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
Now I’ll tell you a little bit about the puzzle number, 1353:
1353 is a composite number.
Prime factorization: 1353 = 3 × 11 × 41
1353 has no exponents greater than 1 in its prime factorization, so √1353 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 1353 has exactly 8 factors.
The factors of 1353 are outlined with their factor pair partners in the graphic below.
1353 is the hypotenuse of a Pythagorean triple:
297-1320-1353 which is 33 times (9-40-41)
If you know what numbers divide evenly into both 3 and 15, then you can solve this puzzle. Only use factor pairs where both numbers are from 1 to 10. Start at the top of the puzzle and work down cell by cell until you have found all the factors!
Here are a few facts about the puzzle number, 1352:
1352 is a composite number.
Prime factorization: 1352 = 2 × 2 × 2 × 11 × 11, which can be written 1352 = 2³ × 13²
1352 has at least one exponent greater than 1 in its prime factorization so √1352 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1352 = (√676)(√2) = 26√2
The exponents in the prime factorization are 3 and 2. Adding one to each exponent and multiplying we get (3 + 1)(2 + 1) = 4 × 3 = 12. Therefore 1352 has exactly 12 factors.
The factors of 1352 are outlined with their factor pair partners in the graphic below.
1352 is the hypotenuse of two Pythagorean triples:
520-1248-1352 which is (5-12-13) times 104
952-960-1352 which is 8 times (119-120-169)
If you want a little help solving the puzzle, I explain how to do it in this video:
Here are some facts about the puzzle number 1351:
1351 is a composite number.
Prime factorization: 1351 = 7 × 193
1351 has no exponents greater than 1 in its prime factorization, so √1351 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 1351 has exactly 4 factors.
The factors of 1351 are outlined with their factor pair partners in the graphic below.
1351 is the hypotenuse of a Pythagorean triple:
665-1176-1351 which is 7 times (95-168-193)
By simply changing two clues of that recently published puzzle that I rejected, I was able to create a love-ly puzzle that can be solved entirely by logic. Can you figure out where to put the numbers from 1 to 12 in each of the four outlined areas that divide the puzzle into four equal sections? If you can, my heart might just skip a beat!
If you need some tips on how to get started on this puzzle, check out this video:
Factors of 1350:
1350 is a composite number.
Prime factorization: 1350 = 2 × 3 × 3 × 3 × 5 × 5, which can be written 1350 = 2 × 3³ × 5²
The exponents in the prime factorization are 1, 3 and 2. Adding one to each and multiplying we get (1 + 1)(3 + 1)(2 + 1) = 2 × 4 × 3 = 24. Therefore 1350 has exactly 24 factors.
Taking the factor pair with the largest square number factor, we get √1350 = (√225)(√6) = 15√6 ≈ 36.74235
Sum-Difference Puzzles:
6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.
1350 has twelve factor pairs. One of the factor pairs adds up to 75, and a different one subtracts to 75. If you can identify those factor pairs, then you can solve the second puzzle!
The second puzzle is really just the first puzzle in disguise. Why would I say that?
More about the Number 1350:
1350 is the sum of consecutive prime numbers two ways: It is the sum of the fourteen prime numbers from 67 to 131, and 673 + 677 = 1350
1350 is the hypotenuse of two Pythagorean triples: 810-1080-1350 which is (3-4-5) times 270 378-1296-1350 which is (7-24-25) times 54
1350 is also the 20th nonagonal number because 20(7 · 20 – 5)/2 = 1350
I was in the mood to make a Find the Factors Challenge Puzzle that used the numbers from 1 to 12 as the factors. I’ve never made such a large puzzle before, but after I made it, I rejected it. All the puzzles I make must meet certain standards: they must have a unique solution, and that solution must be obtainable by using logic. Although the “puzzle” below has a unique solution, and you can fill in a few of the cells using logic, you would have to use guess and check to finish it. Besides that, you wouldn’t be able to know if you guessed right until almost the entire puzzle was completed. Thus, it doesn’t meet my standards.
Even though the puzzle was rejected, there were still some things about it that I really liked. In my next post, I’ll publish a slightly different puzzle that uses some of the same necessary logic that I appreciated but doesn’t rely on guess and check at all. This is NOT the first time I have tweaked a puzzle that didn’t initially meet my standards to make it acceptable. I just thought I would share the process this time. If you try to solve it, you will be able to see the problem with the puzzle yourself.
Now I’ll share some information about the number 1349:
1349 is a composite number.
Prime factorization: 1349 = 19 × 71
1349 has no exponents greater than 1 in its prime factorization, so √1349 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 1349 has exactly 4 factors.
The factors of 1349 are outlined with their factor pair partners in the graphic below.
1349 is the sum of 13 consecutive primes, and it is also the sum of three consecutive primes: 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 1349 443 + 449 + 457 = 1349
Now I’ll write a little bit about the number 1348:
1348 is a composite number.
Prime factorization: 1348 = 2 × 2 × 337, which can be written 1348 = 2² × 337
1348 has at least one exponent greater than 1 in its prime factorization so √1348 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1348 = (√4)(√337) = 2√337
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 1348 has exactly 6 factors.
The factors of 1348 are outlined with their factor pair partners in the graphic below.
1348 is the sum of two squares: 32² + 18² = 1348
1348 is the hypotenuse of a Pythagorean triple: 700-1152-1348 which is 32² – 18², 2(32)(18), 32² + 18²
1348 is also the short leg in a primitive Pythagorean triple: 1348-454275-454277
If you’re not sure how to solve it, I explain how in this youtube video:
Here are some facts about the number 1347:
1347 is a composite number.
Prime factorization: 1347 = 3 × 449
1347 has no exponents greater than 1 in its prime factorization, so √1347 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 1347 has exactly 4 factors.
The factors of 1347 are outlined with their factor pair partners in the graphic below.
1347 isn’t a Lucas number, but as OEIS.org reminds us, 1, 3, 4, 7 happens to be the first four Lucas numbers. Here’s why:
Start with 1, 3, then . . .
1 + 3 = 4
3 + 4 = 7
So what will be the next Lucas number?
1347 is the hypotenuse of a Pythagorean triple:
840-1053-1347 which is 3 times (280-351-449)
I was inspired to make a 3 × 3 Magic Square where every number is different but the numbers in each row, column and diagonal added together equal the same number, 54:
I made it by taking a regular 3 × 3 Magic Square and adding 13 to each of its numbers.
What inspired me to do that? This magical tweet of a palindromic Magic Square for the Year 2019:
Yeah, I know my magic square isn’t quite as impressive. It might be more square than it is magic. It’s also less impressive than this number 2019 spelled out using fifty-one
4 × 4 Magic Squares.
Maybe you will be more impressed by this magic square that has 2019 as its Magic Sum?
I could make that magic sum because 2019 is divisible by 3. Why is 673 in the center? Because 2019÷3 = 673.
673 × 2 = 1346. I’m sharing these magic squares in this post I’ve numbered 1346. In case you haven’t figured it out (1346÷2)×3 = 2019. Happy New Year!
Here’s more about the number 1346:
1346 is a composite number.
Prime factorization: 1346 = 2 × 673
1346 has no exponents greater than 1 in its prime factorization, so √1346 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 1346 has exactly 4 factors.
The factors of 1346 are outlined with their factor pair partners in the graphic below.
1346 is the hypotenuse of a Pythagorean triple:
770-1104-1346 which is 2 times 385-552-673
To solve a level 3 puzzle, start with a clue in the top row of the puzzle and find one other clue in the same row or column. Write the factors for those two clues in the appropriate cells and then work your way down the puzzle cell by cell until you have written the numbers from 1 to 10 in both the first column and the top row.
Level 3 puzzles are a jump from level 2, but still significantly easier than any of the higher level puzzles, because the needed clues are kept in a logical order from the top of the puzzle to the bottom. You can print the puzzles above and work on them or you can access the whole collection in an excel sheet: Level 3’s from 2018. Print them or type the answers directly in excel.
Now I’ll share some information about the number 1345:
1345 is a composite number.
Prime factorization: 1345 = 5 × 269
1345 has no exponents greater than 1 in its prime factorization, so √1345 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 1345 has exactly 4 factors.
The factors of 1345 are outlined with their factor pair partners in the graphic below.
1345 is the sum of two squares in two different ways:
36² + 7² = 1345
33² + 16² = 1345
1345 is also the hypotenuse of FOUR Pythagorean triples:
345-1300-1345 which is 5 times (69-260-269)
504-1247-1345 calculated from 2(36)(7), 36² – 7², 36² + 7²
807-1076-1345 which is (3-4-5) times 269
833-1056-1345 calculated from 33² – 16², 2(33)(16), 33² + 16²
