1608 Rainbow

Today’s Puzzle:

We often think of rainbows around Saint Patrick’s Day. Here is a rainbow puzzle for you to solve. It won’t be all that easy even if I tell you that
13 × 14 = 182,
12 × 13 = 156, and
8 ×  14 = 112.

Good luck!

If you’d like to print the puzzle but not use so much ink, here’s a puzzle with all the same clues:

Factor Rainbow for 1608:

The number 1608 has enough factors to make an impressive factor rainbow:

Factors of 1608:

  • 1608 is a composite number.
  • Prime factorization: 1608 = 2 × 2 × 2 × 3 × 67, which can be written 1608 = 2³ × 3 × 67
  • 1608 has at least one exponent greater than 1 in its prime factorization so √1608 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1608 = (√4)(√402) = 2√402
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1608 has exactly 16 factors.
  • The factors of 1608 are outlined with their factor pair partners in the graphic below.

More about the Number 1608:

1608 is the difference of two squares in four different ways:
403² – 401² = 1608,
203² – 199² = 1608,
137² – 131² = 1608, and
73² – 61² = 1608.

1600 How Would You Describe This Horse Race?

Today’s Puzzle:

Do most of the numbers from 1501 to 1600 have 2 factors, 4 factors, 6 factors, or what? A horse race is a fun way to find the answer to that puzzle!

As I’ve done several times before, I’ve made a horse race for this multiple of 100 and the 99 numbers before it. A horse moves when a number comes up with a particular amount of factors. Some of the races I’ve done in the past have been exciting with several lead changes. In other races, one horse ran quite quickly, leaving all other horses in the dust. One previous horse race resulted in a tie.

How will you describe this horse race? Exciting or boring? Surprizing or predictable? Pick your pony and watch the race to the end before you decide on an adjective.

Click here if you would like the Horse Race to be slightly bigger.

1501 to 1600 Horse Race

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Wow! I’ve not seen that happen before! Visually it looks like 4 won the race, but this horse race is really about finding the mode There are two modes, 4 and 8, for the amount of factors for the numbers from 1501 to 1600. It’s about which amount of factors comes up most often for the entire set of numbers, not which one of those occurs first. Thus, for that reason, I would describe the horse race above as deceptive. That horse race looked at the amount of factors five numbers at a time. Here’s what happens if we look at ten numbers at a time:

1501 to 1600 Horse Race (by tens)

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In the second horse race, it is much more clear that the race ends in a tie, and the mode is BOTH 4 and 8.

Prime Factorization of Numbers from 1501 to 1600:

Of those 100 numbers, 38 have square roots that can be simplified; 62 do not.

Factor Trees for 1600:

1600 has MANY possible factor trees. Some are symmetrical; some are not. Here are two nicely shaped ones:

Factors of 1600:

  • 1600 is a composite number and a perfect square.
  • Prime factorization: 1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5, which can be written 1600 = 2⁶ × 5².
  • 1600 has at least one exponent greater than 1 in its prime factorization so √1600 can be simplified. Taking the factor pair from the factor pair table below with the same number for both factors, we get
    √1600 = (√40)(√40) = 40. However, you could also use
    √1600 = (√4)(√400) = 2 × 20 = 40,
    √1600 = (√16)(√100) = 4 × 10 = 40, or
    √1600 = (√25)(√64) = 5 × 8 = 40.
  • The exponents in the prime factorization are 6 and 2. Adding one to each exponent and multiplying we get (6 + 1)(2 + 1) = 7 × 3 = 21. Therefore 1600 has exactly 21 factors.
  • The factors of 1600 are outlined with their factor pair partners in the graphic below.

More about the Number 1600:

1600 is the sum of two squares:
32² + 24² = 1600.

1600 is the hypotenuse of two Pythagorean triples:
448-1536-1600, calculated from 32² – 24², 2(32)(24), 32² + 24².
It is also (7-24-25) times 64.
960-1280-1600, which is (3-4-5) times 320.

1600 looks square in some other bases:
1600 = 1(40²) + 0(40) + 0(1), so it’s 100₄₀.
1600 =1(39²) + 2(39) + 1(1), so it’s 121₃₉.
1600 =1(38²) + 4(38) + 4(1), so it’s 144₃₈.
1600 =1(37²) + 6(37) + 9(1), so it’s 169₃₇.

Furthermore, 1600 is a repdigit in base 7:
1600 = 4(7³ + 7² + 7¹ + 7º), so it’s 4444₇.

 

How Much of My $1596 Will the Taxman Get?

Today’s Puzzle is a Game Called Taxman:

Tax season in the United States began just a few days ago. Figuring taxes is no fun, but playing Taxman is.

You see 56 cards below. Imagine that they are 56 envelopes, each containing the dollar amount written on them.

You can take whatever envelope you’d like, but the taxman must be able to take at least one envelope when you do. What envelopes will the taxman take when it is his turn? He will take EVERY available envelope that has a number on it that is a factor of the envelope you took. So if you started off taking envelope 56, the taxman would take envelopes 1, 2, 4, 7, 8, 14, and 28. Thus the taxman would start off getting a whole lot more money than you while also limiting envelopes you can choose later. For example, you wouldn’t be able to pick any prime numbers after taking 56  because the 1 would be gone, and the taxman must be able to take one envelope with every choice you make. When you get to the point that there are no more envelopes that you can take, the taxman gets to take ALL the remaining envelopes. You want the taxman to get less money than you do, and hopefully as little money as possible. I’ve included the factors of each of the numbers at the top of each “envelope” to make the game easier for both you and the taxman. You can print “envelopes” from this excel file Taxman & 1537-1544.  You can have a friend be the taxman as you play or you can play both roles.

How I Played the Game:

I took envelopes as I asked myself these questions:

What is the largest prime number on the table? 53. I take 53, and the only envelope the taxman gets is 1.

What is the largest number that is a prime number squared? 49. I take 49, and the only envelope left for the taxman to take is 7.

What is the largest multiple of 7 that has only one other available factor? 35. I take 35, and the only envelope the taxman can still get is 5.

What is the largest multiple of 5 that has only one other available factor? and so forth.

Here is the order I took the envelopes. It is not the only possible order to use, but it was one in which there were only nine cards at the end of the game for the taxman to claim.

The winner of the game is the one with the most money at the end of the game.

How Do We Know Who Won the Game?

We could add up all the money I got and compare it with all the money the taxman got, but that wouldn’t be much fun. We could add up all the taxman money and subtract it from 1596 to find my total, but I prefer a different way: I remove all the taxman’s money from the table and try to mix envelopes to match mine. I start with my envelope with the most cash, 56 = 8 + 1 + 47, so I put the taxman’s envelopes with those numbers next to my 56.

Matching the “envelopes” up like this will take some time, but it feels like playing a game. It may have taken me as long to play this part of the game as it took me to play Taxman because sometimes I had to remove perfectly good sums in order to use all the taxman’s envelopes in a sum. Here is how I matched the envelopes:

I know that I won the game because I had at least one envelope without a sum next to it. I can determine by  how much I won the game by adding 35 + 33 + 46 + 39 + 27 + 38 + 20 + 30 + 42. That sum tells me I won the game by $310.

How much did the taxman get?
(1596-310)/2 = 643.

How much did I get to keep?
643 + 310 = 953.

A Factor Tree for 1596:

If you know that 40² = 1600, and 1600 – 4 = 1596, we might recognize that 1596 is the difference of two perfect squares: 40² – 2². Then we can also know that
1596 = (40 – 2)(40 + 2) = 38 × 42.

Let’s make a factor tree from the factor pair 38 × 42:

Since 1596 has several factors, it has several other possible factor trees.

Factors of 1596:

  • 1596 is a composite number.
  • Prime factorization: 1596 = 2 × 2 × 3 × 7 × 19, which can be written 1596 = 2² × 3 × 7 ×
  • 1596 has at least one exponent greater than 1 in its prime factorization so √1596 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1596 = (√4)(√399) = 2√399
  • 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 1596 has exactly 24 factors.
  • The factors of 1596 are outlined with their factor pair partners in the graphic below.

More about the Number 1596:

The reason that I was dividing up $1596 between me and the taxman is that 1596 is the 56th triangular number. It is the 56th triangular number because
(56)(57)/2 = 1596.

1596 is the difference of two squares in four different ways:

1596 is the sum of the first 15 Fibonacci numbers, which also means it is one number less than the 17th Fibonacci number.

 

 

Sorting the Factors of 1593

Curious Sort of Factors:

1593 has eight factors. Three-fourths of those factors can be made using some or all of its digits. The other two factors seem to be talking about each other. 27 says 177 has two sevens, and 177 says that 27 has one seven. I’ve sorted them into two categories:

Today’s Puzzle:

Can 1593 be expressed using its four digits exactly once and (), +, -, ×, ÷?
Almost, but not quite:
1593 = 3³ × 59¹. We are not allowed to use the 3 twice.
1593 = 531√9. We can use digits as exponents, but we are not allowed to use the square root symbol.
1593 is not a Friedman number because 1593 = 1593 is the only way we can express it using (), +, -, ×, ÷ and only its own digits as numbers or exponents.

Now try this: Can you express each of 1593’s eight factors using all four of its digits exactly once and only (), +, -, ×, ÷? I’ve done a few of them to get you started. (One of them can’t be done, and there is more than one possibility for some of them.)

1 = 1⁵⁹³
3 =
9 = 3(9-5-1)
27 =
59 =
177 =
531 =
1593 = 1593. That’s all we can do for that one.

Factors of 1593:

  • 1593 is a composite number.
  • Prime factorization: 1593 = 3 × 3 × 3 × 59, which can be written 1593 = 3³ × 59.
  • 1593 has at least one exponent greater than 1 in its prime factorization so √1593 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1593 = (√9)(√177) = 3√177.
  • 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 1593 has exactly 8 factors.
  • The factors of 1593 are outlined with their factor pairs in the graphic below.

More about the Number 1593:

Consecutive numbers 1592 and 1593 each have eight factors because they are each a cube times a prime.

1593 is the difference of two squares four different ways:
797² – 796² = 1593,
267² – 264² = 1593,
93² – 84² = 1593, and
43² – 16² = 1593.

1585 is a Centered Triangular Number

Today’s Puzzle:

What geometric shape can 1585 tiny squares be arranged into?

If you answered a 5 × 317 rectangle, you would be right, but there is another shape that is probably more interesting than that rectangle: 1585 tiny squares can be arranged into a centered triangle as illustrated below.

Can you see the three consecutive triangular numbers that make up this centered triangle?

Factors of 1585:

  • 1585 is a composite number.
  • Prime factorization: 1585 = 5 × 317.
  • 1585 has no exponents greater than 1 in its prime factorization, so √1585 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 1585 has exactly 4 factors.
  • The factors of 1585 are outlined with their factor pair partners in the graphic below.

More about the Number 1585:

1585 is the sum of two squares in two different ways:
39² + 8² = 1585
36² + 17² = 1585

1585 is the hypotenuse of FOUR Pythagorean triples:
375-1540-1585, which is 5 times (75-308-317),
624-1457-1585, calculated from 2(39)(8), 39² – 8², 39² + 8²
951-1268-1585, which is (3-4-5) times 317, and
1007-1224-1585, calculated from 36² – 17², 2(36)(17), 36² + 17².

Sharing $15.75 Worth of Puzzles

Pattern Puzzle:

When I saw that 35 × 45 = 1575, I suspected a pattern. I made a chart to see if my suspicions were true, and they were! Can you look at the chart and tell me what that pattern is?

If you were able to see that pattern, then look at each of these. They have patterns because the numbers in 3 × 17, 4 × 16, 5 × 15, 6 × 14, and 7 × 13 have a relationship. What is that relationship?

Factors of 1575:

We can use 35 × 45 = 1575 to make one of its many possible factor trees:

  • 1575 is a composite number.
  • Prime factorization: 1575 = 3 × 3 × 5 × 5 × 7, which can be written 1575 = 3² × 5² × 7.
  • 1575 has at least one exponent greater than 1 in its prime factorization so √1575 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1575 = (√225)(√7) = 15√7.
  • The exponents in the prime factorization are 2, 2, and 1. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1575 has exactly 18 factors.
  • The factors of 1575 are outlined with their factor pair partners in the graphic below.

Can you use a different factor pair to create another factor tree for 1575? Will you always get 1575 = 3² × 5² × 7 in the end?

Difference of Two Squares Puzzle:

1575 is the difference of two squares in NINE different ways:
788² – 787² = 1575,
264² – 261² = 1575,
160² – 155² = 1575,
116² – 109² = 1575,
92² – 83² = 1575,
60² – 45² = 1575,
48² – 27² = 1575,
44² – 19² = 1575, and
40² – 5² = 1575.

In money 1575¢ is represented as $15.75. That’s the same as 63 quarters! Which of those differences of two squares is illustrated using quarters in the image below:

Which of the nine difference of two squares above is illustrated in the following image?

That image illustrates that $15.75 is just one quarter away from the next perfect square dollar amount, $16.00. Both 16 and 1600 are perfect squares. Can you make the rectangle below by moving just one row of quarters from the image above?

Moving that one row could help you notice that
8² – 1² = (8 – 1) × (8 + 1) = 63,
and might be the first step in understanding that  a² – b² = (a + b)(a – b) .

Dividing Mixed Numbers Puzzles:

A quarter is 25¢. The reason a quarter is called a quarter is because it is a quarter or 1/4th of a dollar. We usually write dollar and cents together as decimals. A quarter is $0.25.

Three quarters is 75¢ or $0.75 and is 3/4ths of a dollar.

Two quarters is 50¢ or $0.50 and is 2/4ths or one half of a dollar.

Representing 1575¢ in quarters can help you understand dividing mixed numbers like in the problem below:

The answer to both questions is the same! Now try this one:

You might not find this next example easy, but give it a look:

Why is 13 in the denominator of the answer to both questions when it didn’t appear in either question? Where did the 13 come from?

Now try writing and solving your own problem:

Working with money often seems like more fun than working with numbers. I hope you enjoyed these puzzles today.

 

1574 Avoiding Fractions When Using the Substitution Method to Solve Simultaneous Equations

The Substitution Method

Jo Morgan examines six methods of solving simultaneous linear equations in her wonderful book A Compendium of Mathematical Methods. Six is two more than the number of methods I had known previously. Jo loves pouring over vintage mathematics books to learn how concepts were taught in days gone by. She loves finding resources on Twitter to teach concepts and describes herself as a Resourceaholic. Today is her birthday, and this post is dedicated to her.

In my experience, the substitution method is often the first method taught to solve simultaneous linear equations, and usually, before the accompanying homework assignment is finished, fractions will be part of the solution process. Once students learn other methods, they usually abandon the substitution method with its troublesome fractions. However, the substitution method is really just as good as any other method, and fractions can actually be avoided when using it! I encourage students to use the substitution method and avoid fractions while they work.

In the table below I use the substitution method twelve ways to find one coordinate of the solution. (In each case the other coordinate could be found by substituting the known value back into one of the original equations and solving for the other coordinate.) Notice that only the first column uses fractions; the other two columns do not.

Follow each solution process step by step. Do any of the methods seem less confusing than the others? How will you approach the problem the next time you are asked to solve simultaneous linear equations?

Factors of 1574:

This is my 1574th post. Here is some information about that number:

  • 1574 is a composite number.
  • Prime factorization: 1574 = 2 × 787.
  • 1574 has no exponents greater than 1 in its prime factorization, so √1574 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 1574 has exactly 4 factors.
  • The factors of 1574 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1574:

1574 is in just one Pythagorean triple:
1574-619368-619370, calculated from 2(787)(1), 787²-1², 787²+1².
It can also be calculated from 2(394²-393²), 4(394)(393), 2(394²+393²).

1573 Ring Out Wild Bells

Today’s Puzzle:

Here are the first two verses of Alfred Lord Tennyson’s Ring Out Wild Bells:

Ring out, wild bells, to the wild sky,
The flying cloud, the frosty light;
The year is dying in the night;
Ring out, wild bells, and let him die.

Ring out the old, ring in the new,
Ring, happy bells, across the snow:
The year is going, let him go;
Ring out the false, ring in the true.

I think we are all ready for 2020 to “die” so we can ring in 2021. You can get started with this bell-shaped puzzle. Use logic to find its unique solution.

Print the puzzles or type the solution in this excel file: 12 Factors 1558-1573

Factors of 1573:

This is my 1573rd post, so let’s find the factors of 1573.

  • 1573 is a composite number.
  • Prime factorization: 1573 = 11 × 11 × 13, which can be written 1573 = 11² × 13
  • 1573 has at least one exponent greater than 1 in its prime factorization so √1573 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1573 = (√121)(√13) = 11√13
  • 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 1573 has exactly 6 factors.
  • The factors of 1573 are outlined with their factor pair partners in the graphic below.

More about the number 1573:

1573 is the sum of two squares:
33² + 22² = 1573.

1573 is the hypotenuse of a Pythagorean triple:
605-1452-1573, calculated from 33² – 22², 2(33)(22), 33² + 22².
It is also (5-12-13) times 121.

1572 Twelve Orientations of a Magic Snowflake

Today’s Puzzle:

Without looking, can you decide where the numbers 1 to 13 go on a snowflake so that each of the six edges and each of the six lines of symmetry has the same sum? There is only one solution, but the snowflake can be put into twelve different orientations. Perhaps you solved that puzzle by writing in the numbers when I suggested it a few days ago. You will want to solve that puzzle first.

The twelve orientations are pictured below. They are all pictures of the same snowflake.

Making a six-sided snowflake is more complicated than making a four-sided one. Rather than give step by step pictures of how I made the snowflake, I’ll just show one picture and explain the steps.

  1. Fold a piece of printer paper along a diagonal, cut off the excess to form a square.
  2. Find the center of that diagonal and divide the folded paper into thirds that intersect the center. Cut the raw edge to make the largest possible equilateral triangle that is 6-sheets thick. Unfold. You now have a hexagon.
  3. Fold an edge of the hexagon to the center but only make the crease go from one line of symmetry to the next one. Unfold. Repeat until all six edges have been folded to the center. This will create a medium-size hexagon inside the bigger hexagon.
  4. Fold back into a 6-sheet-thick equilateral triangle. Divide the cut side into thirds and fold at a 90-degree angle to the cut side. Only crease from the edge to the medium-size hexagon. Unfold.
  5. Cut along the fold you just made to create small slits in each of the twelve creases.
  6. Cut a small slit along the folds of each side of the medium-size hexagon.
  7. Fold back into a 6-sheet-thick equilateral triangle. Use decorative scissors to cut along the edge of the cut side of the triangle to make a pretty edge for the snowflake. Unfold until the only fold is along the diagonal.
  8. Using a hole punch but only going halfway on the paper with the punch, make the center punch. It will unfold into a single punch. Fold back into a 6-sheet-thick equilateral triangle. Make a punch through all 6 sheets but keep the punch very close to the center punch. This will create the 7-punch small hexagon in the center of the snowflake.
  9. Using the hole punch and any necessary creases, create hole punches representing the other numbers from 1 to 13. Take care that every edge and every line of symmetry has the same sum.

Factors of 1572:

  • 1572 is a composite number.
  • Prime factorization: 1572 = 2 × 2 × 3 × 131, which can be written 1572 = 2² × 3 ×
  • 1572 has at least one exponent greater than 1 in its prime factorization so √1572 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1572 = (√4)(√393) = 2√
  • 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 1572 has exactly 12 factors.
  • The factors of 1572 are outlined with their factor pair partners in the graphic below.

More about the Number 1572:

1572 is the difference of two squares two different ways:
394² – 392² = 1572 and
134² – 128² = 1572.

Since 1572 is divisible by 3, it is the magic sum of a magic hexagon. There isn’t room to punch enough holes in the hexagon, but you can see where all the numbers go below.

 

1569 A Magic Hexagon Puzzle

Today’s Puzzle:

The hexagon below has six edges and six lines going through its center. Can you write the numbers from 1 to 13 in the boxes so that the sums of the numbers along each of those lines are equal? The solution can be written 12 different ways, but they are all rotations and/or reflections of each other.

It should be helpful to realize that
1 + 13 = 14,
2 + 12 = 14,
3 + 11 = 14,
4 + 10 = 14,
5 + 9 = 14, and
6 + 8 = 14.
That leaves a 7 without a partner. Where do you think the 7 should go? What do you think the magic sum of each of the twelve paths will be?

Should even numbers or odd numbers go in the corners of the hexagon?

Factors of 1569:

This is my 1569th post. What are the factors of 1569?

  • 1569 is a composite number.
  • Prime factorization: 1569 = 3 × 523.
  • 1569 has no exponents greater than 1 in its prime factorization, so √1569 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 1569 has exactly 4 factors.
  • The factors of 1569 are outlined with their factor pair partners in the graphic below.

More about the Number 1569:

1569 is the difference of two squares:
263² – 260².

1569 is also the sum of the six numbers from 259 to 264.

Do you see any relationship between those two facts?

Since 1569 is divisible by 3, it is the magic sum of the magic hexagon shown below: