1391 and Level 1

Many of the clues in this puzzle have double digits. If you know why they do, then you can find all the factors and solve this puzzle! Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here’s some information about the number 1391:

• 1391 is a composite number.
• Prime factorization: 1391 = 13 × 107
• 1391 has no exponents greater than 1 in its prime factorization, so √1391 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 1391 has exactly 4 factors.
• The factors of 1391 are outlined with their factor pair partners in the graphic below. 1391 is the hypotenuse of a Pythagorean triple:
535-1284-1391 which is (5-12-13) times 107

1389 Positive Trinomial Puzzle

Today on Twitter, Mr. Allen requested some good problem-solving resources for quadratics. He made up one himself.

I decided to make one as well. It is similar to my other Find the Factors puzzles. You will have to use logic to solve it, but in many ways, it will be easier to solve than most of my regular puzzles. Like always, there is only one solution. Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Every term is positive so if you already know how to factor trinomials it should be relatively easy to solve. All the factors from (x + 1) to (x + 9) need to appear exactly one time in both the first column and the top row of the puzzle.  Once all the factors are found, the puzzle is solved, but you can find all the products of those factors and write them in the body of the puzzle if you want.

Since this is my 1389th post, here’s a little bit about that number:

• 1389 is a composite number.
• Prime factorization: 1389 = 3 × 463
• 1389 has no exponents greater than 1 in its prime factorization, so √1389 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 1389 has exactly 4 factors.
• The factors of 1389 are outlined with their factor pair partners in the graphic below. 1389 is the difference of two squares in two different ways:
695² – 694² = 1389
233² – 230² = 1389

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

1384 and Level 6

Guessing and checking can be so frustrating! If instead, you study the clues to find a logical place to start this puzzle, you are more likely to be able to find the one and only solution. Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here are a few facts about the number 1384:

• 1384 is a composite number.
• Prime factorization: 1384 = 2 × 2 × 2 × 173, which can be written 1384 = 2³ × 173
• 1384 has at least one exponent greater than 1 in its prime factorization so √1384 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1384 = (√4)(√346) = 2√346
• 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 1384 has exactly 8 factors.
• The factors of 1384 are outlined with their factor pairs in the graphic below. 1384 is the hypotenuse of a Pythagorean triple:
416-1320-1384 which is 8 times (52-165-173)

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)

1382 and Level 4

After solving a couple of level 3 puzzles, you are ready to give a level 4 puzzle a try. It isn’t any more difficult to solve than a level 3, except that the clues are not given in a logical order. Don’t let that stop you from succeeding! Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll share some information about the number 1382:

• 1382 is a composite number.
• Prime factorization: 1382 = 2 × 691
• 1382 has no exponents greater than 1 in its prime factorization, so √1382 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 1382 has exactly 4 factors.
• The factors of 1382 are outlined with their factor pair partners in the graphic below. 1382 is in one Pythagorean triple:
1382-477480-477482 calculated from 2(691)(1), 691² – 1², 691² + 1²

1380 A Different Way to Look at the Logic

This puzzle looks a lot like puzzle #1378. The design is the same, but the clues are not in the same places. I made this puzzle to demonstrate that there is often more than one way to logically find all the factors of a puzzle. If it were a level 4 puzzle, the clues could be anywhere on the puzzle. But since it is a level 3 puzzle, start with the factors of 14 and 8, and then write the factors of 63 in the appropriate places. Continue with the clues in order from top to bottom until all the factors have been found. Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll share some information about the number 1380:

• 1380 is a composite number.
• Prime factorization: 1380 = 2 × 2 × 3 × 5 × 23, which can be written 1380 = 2² × 3 × 5 × 23
• 1380 has at least one exponent greater than 1 in its prime factorization so √1380 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1380 = (√4)(√345) = 2√345
• The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1380 has exactly 24 factors.
• The factors of 1380 are outlined with their factor pair partners in the graphic below. There are MANY possible factor trees for 1380. Here’s one of them: 1380 is the hypotenuse of a Pythagorean triple:
828-1104-1380 which is (3-4-5) times 276

1379 You Can Solve This Magic Square

I knew that 1379 was the magic sum of a 14 × 14 magic square because 14² = 196 and
(196)(197)/(2·14) = 1379.

I know how to solve a 4 × 4 magic square and when the dimensions of the magic square are odd numbers. I can also solve squares whose dimensions can be factored into any of those.  Clearly, the dimensions of a 14 × 14 magic square don’t qualify. I wondered if there was a simple way to solve it. I watched a video that explained how to use a 7 × 7 magic square to solve one that is
14 × 14
. You don’t have to watch the video to solve this magic square. Let me explain:

Notice that I’ve grouped the squares in the 14 × 14 magic square into 4 × 4 sub-grids. Now the magic square can behave more like a 7 × 7 magic square. Also, notice that the highest number in each sub-grid is 4 times the corresponding number in a 7 × 7 magic square. You should notice that many of the numbers appear in order along the diagonal of the 7  × 7 square. The trickiest part for me is always the upper right corner.

Unfortunately, the 4 × 4 sub-grids are not all the same. In the video, they were labeled X, Y, and Z. To make it simpler, I’ve color-coded them so that you can know how to place the four numbers in each sub-grid from lowest to highest. You can use this excel file,10 Factors 1373-1388 to first solve the 7 × 7 magic square and then use it and the three squares above to solve the 14 x 14 one. I would encourage you to give it a try! It is so satisfying to succeed!

Here’s a little more about the number 1379:

• 1379 is a composite number.
• Prime factorization: 1379 = 7 × 197
• 1379 has no exponents greater than 1 in its prime factorization, so √1379 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 1379 has exactly 4 factors.
• The factors of 1379 are outlined with their factor pair partners in the graphic below. 1379 is the hypotenuse of a Pythagorean triple:
196-1365-1379 which is 7 times (28-195-197)

1378 and Level 3

The first row with a clue has a 14 in it. Use that 14 and the clue in the same column, to figure out where to put one of the factor pairs of 14 in this puzzle.  Only use factor pairs where both numbers are from 1 to 10. Then work your way down the puzzle, row by row until you have found all the factors of this level 3 puzzle. The completed puzzle will look like a multiplication table with the factors in a different order. Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here is some information about the puzzle number, 1378:

• 1378 is a composite number.
• Prime factorization: 1378 = 2 × 13 × 53
• 1378 has no exponents greater than 1 in its prime factorization, so √1378 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 1378 has exactly 8 factors.
• The factors of 1378 are outlined with their factor pair partners in the graphic below. 1378 is the 52nd triangular number because (52)(53)/2 = 1378.

1378 is the hypotenuse of FOUR Pythagorean triples:
222-1360-1378 which is 2 times (111-680-689)
530-1272-1378 which is (5-12-13) times 106
728-1170-1378 which is 26 times (28-45-53)
800-1122-1378 which is 2 times (400-561-689)