Puzzle

1622 A Blue Egg for Your Easter Basket

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Today’s Puzzle:

These somewhat tricky level-5 puzzles are probably better suited for middle school and up than younger kids. Use logic on every step and you should be able to find its unique solution.

Math Eggs from Twitter:

Here are some Easter egg puzzles I saw on Twitter. Some are perfect for the littles and others are for older kids. Easter egg hunts can be fun for anyone of any age.

Factors of 1622:

  • 1622 is a composite number.
  • Prime factorization: 1622 = 2 Γ— 811.
  • 1622 has no exponents greater than 1 in its prime factorization, so √1622 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 Γ— 2 = 4. Therefore 1622 has exactly 4 factors.
  • The factors of 1622 are outlined with their factor pair partners in the graphic below.

More about the Number 1622:

1622 is the sum of four consecutive numbers:
409 + 410 + 411 + 412 = 1622.

1621 Give This Purple Egg a Crack!

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Today’s Puzzle:

Here’s a level-5 purple Easter egg for you to try. All you need to do is write the numbers from 1 to 12 in both the first column and in the top row so that those numbers and the given clues function like a multiplication table. Go ahead. Give it a crack!

Factors of 1621:

  • 1621 is a prime number.
  • Prime factorization: 1621 is prime.
  • 1621 has no exponents greater than 1 in its prime factorization, so √1621 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1621 has exactly 2 factors.
  • The factors of 1621 are outlined with their factor pair partners in the graphic below.

How do we know that 1621 is a prime number? If 1621 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1621. Since 1621 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1621 is a prime number.

More about the Number 1621:

OEIS.org informs us that 1621 is in an interesting group of prime numbers.

I have verified it. They really are all prime!

1621 is the sum of two squares:
39Β² + 10Β² = 1621.

1621 is the hypotenuse of a Pythagorean triple:
780-1421-1621, calculated from 2(39)(10), 39Β² – 10Β², 39Β² + 10Β².

Here’s another way we know that 1621 is a prime number: SinceΒ its last two digits divided by 4 leave a remainder of 1, and 39Β² + 10Β² = 1621 with 39 and 10 having no common prime factors, 1621 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1621. Since 1621 is not divisible by 5, 13, 17, 29, or 37, we know that 1621 is a prime number.

Math Happens When Factors of 1620 Make Sum-Difference

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Math Happens Puzzle:

Math Happens put another one of my puzzles in the Austin Chronicle and the Orange Leader!

You can see the puzzle on page 23 of this e-edition or this pdf of the newspaper. You can also find links to all of my Sum-Difference puzzles here.

Here is another one of Math Happen’s amazing puzzles. The tabletops shown are exactly the same. Click on it to see proof!

And how about this way:

Math Happens in many different ways as you can see inΒ their blog post from February 5. You can also look for Math Happens on a page in the middle of each of theseΒ  2020 issuesΒ orΒ 2021 issuesΒ of the Austin Chronicle newspaper online.

Would you like puzzles like these in your community newspaper? Have your paper contact Math Happens on Twitter and make it happen!

Today’s Sum-Difference Puzzles:

Just like the number 6 in the newspaper puzzle above, 180 and 1620 both have factor pairs that make sum-difference. To help you solve these puzzles, I’ve listed all of their factor pairs in the graphics below the puzzle.

As shown below, 180 has nine factor pairs. One of those pairs adds up to 41, and another one subtracts to 41. Put the factors in the appropriate boxes in the first puzzle.

The needed factors for the second puzzle are multiples of the numbers in the first puzzle. 1620 has fifteen factor pairs. One of the factor pairs adds up to Β­123, and a different one subtracts to 123. If you can identify those factor pairs, then you can solve the second puzzle!

What Are the Factors of 1620?

  • 1620 is a composite number.
  • Prime factorization:Β 1620 = 2 Γ— 2 Γ— 3 Γ— 3 Γ— 3 Γ— 3 Γ— 5, which can be written 1620 = 2Β² Γ— 3⁴ Γ— 5.
  • 1620 has at least one exponent greater than 1 in its prime factorization so √1620 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1620 = (√324)(√5) = 18√5.
  • The exponents in the prime factorization are 4, 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(4 + 1)(1 + 1) = 3 Γ— 5 Γ— 2 = 30. Therefore 1620 has exactly 30 factors.
  • The factors of 1620 are outlined with their factor pair partners in the graphic above.

Factor Trees for 1620:

Here are two of the MANY possible factor trees for 1620:

Hint: I chose to build these factor trees with those two sets of factor pairs for a reason.

More about the Number 1620:

1620 is the sum of the interior angles of a hendecagon (11-sided polygon) because
(11 – 2)180 = 1620.

1620 is the sum of two squares:
36Β² + 18Β² = 1620.

1620 is the hypotenuse of a Pythagorean triple:
972-1296-1620, calculated from 36Β² – 18Β², 2(36)(18), 36Β² + 18Β².
It is also (3-4-5) times 324.

This is only some of the math that happens with the number 1620.

1619 A Pink Egg Hidden in the Grass

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Today’s Puzzle:

Easter is less than two weeks away. This pink puzzle is the first of three level-5 Easter eggs hidden amongst some blades of grass for you to find and solve. The puzzle might be a little tricky, but use logic every step of the way, and you’ll be able to find the unique solution:

Factors of 1619:

  • 1619 is a prime number.
  • Prime factorization: 1619 is prime.
  • 1619 has no exponents greater than 1 in its prime factorization, so √1619 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1619 has exactly 2 factors.
  • The factors of 1619 are outlined with their factor pair partners in the graphic below.

How do we know that 1619 is a prime number? If 1619 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1619. Since 1619 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1619 is a prime number.

More about the Number 1619:

1619 is the sum of two consecutive numbers:
809 + 810 = 1619.

1619 is also the difference of two consecutive squares:
810Β² – 809Β² = 1619.

What do you think about that?

1618 Math Happens in the Austin Chronicle

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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 theseΒ  2020 issues orΒ 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.

You can have your local newspaper contact them through Twitter!

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 Γ— 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 Γ— 2 = 4. Therefore 1618 has exactly 4 factors.
  • The factors of 1618 are outlined with their factor pair partners in the graphic below.

More about the Number 1618:

1618 = 2 Γ— 809, and 2809 is a perfect square. Thank you OEIS.org for that fun fact!

1618 is the sum of two squares:
33Β² + 23Β² = 1618.

1618 is the hypotenuse of a Pythagorean triple:
560-1518-1618, calculated from 2(33)(23), 33Β² – 23Β², 33Β² + 23Β².
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!

1616 Centering the Pendulum

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Today’s Puzzle:

Centering the Pendulum is Bill Davidson’s podcast about educators and how they inspire students to learn mathematics. All of his podcasts are wonderful and more than worth the 15 minutes or so needed to listen to each one.Β  I am quite honored that Find the Factors is the subject of his fourth podcast, and you can listen to it here.

To mark this occasion, I’ve made a mystery level puzzle that resembles a swinging pendulum that hopefully is centered! The puzzle might or might not be a little tricky, but just use logic every step of the way, and you should be fine.

Using logic, write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table. Have fun!

Factors of 1616:

  • 1616 is a composite number.
  • Prime factorization: 1616 = 2 Γ— 2 Γ— 2 Γ— 2 Γ— 101, which can be written 1616 = 2⁴ Γ— 101.
  • 1616 has at least one exponent greater than 1 in its prime factorization so √1616 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1616 = (√16)(√101) = 4√101.
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 Γ— 2 = 10. Therefore 1616 has exactly 10 factors.
  • The factors of 1616 are outlined with their factor pair partners in the graphic below.

More about the Number 1616:

1616 is the sum of two squares:
40Β² + 4Β² = 1616.

1616 is also the hypotenuse of a Pythagorean triple:
320-1584-1616, which is 16 times (20-99-101)
and can be calculated from 2(40)(4), 40Β² – 4Β², 40Β² + 4Β².

1616 is also the difference of two squares in three ways:
405Β² – 403Β² = 1616,
204Β² – 200Β² = 1616, and
105Β² – 97Β² = 1616.

1615 Should Today Be Root Ten Day?

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Today’s Puzzle:

In a recent post, I compared Ο€ or (3.14…) with √10 or (3.16…). Steve Morris lives in England where today’s date is written 16-3, not 3-16. He jokingly commented, “So I guess Tuesday (16 March) should be Root Ten Day!” Seriously, day-month-year makes more sense as a writing convention than month-day-year.

Should today be Root Ten Day?
14 March has long been embraced as pi day in the United States, but should 16 March also be a quasi-holiday where kids eat roots like ten French fries or ten carrot sticks?

I remember one of my college professors telling his class that
√2 is about 1.4, and Valentines day is February 14,
√3 is about 1.7, and Saint Patrick’s day is March 17.

To which we could add
√1 is 1, and New Year’s Day is January 1, and
√10 is about 3.1, and Halloween is October 31. (I realize there is a rounding issue with that one.)

Oops. That could be said about all the fake holidays I’ve listed above.

And here’s a more serious thought:

Well, however you want to remember what √10 is or not, I decided to make today’s puzzle look like a square root sign for the fun of it. Write the numbers from 1 to 12 in both the first column and the top row so that the puzzle functions like a multiplication table.

Factors of 1615:

  • 1615 is a composite number.
  • Prime factorization: 1615 = 5 Γ— 17 Γ— 19.
  • 1615 has no exponents greater than 1 in its prime factorization, so √1615 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 Γ— 2 Γ— 2 = 8. Therefore 1615 has exactly 8 factors.
  • The factors of 1615 are outlined with their factor pair partners in the graphic below.

More about the Number 1615:

1615 is the hypotenuse of FOUR Pythagorean triples:
247-1596-1615, which is 19 times (13-84-85),
684-1463-1615, which is 19 times (36-77-85),
760-1425-1615, which is (8-15-17) times 95, and
969-1292-1615, which is (3-4-5) times 323.

1614 and Level 1

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Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that the puzzle functions like a multiplication table.

Factors of 1614:

1614 is even so it’s divisible by 2.

Here’s the weird thought process I used to determine its divisibility by 3:
161514 is made from three consecutive numbers, so it’s divisible by 3. We know that 15 is divisible by 3, so we can remove it. That means 1614 is divisible by 3.

Since 1614 is divisible by both 2 and 3, it is divisible by 6.

  • 1614 is a composite number.
  • Prime factorization: 1614 = 2 Γ— 3 Γ— 269.
  • 1614 has no exponents greater than 1 in its prime factorization, so √1614 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 Γ— 2 Γ— 2 = 8. Therefore 1614 has exactly 8 factors.
  • The factors of 1614 are outlined with their factor pair partners in the graphic below.

More about the Number 1614:

1614 is the hypotenuse of a Pythagorean triple:
414-1560-1614, which is 6 times (69-260-269).

1614 is also a leg in two Pythagorean triples because
2(807)(1) = 1614, and
2(269)(3) = 1614.

You can calculate those Pythagorean triples by letting a be the first number in parenthesis for each of those equations, and b be the second number in parenthesis. Then substitute those values in the three expressions below, and you will have some Pythagorean triples!
2(a)(b), aΒ² – bΒ², aΒ² + bΒ².

1612 Celtic Knot

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Today’s Puzzle:

Many Celtic knots have no beginning and no end, so they are mathematically interesting. This one is like a Trinity Knot, and it doubles as a mystery-level puzzle. Solving it might be a little tricky, but it will still be lots of fun.

Print the puzzles or type the solution in this excel file: 14 Factors 1604-1612.

Factors of 1612:

  • 1612 is a composite number.
  • Prime factorization: 1612 = 2 Γ— 2 Γ— 13 Γ— 31, which can be written 1612 = 2Β² Γ— 13 Γ— 31.
  • 1612 has at least one exponent greater than 1 in its prime factorization so √1612 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1612 = (√4)(√403) = 2√403.
  • 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 Γ— 2 Γ— 2 = 12. Therefore 1612 has exactly 12 factors.
  • The factors of 1612 are outlined with their factor pair partners in the graphic below.

More about the Number 1612:

1612 can be written as a palindromic product in two different ways:
2 Γ— 13 Γ— 31 Γ— 2,
26 Γ— 62.

1612 is the hypotenuse of a Pythagorean triple:
620- 1488-1612, which is (5-12-13) times 124.

1612 can be written as the difference of two squares in two ways:
404Β² – 402Β² = 1612, and
44Β² – 18Β² = 1612.

 

 

1611 A Little Blarney?

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Today’s Puzzle:

Many travelers to Ireland hang upsidedown to kiss the Blarney Stone at the top of Blarney castle. There is even a Sherlock Holmes mystery about someone who appears to have fallen to his death while trying to kiss the Blarney Stone. If I said this mystery level puzzle represents the Blarney Stone, would that just be a bunch of blarney?

Print the puzzles or type the solution in this excel file: 14 Factors 1604-1612.

Factors of 1611:

  • 1611 is a composite number.
  • Prime factorization: 1611 = 3 Γ— 3 Γ— 179, which can be written 1611 = 3Β² Γ— 179.
  • 1611 has at least one exponent greater than 1 in its prime factorization so √1611 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1611 = (√9)(√179) = 3√179.
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 Γ— 2 = 6. Therefore 1611 has exactly 6 factors.
  • The factors of 1611 are outlined with their factor pair partners in the graphic below.

More about the number 1611:

1611 is the difference of two squares in three different ways:
806Β² – 805Β² = 1611,
270Β² – 267Β² = 1611, and
94Β² – 85Β² = 1611.