today is my sister’s birthday, but the cake is tipped over and there’s a big hole in it! And what happened to the candle? Can you solve this mystery? Happy birthday, Sue! Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Now I’ll share some facts about the puzzle number 1416:

• 1416 is a composite number.
• Prime factorization: 1416 = 2 × 2 × 2 × 3 × 59, which can be written 1416 = 2³ × 3 × 59
• 1416 has at least one exponent greater than 1 in its prime factorization so √1416 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1416 = (√4)(√354) = 2√354
• The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1416 has exactly 16 factors.
• The factors of 1416 are outlined with their factor pair partners in the graphic below. 1416 is the difference of two squares four ways:
355² – 353²  = 1416
179² – 175²  = 1416
121² – 115²  = 1416
65² – 53²  = 1416

# 1415 and Level 6

Very likely when you look at this puzzle common factors of 40 and 10, 8 and 16, 9 and 18, and 20 and 40 will pop into your head. Will they be the right common factors that work with all the other clues in the puzzle to produce a unique solution? Let logic be your guide when finding the factors. Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Now I’ll tell you something about the puzzle number, 1415:

• 1415 is a composite number.
• Prime factorization: 1415 = 5 × 283
• 1415 has no exponents greater than 1 in its prime factorization, so √1415 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 1415 has exactly 4 factors.
• The factors of 1415 are outlined with their factor pair partners in the graphic below. 1415 is also the hypotenuse of a Pythagorean triple:
849-1132-1415 which is (3-4-5) times 283.

# 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)

# 1413 and Level 4

You find a rhythm as you solve this level 4 puzzle. The logic is quite easy for most of it, but there is at least one place that will require you to think things through before proceeding. Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Now I’ll share a few facts about the puzzle number, 1413:

• 1413 is a composite number.
• Prime factorization: 1413 = 3 × 3 × 157, which can be written 1413 = 3² × 157
• 1413 has at least one exponent greater than 1 in its prime factorization so √1413 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1413 = (√9)(√157) = 3√157
• 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 1413 has exactly 6 factors.
• The factors of 1413 are outlined with their factor pair partners in the graphic below. 1413 is the sum of two squares:
33² + 18² = 1413

That means that 1413 is the hypotenuse of a Pythagorean triple:
765-1188-1413 calculated from 33² – 18², 2(33)(18), 33² + 18².
It is also 9 times (85-132-157)

# 1412 and Level 3

If you know the greatest common factor of 56 and 48, then you have taken the first step in solving this puzzle. Once you put the factors of 56 and 48 in the appropriate cells, work down from the top of the puzzle to the bottom, cell by cell, until you have put all the numbers from 1 to 10 in both the first column and the top row. Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Here are a few facts about the puzzle number, 1412:

• 1412 is a composite number.
• Prime factorization: 1412 = 2 × 2 × 353, which can be written 1412 = 2² × 353
• 1412 has at least one exponent greater than 1 in its prime factorization so √1412 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1412 = (√4)(√353) = 2√353
• 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 1412 has exactly 6 factors.
• The factors of 1412 are outlined with their factor pair partners in the graphic below. 1412 is the sum of two squares:
34² + 16² = 1412

1412 is the hypotenuse of a Pythagorean triple:
900-1088-1412 calculated from 34² – 16², 2(34)(16), 34² + 16²

# 1411 and Level 2

Four of the fourteen clues, 18, 24, 16, and 40, appear twice in this puzzle, but do they lead you to the same factors? Where do the factors from 1 to 10 belong that will make this puzzle function like a multiplication table? Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Now I’ll write a little bit about the puzzle number, 1411:

• 1411 is a composite number.
• Prime factorization: 1411 = 17 × 83
• 1411 has no exponents greater than 1 in its prime factorization, so √1411 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 1411 has exactly 4 factors.
• The factors of 1411 are outlined with their factor pair partners in the graphic below. 1411 is the hypotenuse of a Pythagorean triple:
664-1245-1411 which is (8-15-17) times 83.

# 1410 and Level 1

Start the school year off right with a quick review of the multiplication table. You can actually construct an entire 10 × 10 table with only the nine clues in this puzzle. Figure out where the numbers 1 to 10 go in both the first column and the top row and amaze yourself with how much you remember! Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Here are some facts about the puzzle number, 1410:

• 1410 is a composite number.
• Prime factorization: 1410 = 2 × 3 × 5 × 47
• 1410 has no exponents greater than 1 in its prime factorization, so √1410 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 1410 has exactly 16 factors.
• The factors of 1410 are outlined with their factor pair partners in the graphic below. 1410 is the hypotenuse of a Pythagorean triple:
846-1128-1410 which is (3-4-5) times 282.

# 1403 Multiplication Table Challenge

Just because you’re not in elementary school anymore doesn’t mean that the multiplication table can’t be a challenge. This one certainly is. Can you write the numbers 1 to 10 in the four factor areas so that this multiplication table works with the given clues? Don’t get discouraged; it will probably take you at least 15 minutes just to put those factors in the right places. Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll share some information about the puzzle number, 1403:

• 1403 is a composite number.
• Prime factorization: 1403 = 23 × 61
• 1403 has no exponents greater than 1 in its prime factorization, so √1403 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 1403 has exactly 4 factors.
• The factors of 1403 are outlined with their factor pair partners in the graphic below. 1403 is the hypotenuse of a Pythagorean triple:
253-1380-1403 which is 23 times (11-60-61)

# 1402 Mystery Level

Mystery level puzzles may be very difficult or relatively easy. How much trouble will this one be? You’ll have to try it to see! Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll tell you a little bit about the number 1402:

• 1402 is a composite number.
• Prime factorization: 1402 = 2 × 701
• 1402 has no exponents greater than 1 in its prime factorization, so √1402 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 1402 has exactly 4 factors.
• The factors of 1402 are outlined with their factor pair partners in the graphic below. 1402 is the hypotenuse of a Pythagorean triple:
31² + 21² = 1402

1402 is the hypotenuse of a Pythagorean triple:
520-1302-1402 which is 2 times (260-651-701)
and can also be calculated from 2(31)(21), 31² – 21², 31² + 21²

# 1401 Roasting Over an Open Fire

I went camping last week. My family roasted hotdogs. Some people refer to them as mystery meat. Others roasted marshmallows. I was surprised to learn that almost all brands of marshmallows have blue dye in them.  I’m told that without that blue dye the marshmallows will lose their whiteness as they sit on store shelves. Why they have to be that white is a mystery to me.

Here’s a mystery level puzzle for you to solve. It looks a lot like the utensil that was used to roast the hotdogs and marshmallows. Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll tell you something about the number 1401:

• 1401 is a composite number.
• Prime factorization: 1401 = 3 × 467
• 1401 has no exponents greater than 1 in its prime factorization, so √1401 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 1401 has exactly 4 factors.
• The factors of 1401 are outlined with their factor pair partners in the graphic below. 1401 is the difference of two squares in two different ways. Can you figure out what those ways are?