Number

1388 Mystery Level

/ ivasallay / Leave a comment

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

The Shape of 1387 Tiny Squares

/ ivasallay / Leave a comment

22³ – 21³ = 1387, and that’s why it is the 22nd hexagonal number.

1387 is also the 19th decagonal number. Why? Because 4(19²) – 3(19) = 1387.

Here’s more about the number 1387:

  • 1387 is a composite number.
  • Prime factorization: 1387 = 19 × 73
  • 1387 has no exponents greater than 1 in its prime factorization, so √1387 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 1387 has exactly 4 factors.
  • The factors of 1387 are outlined with their factor pair partners in the graphic below.

 

1387 is the hypotenuse of a Pythagorean triple:
912-1045-1387 which is 19 times (48-55-73)

1386 What You Need to Know About the Multiplication Game

/ ivasallay / Leave a comment

Helpful Hints about the Multiplication Game:

I recently wrote about Hooda Math’s Multiplication Game. There’s a couple of things I didn’t tell you in that post.

First of all, you don’t have to use a computer to play the game. (However, using one the first time you play will help you understand how to play). You can print a game board to play. I’ve created a game board below that you could use. Each player can use different items such as beads, pennies, nickels, and dimes as markers to mark the factors used and to claim the resulting products on the game board.

The second thing you should know is that getting four squares in a row, horizontally, vertically or diagonally is NOT equally likely every place on the board. If one particular number is all you need to get a win, you are less likely to get that number if it only has one factor (like the numbers marked in yellow have). As far as this game is concerned, the products have the number of factors that I’ve indicated, even though in reality most of them have more than that.

You can’t win unless your opponent gives you one of the factors you need to claim that winning space. If 4 of the 9 possible factors will get it for you, the odds are much better your opponent will give you what you need than if only 1 of the 9 possible factors will do it.

If you know which numbers have four possible factors, you may have an advantage over someone who thinks this game is really just a variation of tic-tac-toe. Of course, those products with four factors could also make you more likely to get blocked as well! And if you use my colorful game board, your opponent will know just as much as you do about how many ways they can get each square.

1386 Factor Cake:

Since the biggest prime factor of 1386 is 11, it makes an especially festive factor cake!

Factors of 1386:

Now I’ll share some information about the number 1386:

  • 1386 is a composite number.
  • Prime factorization: 1386 = 2 × 3 × 3 × 7 × 11, which can be written 1386 = 2 × 3² × 7 × 11
  • 1386 has at least one exponent greater than 1 in its prime factorization so √1386 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1386 = (√9)(√154) = 3√154
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1386 has exactly 24 factors.
  • The factors of 1386 are outlined with their factor pair partners in the graphic below.

Sum-Difference Puzzle:

1386 has twelve factor pairs. One of the factor pairs adds up to 85, and a different one subtracts to 85. If you can identify those factor pairs, then you can solve this puzzle!

One More Fact about the Number 1386:

OEIS.org also noted that 1 + 3⁴ + 8 + 6⁴ = 1386.

1385 Mystery Level

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

1381 Hooda Math’s Multiplication Game

/ ivasallay / Leave a comment

The school year is almost over, and class periods were only twenty-five minutes long today. I went online looking for math games that would benefit my students and I found a winner with Hooda Math’s Multiplication Game.

If you count the multiplication facts in a 9×9 multiplication table, you will see 99 facts, but many of the products are duplicated in the table. Every yellow square below is also in white elsewhere in the table:

There are actually only 36 unique products in the multiplication table above. Hooda Math has cleverly arranged those 36 products in a 6×6 grid that becomes the game board. In this two-person game, students take turns moving one of two arrows to a number from 1 to 9 at the bottom of the screen and claiming the square that contains the product of the numbers. The catch is that players must keep one of the numbers chosen by the previous player and cannot claim a product that has already been claimed by either player. (Player 1 cannot score on his first turn.) One student is green and the other is purple and the first to claim four squares in a row is the winner. The rules on the website are VERY short and simple.

Students played this game today. I played it as well. Sometimes I won, and sometimes I lost, but the losses are more interesting than the wins:

In one game, my opponent took the square that I needed to get four in a row vertically for the win. All she was trying to do was block me from winning, however, when she took that square, the game declared her the winner. We were puzzled why she was the winner until she figured out that making that move gave her four in a row diagonally. That’s when we found out players can win by getting four in a row diagonally as well as vertically or horizontally.

In another game, I had two possible moves that would have made me be the winner. I just needed my opponent to choose a 1, 6, or 8 as their other factor, and I would win with 1 × 1 = 1 or 6 × 8 = 48. Unfortunately, he knew to beware of the numbers that would make me win. One of the arrows was pointing to 5, and he made the other arrow point to 5. By now there were no other products left on the board that were divisible by 5, so I couldn’t win because I couldn’t move either of the arrows.

That’s how I didn’t win the game either of those times, but I had a lot of fun anyway, and you will, too!

This is my 1381st post. Here’s some information about that number:

  • 1381 is a prime number.
  • Prime factorization: 1381 is prime.
  • 1381 has no exponents greater than 1 in its prime factorization, so √1381 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1381 has exactly 2 factors.
  • The factors of 1381 are outlined with their factor pair partners in the graphic below.

How do we know that 1381 is a prime number? If 1381 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1381. Since 1381 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1381 is a prime number.

1381 is the 24th Centered Pentagonal Number because
5(23)(24)/2 + 1 = 1381. In formula means that it is one more than five times the 23rd triangular number, as I’ve illustrated below:

1381 is the sum of two squares:
34² +15² = 1381

1381 is the hypotenuse of a Pythagorean triple:
931-1020-1381 calculated from 34² -15², 2(34)(15), 34² +15²

Here’s another way we know that 1381 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 15² = 1381 with 34 and 15 having no common prime factors, 1381 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1381. Since 1381 is not divisible by 5, 13, 17,  29, or 37, we know that 1381 is a prime number.

 

 

1380 A Different Way to Look at the Logic

/ ivasallay / 2 Comments

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

/ ivasallay / Leave a comment

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)