# 1651 Multiplication Fun

### Today’s Puzzle:

Look how much fun these kids are having doing multiplication!

A game like that can help kids get ready to solve a fun puzzle based on the multiplication table.

Write each number from 1 to 10 in both the first column and the top row so that those numbers and the given clues become a multiplication table. ### Factors of 1651:

• 1651 is a composite number.
• Prime factorization: 1651 = 13 × 127.
• 1651 has no exponents greater than 1 in its prime factorization, so √1651 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 1651 has exactly 4 factors.
• The factors of 1651 are outlined with their factor pair partners in the graphic below. ### More About the Number 1651:

1651 is the hypotenuse of a Pythagorean triple:
635-1524-1651, which is (5-12-13) times 127.

1651 is the 26th heptagonal number because
5(26²)/2 – 3(26)/2 = 1651.

1651 is a nice-looking palindrome in base 2:
1651₁₀ = 11001110011₂.
That just means that
2¹⁰ + 2⁹ + 2⁶ + 2⁵ + 2⁴+ 2¹+ 2⁰ = 1024 + 512 + 64 + 32 + 16 + 2 + 1 = 1651.

# 1650 Wrinkles in the Multiplication Table

### Today’s Puzzle:

Are you familiar with the book A Wrinkle in Time? Kat of The Lily Cafe’s blog loves books and recently compared Meg in that book to her six-year-old son. She wrote a post titled Am I Raising a Meg? Her six-year-old LOVES math and is very much interested in multiplication and division. When Mom thought he was playing a game on her phone, he was actually playing with the calculator app! I felt so happy inside as I read that!

I wonder if they have discovered the storybooks in the Math Book Magic blog. Such books could combine Mom’s love for reading with her son’s love of math.

Someday her son might like to solve a “wrinkled” multiplication table puzzle like this one that has only nine clues.

Write all the numbers 1 to 10 in both the first column and the top row so that those numbers and the given clues become a multiplication table. ### Factor Cake for 1650:

This is my 1650th post.
1650 is divisible by 2 and by 5 because it ends with a 0.
1650 is divisible by 3 because 1 + 6 + 5 + 0 = 12, a number divisible by 3.
1650 is divisible by 11 because 1 – 6 + 5 – 0 = 0, a number divisible by 11.

I think we can make a lovely factor cake for 1650: ### Factors of 1650:

• 1650 is a composite number.
• Prime factorization: 1650 = 2 × 3 × 5 × 5 × 11, which can be written 1650 = 2 × 3 × 5² × 11.
• 1650 has at least one exponent greater than 1 in its prime factorization so √1650 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1650 = (√25)(√66) = 5√66.
• The exponents in the prime factorization are 1, 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(2 + 1)(1 + 1) = 2 × 2 × 3 × 2 = 24. Therefore 1650 has exactly 24 factors.
• The factors of 1650 are outlined with their factor pair partners in the graphic below. ### More About the Number 1650:

1650 is the hypotenuse of TWO Pythagorean triples:
462-1584-1650, which is (7-24-25) times 66, and
990-1320-1650, which is (3-4-5) times 330.

# 1649 Tweaking a Puzzle Posed by Sunil Singh @Mathgarden

### Today’s Puzzle:

A couple of days ago on Twitter, I saw an interesting puzzle posed by Sunil Singh @Mathgarden.

After I found one of several of its solutions, I wondered if I could add a bridge that would use all nine numbers from 1 to 9 in the solution, so I tweaked it. I decided to move the puzzle to the ocean when I added that extra bridge. I was able to solve this problem using logic and addition facts, rather than algebra. Try solving it yourself. If you want to see any of the steps I used to solve the puzzle, scroll down to the end of the post.

### Factors of 1649:

• 1649 is a composite number.
• Prime factorization: 1649 = 17 × 97.
• 1649 has no exponents greater than 1 in its prime factorization, so √1649 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 1649 has exactly 4 factors.
• The factors of 1649 are outlined with their factor pair partners in the graphic below. ### More About the Number 1649:

1649 is the sum of two squares in TWO different ways:
32² + 25² = 1649, and
40² + 7² = 1649.

1649 is the hypotenuse of FOUR Pythagorean triples:
399-1600-1649, calculated from 32² – 25², 2(32)(25), 32² + 25²,
560-1551-1649, calculated from 2(40)(7), 40² – 7², 40² + 7²,
776 1455 1649, which is (8-15-17) times 97, and
1105 1224 1649, which is 17 times (65-72-97).

### Some Logical Steps to Solve Today’s Puzzle:

I found four different ways to write the solution but they are all rotations or reflections of each other. Here are the steps to one of those four ways:

1st Step: The biggest number that can be used is 9. Since every island must be included in at least two sums, neither 8 nor 9 can be an addend; they both must be sums. 7 must be an addend in their sums because adding any number to 7 will yield 8, 9, or some larger forbidden number. Thus 8 and 9 are bridges that connect to island 7. I chose to write 7, 8, and 9 in the top right section, but 8 and 9 could change places with each other. I could have also chosen to write those numbers in the bottom left area.

2nd Step: 1 + 7 = 8, and 7 + 2 = 9. 3rd Step: 1 + 2 = 3. 4th Step: The last island must be the smallest remaining number (4) because the smallest remaining number can’t be the sum of a bigger number and the number on either adjacent island. Final Step: 1 + 4 = 5, and 4 + 2 = 6. Did you enjoy this puzzle? How did my steps compare to the steps that you took?

Please, check the comments for another solution.

# 1648 A Pythagorean Triple Logic Puzzle with a Triangular Card Deck

### Today’s Puzzle:

It’s been a few years since I’ve made one of these Pythagorean triple logic puzzles. The triangles in it are shaped a little different than in years past because I also wanted to make a deck of Pythagorean triple cards that are shaped like equilateral triangles or at least as close as I can get to equal sides. This puzzle won’t be easy, but do give it a try!

PUZZLE DIRECTIONS: This puzzle is NOT drawn to scale. Although all of the angles may look like 60-degree angles, none of them are. The marked angles are 90 degrees. Lines that look parallel are NOT parallel. Although side lengths look equal, they are NOT equal. Most rules of geometry do not apply here: in fact, non-adjacent triangles in the drawing might actually overlap.

No geometry is needed to solve this puzzle. All that is needed is logic and the table of Pythagorean triples under the puzzle. The puzzle only uses triples in which each leg and each hypotenuse is less than 100 units long. The puzzle has only one solution.  Print the puzzles or type the solution in this excel file: 12 Factors 1639-1648

### Triangular Card Deck of Pythagorean Triples:

You can make this deck of 50 playing cards to help you solve the puzzle or perhaps to play a domino-type game. Print each group of 25 cards on a separate sheet of paper. Cut the cards out along the solid lines and fold the cards on the dotted lines. Use a glue stick to keep the front of each card attached to its back. Laminate the cards, if desired. The right angles are the only angles marked on the cards, and the hypotenuses are all along the folded edges. Note: Some sides will not match with any other side in the deck, and the 57-76-95 triangle does not match sides with ANY triangle with a hypotenuse less than 100.  ### Factors of the Number 1648:

• 1648 is a composite number.
• Prime factorization: 1648 = 2 × 2 × 2 × 2 × 103, which can be written 1648 = 2⁴ × 103.
• 1648 has at least one exponent greater than 1 in its prime factorization so √1648 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1648 = (√16)(√103) = 4√103.
• 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 1648 has exactly 10 factors.
• The factors of 1648 are outlined with their factor pair partners in the graphic below. ### More About the Number 1648:

1648 is not the hypotenuse of any Pythagorean triples because none of its prime factors leave a remainder of 1 when divided by 4.

1648 is the difference of two squares three different ways:
413² – 411² = 1648,
208² – 204² = 1648, and
107² – 99² = 1648.

That means 1648 is a leg in THREE Pythagorean triples calculated from
413² – 411², 2(413)(411), 413² + 411²;
208² – 204², 2(208)(204), 208² + 204²; and
107² – 99², 2(107)(99), 107² + 99².

1648 can be expressed as 2(824)(1), 2(412)(2), 2(206)(4), as well as 2(103)(8).

That means 1648 is a leg in FOUR Pythagorean triples calculated from
2(824)(1), 824² – 1², 824² + 1²;
2(412)(2), 412² – 2², 412² + 2²;
2(206)(4), 206² – 4², 206² + 4²; and
2(103)(8), 103² – 8², 103² + 8².

Sometimes those formulas produce duplicate triples, but not this time. 1648 is in SEVEN Pythagorean triples!
1648-339486-339490,
1648-84864-84880,
1648-21186-21250,
1648-678975-678977,
1648-169740-169748,
1648-42420-42452,
1648-10545-10673.

Aren’t you glad that today’s puzzle was limited to triples with hypotenuses less than 100? I certainly wouldn’t want any of those triples with 1648 as a side to be part of any puzzle! But 2-digit sides, 16 and 48, helped to make today’s puzzle, and I do hope you were able to solve it today!

# 1647 A Fun Mystery

### Today’s Puzzle:

I’ve never made a puzzle quite like this one before, and even though the logic needed is a little tricky, I found it quite enjoyable. Share it with a friend and see if you don’t have some great discussions such as if the common factor of 12 and 36 is 3, 4, 6, or 12. ### Factors of 1647:

1 + 6 + 4 + 7 = 18, a multiple of 3 and 9, so 1647 is divisible by both 3 and 9.

• 1647 is a composite number.
• Prime factorization: 1647 = 3 × 3 × 3 × 61, which can be written 1647 = 3³ × 61.
• 1647 has at least one exponent greater than 1 in its prime factorization so √1647 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1647 = (√9)(√183) = 3√183.
• 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 1647 has exactly 8 factors.
• The factors of 1647 are outlined with their factor pair partners in the graphic below. ### More About the Number 1647:

1647 is the hypotenuse of a Pythagorean triple:
297-1620-1647, which is 27 times (11-60-61).

1647 is a repdigit in base 13:
1647₁₀ = 999₁₃ because
9(13² + 13¹ + 13º) =
9(169 + 13 + 1) =
9(183) = 1647.

# 1646 Mystery Level

### Today’s Puzzle:

It’s a mystery if this puzzle is easy, difficult, or half easy and half difficult. The only way for you to know is to use logic to try solving it yourself! ### Factors of 1646:

• 1646 is a composite number.
• Prime factorization: 1646 = 2 × 823.
• 1646 has no exponents greater than 1 in its prime factorization, so √1646 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 1646 has exactly 4 factors.
• The factors of 1646 are outlined with their factor pair partners in the graphic below. ### More About the Number 1646:

823 is part of the prime decade: 821, 823, 827, 829.
You can be sure that each of those primes doubled: 1642, 1646, 1654, and 1658 will have exactly four factors.

1646 is in only one Pythagorean triple:
1646-677328-677330, calculated from 2(823)(1), 823² – 1², 823² + 1².

# 1645 and Level 6

### Today’s Puzzle:

The number 36 appears as a clue in this puzzle three times. None of those 36’s will be 3 × 12 because 21 and 33 must use both 3’s. That means two of the 36’s will be 4 × 9, and one of them will be 6 × 6. Can both of the 36’s associated with the 24 be 4 × 9? Answering that question will help you find the logic needed to know which common factor to use for 72 and 36. ### Factors of 1645:

• 1645 is a composite number.
• Prime factorization: 1645 = 5 × 7 × 47.
• 1645 has no exponents greater than 1 in its prime factorization, so √1645 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 1645 has exactly 8 factors.
• The factors of 1645 are outlined with their factor pair partners in the graphic below. ### More About the Number 1645:

1645 is the hypotenuse of a Pythagorean triple:
987-1316-1645, which is (3-4-5) times 329.

# 1644 Level 5 Puzzles Are Not So Easy to Solve

### Today’s Puzzle:

This puzzle isn’t so easy to solve. For example, the common factor of clues 48 and 24 might be 4, 6, 8, or 12. Which one should you use? Logic will help answer that question. Give this puzzle a try? ### Factors of 1644:

1644 has 12 factors and is divisible by 12.

• 1644 is a composite number.
• Prime factorization: 1644 = 2 × 2 × 3 × 137, which can be written 1644 = 2² × 3 × 137.
• 1644 has at least one exponent greater than 1 in its prime factorization so √1644 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1644 = (√4)(√411) = 2√411.
• 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 1644 has exactly 12 factors.
• The factors of 1644 are outlined with their factor pair partners in the graphic below. ### More about the Number 1644:

1644 is the hypotenuse of a Pythagorean triple:
1056-1260-1644 which is 12 times (88-105-137).

# 1643 Thrift Store Find

### Today’s Puzzle:

Yesterday I walked to a thrift store located about a mile and a half from my house. I looked through their picture books and was delighted to find all of these: I love reading math picture books to my grandchildren. My eighteenth (nine girls and nine boys) grandchild was born earlier this month, so each book will be treasured.

The display of books made me think of some math questions:

How many books do you see? How did you count them?

How many of the books have you read? How many haven’t you read?

If you’ve read X number of the books, but haven’t read Y of them, write an expression for the total number of books.

The thrift store price of each book was \$1.29. How much did I expect to pay for all these books?

I wasn’t aware of the store’s “buy four get one free” policy before I checked out. How many free books did I get? Could I have done better than that?

I was charged 7.25% sales tax. What was the total tax I paid? What was the total amount I paid for the books?

I didn’t have time to look through every shelf of books yesterday, so today I went back and found a few more treasures. (I also bought four non-mathy books that are not pictured.) Ten Apples Up On Top was a book I read to my children when they were young. We loved it so much that eventually, it fell apart. Some of my grandchildren haveTen Tiny Tickles in their home libraries, but now they can enjoy it at my house as well. What kind of math can be found in the books I bought?

1. One Hundred Days (Plus One) – What number comes after 100?
2. The Stephen Cartwright 123 – Count to 21.
3. Boom Chicka Rock – Hour hand clock math.
4. Numbears – Count to 12.
5. Ten Happy Whales – Adding one more; Introduction to addition.
6. 10 Black Dots – Count to 10.
7. One Hundred Hungry Ants – Dividing 100 by 2, 4, 5, and 10.
8. A Remainder of One – Dividing 25 by 2, 3, or 4 leaves a remainder of one, but dividing by 5 does not.
9. Sixteen Runaway Pumpkins – Powers of 2.
10. 10 for dinner – Ways to add four or five numbers to make 10.
11. Great Estimations – Estimating tens, hundreds, or thousands of objects.
12. Baby Counts – Count to 4.
13. Counting Farm – Count to 10.
14. 123 First Board Book – Count to 10, 20, 50, 100.
15. Ten Apples Up On Top – Count to 10 (Count by 10’s also).
16. Cookie’s Week – Calendar math – Days of the week.
17. Millions of Snowflakes – Count to 5.
18. A House for Hermit Crab – Calendar Math – Months of the Year.
19. Ten Tiny Tickles – Count to 10.
20. Each Orange Had 8 Slices – Word problems, multiplying three numbers together.
21. One Less Fish – Counting down from 12 to 0.
22. Lunch Money and Other Poems about School has three poems about math: Eight-Oh-Three (clock math), Lunch Money (types of coins), Math My Way (should 2 + 2 be 22 instead of 4? What about 3 + 3 and 4 + 4?)
23. Nine O’Clock Lullaby – Clock math, time zones around the world.
24. One Monday Morning – Calendar Math, Days of the week.
25. A Quarter from the Tooth Fairy – Different ways to make 25¢.

All of these books have charming pictures, and I was confident I would enjoy reading them aloud. So many books, so little time to spend with grandkids! We didn’t read even half of them today, but we enjoyed the ones we did read very much. I will mention only three of them here:

1. Baby Counts has only ten words in it, but the pictures made my 2-year-old grandson laugh so much that we reread it as soon as we finished it several times. He already knew how to count to four and delighted in counting along with me.
2. Ten Apples Up On Top is as captivating now as it was when my children were young.
3. I was surprised how much kids who have never heard of the tooth fairy enjoyed A Quarter from the Tooth Fairy.

I look forward to enjoying all of the books with my grandkids when they visit me in the future.

I looked but didn’t find any math-related games at this thrift store this time. One of the people I follow on Twitter reminds us that some great games can often be found there.

### Factors of 1643:

• 1643 is a composite number.
• Prime factorization: 1643 = 31 × 53.
• 1643 has no exponents greater than 1 in its prime factorization, so √1643 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 1643 has exactly 4 factors.
• The factors of 1643 are outlined with their factor pair partners in the graphic below. ### More about the number 1643:

1643 is the hypotenuse of a Pythagorean triple:
868-1395-1643, which is 31 times (28-45-53).

From OEIS.org, we learn that not only is
1643 = 31 × 53, but also
1643₁₀ = 3153₈.

# 1642 and Level 4

### Today’s Puzzle:

There are several clues in this puzzle with more than one factor pair, but if you use logic every step of the way, you can still write the factors 1 to 12 in the appropriate places to turn the puzzle into a multiplication table. ### Factors of 1642:

• 1642 is a composite number.
• Prime factorization: 1642 = 2 × 821.
• 1642 has no exponents greater than 1 in its prime factorization, so √1642 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 1642 has exactly 4 factors.
• The factors of 1642 are outlined with their factor pair partners in the graphic below. ### More About the Number 1642:

1642 is the sum of two squares:
39² + 11² = 1642.

1642 is the hypotenuse of a Pythagorean triple:
858-1400-1642, calculated from 2(39)(11), 39² – 11², 39² + 11².
That triple is also 2 times (429-700-821).