# 1425 Jack O’Lantern

Most of my puzzles don’t have as many clues as this Jack O’lantern puzzle has. Those extra clues could make it easier to solve. On the other hand, some of the clues might still be tricky. Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Here are some facts about the puzzle number, 1425:

• 1425 is a composite number.
• Prime factorization: 1425 = 3 × 5 × 5 × 19, which can be written 1425 = 3 × 5² × 19
• 1425 has at least one exponent greater than 1 in its prime factorization so √1425 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1425 = (√25)(√57) = 5√57
• 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 1425 has exactly 12 factors.
• The factors of 1425 are outlined with their factor pair partners in the graphic below. 1425 is the hypotenuse of TWO Pythagorean triples:
399-1368-1425 which is (7-24-25) times 57
855-1140-1425 which is (3-4-5) times 285

# 1422 Candy Corn Mystery

You can begin this candy corn puzzle easily enough, but the logic needed to solve it is a bit complicated. Good luck with this one! Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Now I’ll tell you some facts about the puzzle number, 1422:

• 1422 is a composite number.
• Prime factorization: 1422 = 2 × 3 × 3 × 79, which can be written 1422 = 2 × 3² × 79
• 1422 has at least one exponent greater than 1 in its prime factorization so √1422 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1422 = (√9)(√158) = 3√158
• 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 1422 has exactly 12 factors.
• The factors of 1422 are outlined with their factor pair partners in the graphic below. 1422 is palindrome 1K1 in BASE 29 (K is 20 base 10)
because 1(29²) + 20(29¹) + 1(29º) = 1422.

# 1417 Mystery Puzzle

How hard is today’s puzzle? It’s a little harder just because I’m not telling what the level number is. Are you going to let that stop you from finding the unique solution? I hope not! Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

1417 is just the puzzle number, but in case you want to know something about it, here are some facts:

• 1415 is a composite number.
• Prime factorization: 1415 = 13 × 109
• 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. 1417 is the sum of two squares in two different ways:
29² + 24² = 1417
36² + 11² = 1417

1417 is the hypotenuse of FOUR Pythagorean triples:
265-1392-1417 calculated from 29² – 24², 2(29)(24), 29² + 24²
545-1308-1417 which is (5-12-13) times 109
780-1183-1417 which is 13 times (60-91-109)
792-1175-1417 calculated from 2(36)(11) , 36² – 11² , 36² + 11²

# 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

# 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?

# 1388 Mystery Level

Sometimes puzzles start out easy enough but get a little more complicated later on. Does that happen with this puzzle? There’s only one way to find out! Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here are some facts about the number 1388:

• 1388 is a composite number.
• Prime factorization: 1388 = 2 × 2 × 347, which can be written 1388 = 2² × 347
• 1388 has at least one exponent greater than 1 in its prime factorization so √1388 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1388 = (√4)(√347) = 2√347
• 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 1388 has exactly 6 factors.
• The factors of 1388 are outlined with their factor pair partners in the graphic below. 1388 is the difference of two squares:
348² – 346² = 1388

# 1385 Mystery Level

You can suspect that the common factor of 9 and 6 is either 1 or 3, but don’t jump to conclusions about which one will satisfy this mystery! There’s important evidence elsewhere in the puzzle that you should consider first. Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll share some facts about the puzzle number, 1385:

• 1385 is a composite number.
• Prime factorization: 1385 = 5 × 277
• 1385 has no exponents greater than 1 in its prime factorization, so √1385 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 1385 has exactly 4 factors.
• The factors of 1385 are outlined with their factor pair partners in the graphic below. 1385 is the sum of two squares in two different ways:
32² + 19² = 1385
37² + 4² = 1385

1385 is the hypotenuse of a Pythagorean triple:
296-1353-1385 calculated from 2(37)(4), 37² – 4², 37² + 4²
575-1260-1385 which is 5 times (115-252-277)
663-1216-1385 calculated from 32² – 19², 2(32)(19), 32² + 19²
831-1108-1385 which is (3-4-5) times 277

# 1365 Shamrock Mystery

Beautiful shamrocks with their three heart-shaped leaves are not difficult to find. Finding the factors in this shamrock-shaped puzzle might be a different story.  Sure, it might start off to be easy, but after a while, you might find it a wee bit more difficult, unless, of course, the luck of the Irish is with you! Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Now I’ll share some information about the number 1365:

• 1365 is a composite number.
• Prime factorization: 1365 = 3 × 5 × 7 × 13
• The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1365 has exactly 16 factors.
• Factors of 1365: 1, 3, 5, 7, 13, 15, 21, 35, 39, 65, 91, 105, 195, 273, 455, 1365
• Factor pairs: 1365 = 1 × 1365, 3 × 455, 5 × 273, 7 × 195, 13 × 105, 15 × 91, 21 × 65, or 35 × 39
• 1365 has no square factors that allow its square root to be simplified. √1365 ≈ 36.94591 1365 is the hypotenuse of FOUR Pythagorean triples:
336-1323-1365 which is 21 times (16-63-65)
525-1260-1365 which is (5-12-13) times 105
693-1176-1365 which is 21 times (33-56-65)
819-1092-1365 which is (3-4-5) times 273

1365 looks interesting in some other bases:
It’s 10101010101 in BASE 2,
111111 in BASE 4,
2525 in BASE 8, and
555 in BASE 16

I’m feeling pretty lucky that I noticed all those fabulous number facts! If you haven’t been so lucky finding the factors of the puzzle, the same puzzle but with more clues might help: # 1356 Mystery

The clues that appear in this puzzle are all you need to find all the factors. Seriously. Some mysteries are easier to solve than others. Give this one a try! Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

Here is some information about the puzzle number, 1356:

• 1356 is a composite number.
• Prime factorization: 1356 = 2 × 2 × 3 × 113, which can be written 1356 = 2² × 3 × 113
• 1356 has at least one exponent greater than 1 in its prime factorization so √1356 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1356 = (√4)(√339) = 2√339
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1356 has exactly 12 factors.
• The factors of 1356 are outlined with their factor pair partners in the graphic below. 1356 is the hypotenuse of a Pythagorean triple:
180-1344-1356 which is 12 times (15-112-113)