Puzzles

1628 A Simple Cross

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Today’s Puzzle:

A simple cross is an appropriate symbol for Good Friday. Write the numbers from 1 to 10 in the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1628:

  • 1628 is a composite number.
  • Prime factorization: 1628 = 2 × 2 × 11 × 37, which can be written 1628 = 2² × 11 × 37.
  • 1628 has at least one exponent greater than 1 in its prime factorization so √1628 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1628 = (√4)(√407) = 2√407.
  • 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 1628 has exactly 12 factors.
  • The factors of 1628 are outlined with their factor pair partners in the graphic below.

More about the Number 1628:

1628 is the hypotenuse of a Pythagorean triple:
528-1540-1628, which is (12-35-37) times 44.

1628 is the difference of two squares in two different ways:
408² – 406² = 1628, and
48² – 26²  = 1628.

1623 Easter Basket Challenge Puzzle

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Today’s Puzzle:

Since I’ve recently made puzzles with a pink, purple, or blue Easter egg as well as some blades of grass blowing in the spring wind, it only makes sense that I would also give you an Easter basket in which to hold those other puzzles.

The puzzle is solved if you have written the numbers 1 to 10 in each of the boldly outlined areas of the puzzle, and if those numbers work with the clues to form four multiplication tables.

Print the puzzles or type the solution in this excel file: 12 Factors 1614-1623.

If you need a little help, here’s the same puzzle with the factor pairs for the clues written in.

And if you want even more help, here’s a 2 1/2 minute video on how to get started. I assume you already know the directions on how to solve this kind of puzzle that I gave at the top of this post.

Factors of 1623:

  • 1623 is a composite number.
  • Prime factorization: 1623 = 3 × 541.
  • 1623 has no exponents greater than 1 in its prime factorization, so √1623 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 1623 has exactly 4 factors.
  • The factors of 1623 are outlined with their factor pair partners in the graphic below.

More About the Number 1623:

1623 is the hypotenuse of a Pythagorean triple:
1023-1260-1623, which is 3 times (341-420-541).

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 × 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 × 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.

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:
810² – 809² = 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 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.

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 × 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.

More about the Number 1618:

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!

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.  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 × 2 × 2 × 2 × 101, which can be written 1616 = 2⁴ × 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 × 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:
40² + 4² = 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), 40² – 4², 40² + 4².

1616 is also the difference of two squares in three ways:
405² – 403² = 1616,
204² – 200² = 1616, and
105² – 97² = 1616.

1615 Should Today Be Root Ten Day?

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Today’s Puzzle:

In a recent post, I compared π 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 × 17 × 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 × 2 × 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.

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 × 2 × 13 × 31, which can be written 1612 = 2² × 13 × 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 × 2 × 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 × 13 × 31 × 2,
26 × 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:
404² – 402² = 1612, and
44² – 18² = 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 × 3 × 179, which can be written 1611 = 3² × 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 × 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:
806² – 805² = 1611,
270² – 267² = 1611, and
94² – 85² = 1611.

1610 Four-Leaf Clovers

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Today’s Puzzle:

Sometimes four-leaf clovers are associated with Saint Patrick’s Day. Four-leaf clovers are supposed to be lucky, but you might not feel so lucky as you work on solving this puzzle. I assure you, there is a logical way to proceed on each step!

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

Factors of 1610:

  • 1610 is a composite number.
  • Prime factorization: 1610 = 2 × 5 × 7 × 23.
  • 1610 has no exponents greater than 1 in its prime factorization, so √1610 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1610 has exactly 16 factors.
  • The factors of 1610 are outlined with their factor pair partners in the graphic below.

More about the Number 1610:

1610 is the hypotenuse of a Pythagorean triple:
966-1288-1610, which is (3-4-5) times 322.

1610 is not the sum of two squares or the difference of two squares. 1610 is a leg in some Pythagorean triples because
2(805)(1) = 1610,
2(161)(5) = 1610,
2(115)(7) = 1610, and
2(35)(23) = 1610.

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), a² – b², a² + b².