1698 A Little Virgács and Candy

Today’s Puzzle:

If you were a child in Hungary, you might have found some virgács and some candy in your boot this morning. Mikulás (St. Nick) would have given you the candy because of how good you’ve been this year, and the virgács for those times you weren’t so good.

This virgács and candy puzzle is like a mixed-up multiplication table. It is a lot easier to solve because I made it a level 3 puzzle. First, find the common factor of 56 and 72 that will allow only numbers between 1 and 12 to go in the first column. Put the factors in the appropriate cells, then work your way down the puzzle, row by row until each number from 1 to 12 is in both the first column and the top row. Factors of 1698:

• 1698 is a composite number.
• Prime factorization: 1698 = 2 × 3 × 283.
• 1698 has no exponents greater than 1 in its prime factorization, so √1698 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 1698 has exactly 8 factors.
• The factors of 1698 are outlined with their factor pair partners in the graphic below. 1698 = 2(849)(1), so it is a leg in the Pythagorean triple calculated from
2(849)(1), 849² – 1², 849² + 1².

1698 = 2(283)(3), so it is a leg in the Pythagorean triple calculated from
2(283)(3), 283² – 3², 283² + 3².

1686 Some Candy Corn for You to Chew on

Today’s Puzzle:

Candy corn probably isn’t your favorite Halloween treat, but this candy corn puzzle could give you something satisfying to chew on. Give it a try!

Find the common factor of 33 and 66, write the factors in the appropriate cells. Since this is a level 3 puzzle, you can then work from the top of the puzzle row by row until you have found all the factors. The numbers from 1 to 12 must appear once in both the first column and the top row. Here’s the same puzzle without any color if you prefer: Factors of 1686:

• 1686 is a composite number.
• Prime factorization: 1686 = 2 × 3 × 281.
• 1686 has no exponents greater than 1 in its prime factorization, so √1686 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 1686 has exactly 8 factors.
• The factors of 1686 are outlined with their factor pair partners in the graphic below. 1686 is the hypotenuse of a Pythagorean triple:
960-1386-1686, which is 6 times (160-231-281).

1686 is also a leg in these two Pythagorean triples:
1686-710648-710650, calculated from 2(843)(1), 843² – 1², 843² + 1² and
1686-78952-78970, calculated from 2(281)(3), 281² – 3², 281² – 3².

1675 You CAN Solve This Level 3 Puzzle!

Great news for math enthusiasts, students, and teachers everywhere! The #148 Playful Math Education Carnival was published today at Math Book Magic!

Today’s puzzle:

I am confident that you can solve this level 3 puzzle! Here’s how: Using only the numbers from 1 to 10, write the factors of 27 and 6 in the appropriate cells. Next, write 18’s factors. Then, since this is a level 3 puzzle, write the factors of 30, 50, 40, 36, 14, 56, and 8, in that order, until all the numbers from 1 to 10 appear in the first column as well as in the top row. As always, there is only one solution. What did I tell you? You could solve it!

Factors of 1675:

• 1675 is a composite number.
• Prime factorization: 1675 = 5 × 5 × 67, which can be written 1675 = 5² × 67.
• 1675 has at least one exponent greater than 1 in its prime factorization so √1675 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1675 = (√25)(√67) = 5√67.
• 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 1675 has exactly 6 factors.
• The factors of 1675 are outlined with their factor pair partners in the graphic below. Did you notice that 1675 = 5 × 67 × 5?

How many quarters are in \$16.75?
Well, 17 × 4 quarters = 68 quarters = \$17.00.
Subtracting one quarter from both sides of the equation, we get
67 quarters = \$16.75.

1675 is the hypotenuse of TWO Pythagorean triples:
469-1608-1675, which is (7-24-25) times 67, and
1005-1340-1675, which is (3-4-5) times 335.

1675 is the difference of two squares in THREE different ways:
838² – 837² =  1675,
170² – 165² =  1675, and
46² – 21²  =  1675.

1666 Demystifying a Tricky Puzzle

Today’s Puzzle:

This puzzle isn’t as tricky as it could be simply because I arranged its clues into a level 3 puzzle. That means that after you write the factors of 99 and 18 in the appropriate boxes in the first column and top row, you work your way down the puzzle clue by clue in the order they appear. Still, you will have to think about what to do with the 70, but I think you can handle it! Factors of 1666:

• 1666 is a composite number.
• Prime factorization: 1666 = 2 × 7 × 7 × 17, which can be written 1666 = 2 × 7² × 17.
• 1666 has at least one exponent greater than 1 in its prime factorization so √1666 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1666 = (√49)(√34) = 7√34.
• 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 1666 has exactly 12 factors.
• The factors of 1666 are outlined with their factor pair partners in the graphic below. 1666 is the sum of two squares because ALL of its odd prime factors either leave a remainder of 1 when divided by 4 OR have an even exponent:
1666 = 2 × 7² × 17,
17÷4 = 4 R1, The exponent on 7² is even.
What are the two squares?
35² +  21² = 1666.

1666 is the hypotenuse of a Pythagorean triple :
784-1470-1666 which is (8-15-17) times 98 and
can also be calculated from 35² –  21², 2(35)(21), 35² +  21².

1652 Start at the Top and Work Your Way Down to the Bottom

Today’s Puzzle:

This is a level 3 puzzle so the clues are given in a logical order starting from the top of the puzzle. Begin by writing the factors of 20 and 32 in the appropriate cells. Then write the rest of the numbers so that both the first column and the top row have all the numbers from 1 to 10, and the written numbers are the factors of the given clues. Factors of 1652:

• 1652 is a composite number.
• Prime factorization: 1652 = 2 × 2 × 7 × 59, which can be written 1652 = 2² × 7 × 59.
• 1652 has at least one exponent greater than 1 in its prime factorization so √1652 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1652 = (√4)(√413) = 2√413.
• 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 1652 has exactly 12 factors.
• The factors of 1652 are outlined with their factor pair partners in the graphic below. 1652 is the difference of two squares two different ways:
414² – 412² = 1652 and
66² – 52² = 1652.

1641 and Level 3

Today’s Puzzle:

This is a level 3 puzzle so the clues have been placed so that you can know what order you should use the clues. Place the factors of 90 and 30 in the appropriate cells, then work your way down the puzzle cell by cell filling in the factors of the clues as you go. Factors of 1641:

1 + 4 + 1 = 6, so 1641 is divisible by 3. (It isn’t necessary to include multiples of 3 in the sum to determine divisibility by 3.)

• 1641 is a composite number.
• Prime factorization: 1641 = 3 × 547.
• 1641 has no exponents greater than 1 in its prime factorization, so √1641 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 1641 has exactly 4 factors.
• The factors of 1641 are outlined with their factor pair partners in the graphic below. 1641 is the difference of two squares in two different ways:
821² – 820² = 1641, and
275² – 272² = 1641.

From OEIS.org we learn that the number formed from 1²6²4²1² is a perfect square:
Recall that 1² = 1; 6² = 36; 4² = 16; and 1² = 1. Those squares form the number, 136161.
Sure enough, 136161 = 369².

1630 and Level 3

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so those numbers and the given clues make the puzzle function like a multiplication table. Because this is a level 3 puzzle, first write the factors for 72 and 90. Then work your way down the puzzle row by row until you have found all the factors. Factors of 1630:

• 1630 is a composite number.
• Prime factorization: 1630 = 2 × 5 × 163.
• 1630 has no exponents greater than 1 in its prime factorization, so √1630 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 1630 has exactly 8 factors.
• The factors of 1630 are outlined with their factor pair partners in the graphic below. 1630 is the hypotenuse of a Pythagorean triple:
978-1304-1630, which is (3-4-5) times 326.

1618 Math Happens in the Austin Chronicle

Math Happens!

Several years before I started blogging, I tried to get my puzzles in newspapers, but the publishers of those newspapers just ignored them. Because of that, it is even sweeter to me that Math Happens put one of them in the Austin Chronicle! You can see it in the newspaper on page 25 of this pdf or in this cool page-turning e-edition. Math Happens in many different ways as you can see in their blog post from February 5. You can also look for Math Happens on a page in the middle of each of these  2020 issues or 2021 issues of the Austin Chronicle newspaper online.

Math Happens also in the Orange Leader, and they would love to also be in your local community newspaper.

Today’s Puzzle:

Spring happens in just a few days! Today’s puzzle represents grasses blowing in a spring wind, readily anticipating the hiding of Easter eggs. It’s a level 3 puzzle, so start by finding the factors of the clue at the top of the puzzle (and the clue that goes with it), and work your way down cell by cell until you have written all the numbers from 1 to 12 in both the factor column and the factor row. You can do this! Factors of 1618:

• 1618 is a composite number.
• Prime factorization: 1618 = 2 × 809.
• 1618 has no exponents greater than 1 in its prime factorization, so √1618 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 1618 has exactly 4 factors.
• The factors of 1618 are outlined with their factor pair partners in the graphic below. 1618 = 2 × 809, and 2809 is a perfect square. Thank you OEIS.org for that fun fact!

1618 is the sum of two squares:
33² + 23² = 1618.

1618 is the hypotenuse of a Pythagorean triple:
560-1518-1618, calculated from 2(33)(23), 33² – 23², 33² + 23².
It is also 2 times (280-759-809).

1618 is the 22nd centered heptagonal number because it is one more than seven times the 21st triangular number:
7(21)(22)/2 + 1 = 1618.

1618 has exactly four factors. The last number with exactly four factors was 1603. That’s the biggest gap so far between two numbers with exactly four factors!
(It will be interesting to see who will win the horse race for the current set of 100 numbers. So far, the horses for 2 factors and 8 factors are each running twice as fast as the horse for 4 factors, and 1619 will be a prime number, giving 2 factors the lead!)

A lot of math is happening with this number!

1607 Shillelagh

Today’s Puzzle:

A Shillelagh is an Irish wooden walking stick. This Shillelagh is keeping with our Saint Patrick’s Day theme, but it is a Find the Factors 1 to 14 puzzle.  Brutal! It will be a whole lot less tricky for you to solve because I made it a level 3 puzzle: The logic needed to solve the puzzle is built in. Just start with the clue at the top of the puzzle and work your way down cell by cell until you have found all the factors. So crack on! Print the puzzles or type the solution in this excel file: 14 Factors 1604-1612.

Factors of 1607:

• 1607 is a prime number.
• Prime factorization: 1607 is prime.
• 1607 has no exponents greater than 1 in its prime factorization, so √1607 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1607 has exactly 2 factors.
• The factors of 1607 are outlined with their factor pair partners in the graphic below. How do we know that 1607 is a prime number? If 1607 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1607. Since 1607 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1607 is a prime number.

1607 is the sum of two consecutive numbers:
803 + 804 = 1607.

1607 is also the difference of two consecutive numbers:
804² – 803² = 1607.

Did you notice what happened there? Try this next one:

1607² = 2582449.

1607²/2 = 1291224.5.

(1607-1291224-1291225) is a primitive Pythagorean triple.

Cool, isn’t it?

1597 and Level 3

Today’s Puzzle:

You can solve this level 3 puzzle! Each number from 1 to 10 must appear in both the first column and the top row.

What is the greatest common factor of 24 and 56? Write that number above the column in which those clues appear. Write the corresponding factors in the first column. Next, starting with 72, write the factors of each clue going down the puzzle row by row until you have found all the factors. Factors of 1597:

• 1597 is a prime number.
• Prime factorization: 1597 is prime.
• 1597 has no exponents greater than 1 in its prime factorization, so √1597 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1597 has exactly 2 factors.
• The factors of 1597 are outlined with their factor pair partners in the graphic below. How do we know that 1597 is a prime number? If 1597 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1597. Since 1597 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1597 is a prime number.