Mathematics

1569 A Magic Hexagon Puzzle

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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:

1566 How to Make a Magic Square Snowflake

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Today’s Puzzle:

Can you place all the numbers from one to nine in a 3 × 3 grid so that the numbers on each of the four edges and the numbers on each of the four lines of symmetry (where the fold lines are) have the same sum?

Here are some hints to get you started:

Begin by grouping and adding the numbers in this fashion:
Odd numbers 1 + 9 = 10,
Even numbers 2 + 8 = 10.
Odd numbers 3 + 7 = 10,
Even numbers 4 + 6 = 10, leaving you with just
5, which has no partner so it must go in the center of the magic square.

Those four sums plus 5 will form the four lines of symmetry, each with a magic sum of 10 + 5 = 15. Where do you put the sums in the magic square? Notice that the only way the edges can equal an odd number like 15 is to add two even numbers and one odd. (Adding two odd numbers and an even will never give you an odd number.) That means even numbers must go in the corners and an odd number will go between them.

Place the 1 in the middle of any edge. Place the nine in the middle on the opposite edge. What numbers will go on the edge with the 1? They must be two numbers that add up to 15 – 1 = 14. Only two of the even numbers add up to 14. Place them on the square on either side of the 1. Place their partners from the sums to 10 that are written above on their same lines of symmetry. Now you only have two odd numbers left to place. Place them so that those edges add up to 15.

Using a hole punch to make the numbers instead of writing numerals makes the snowflake even better: There are eight different orientations for the numbers to be on the snowflake, but all eight ways really only form one possible snowflake.

Keep the grid you’ve made with your numerals. You will need it to know where to punch your holes in your snowflake.

How to Make Your Own Magic Square Snowflake:

You will need an 8 1/2 by 11 sheet of white printer paper, scissors, and a hole punch. Decorative scissors are optional.

  1. Begin with an 8 1/2 by 11 sheet of white printer paper. Follow  Paula Krieg’s instructions on how to fold and cut it into an 8 1/2 × 8 1/2 perfect square.
  2. Make a crease along both diagonals of your square. Unfold.
  3. Follow EZ Origami’s instructions on how to fold your square into thirds. Unfold.
  4. Turn your square 90° and follow EZ Origami’s instructions again to divide your square into thirds in the other direction.
  5. Fold snowflake in half vertically and then horizontally. Unfold.
  6. If desired, use decorative scissors to give your snowflake a pretty edge. You can cut two edges at the same time. Unfold.
  7. Make a narrow cut along the folds of the smaller square edges, taking care not to cut where the folds intersect. Unfold. (Do not use decorative scissors for this step.)
  8. Use the grid with numerals that you made earlier and a hole punch to place the right number of holes in each of the nine squares. An ordinary hole punch with handles will work, but I used the decorative one pictured below.

I used the following facts to make my holes with my hole punch in the appropriate squares. Look at the snowflake picture above if clarification is needed:
1. Make a single punch in the center of its square.
2(1) = 2. Fold paper on diagonal, make a single punch above the diagonal.
2(3(1/2)) = 3. Fold paper on diagonal, make three half punches on the diagonal. Unfold.
2(2) = 4. Fold paper on diagonal, make two punches above the diagonal. Unfold.
4(1 + 1/4) = 5. The center square has four smaller squares. Fold center square to make one small square with four thicknesses, make 1/4 punch at corner of folds in center of snowflake. Make a single punch in center of small, thick square. Unfold.
2(3) = 6. Fold paper on diagonal, make three punches above the diagonal. Unfold.
2(3 + 1/2) = 7. Fold paper on diagonal, make half punch on center of diagonal and three punches above the diagonal. Unfold.
2(4) = 8. Fold paper on diagonal, make four punches above the diagonal. Unfold.
2((3 + 3(1/2)) = 9. Fold paper on diagonal, make three half punches on diagonal and three punches above the diagonal. Unfold.

Finding the Prime Factorization of 1566:

Since this is my 1566th post, I’ll explain how I find its prime factorization using as few divisions as I possibly can.

I know that 1566 can be divided by 9 because 1 + 5 + 6 + 6 = 18, a multiple of 9.
1566 ÷ 9 = 174.

I know that 174 can be divided by 6 because it is even and 1 + 7 + 4 = 12, a multiple of 3.
174 ÷ 6 = 29.

I show my work here:

Now all that’s left to do is put the prime factors in numerical order with exponents.

Factors of 1566:

  • 1566 is a composite number.
  • Prime factorization: 1566 = 2 × 3 × 3 × 3 × 29, which can be written 1566 = 2 × 3³ × 29.
  • 1566 has at least one exponent greater than 1 in its prime factorization so √1566 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1566 = (√9)(√174) = 3√174.
  • The exponents in the prime factorization are 1,3 and 1. Adding one to each exponent and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 4 × 2 = 16. Therefore 1566 has exactly 16 factors.
  • The factors of 1566 are outlined with their factor pair partners in the graphic below.

More about the Number 1566:

1566 is the hypotenuse of a Pythagorean triple:
1080-1134-1566 which is (20-21-29) times 54.

Since 1566 is a multiple of 3, it is the magic sum of this 3 × 3 magic square:

That can happen for ANY multiple of 3. Multiples of 3 that are less than 15 must have negative numbers in their magic squares in order to be the magic sum. For example, zero is a multiple of 3 because 3 × 0 = 0. Here’s how zero is the magic sum of a magic square:

 

1556 Stacks Up Nicely!

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What is special about the number 1556?

What makes 1556 stack up?

From OEIS.org we learn that
2² + 3² + 5² + 7² + 11² + 13² + 17² + 19² + 23² =  1556.
Yes, that’s the sum of the squares of the first nine prime numbers.
Those perfect squares can be stacked on top of each other as I illustrate in the graphic below:

Factors of 1556:

1556 (and every other whole number whose last two digits are 56) is divisible by 4:

  • 1556 is a composite number.
  • Prime factorization: 1556 = 2 × 2 × 389, which can be written 1556 = 2² × 389.
  • 1556 has at least one exponent greater than 1 in its prime factorization so √1556 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1556 = (√4)(√389) = 2√389.
  • 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 1556 has exactly 6 factors.
  • The factors of 1556 are outlined with their factor pair partners in the graphic below.

More about the Number 1556:

1556 is the sum of two squares:
34² + 20² = 1556.

1556 is the hypotenuse of a Pythagorean triple:
756 -1360-1556, which is 4 times (189-340-389).
It can also be calculated from 34² – 20², 2(34)(20), 34² + 20².

1556 is also the difference of two squares:
390²  – 388²  = 1556.

1554 What Patterns Do You See?

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Today’s Puzzle:

I like multiples of 111, including 1554. What cool patterns do you notice if a 2-digit number is multiplied by 111 as shown in the graphic below:

A Factor Tree for 1554:

Here’s a factor tree for 1554 that begins with the factor pair 14 × 111:

Factors of 1554:

  • 1554 is a composite number.
  • Prime factorization: 1554 = 2 × 3 × 7 × 37.
  • 1554 has no exponents greater than 1 in its prime factorization, so √1554 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 1554 has exactly 16 factors.
  • The factors of 1554 are outlined with their factor pair partners in the graphic below.

More about the Number 1554:

1554 is the hypotenuse of a Pythagorean triple:
504-1470-1554, which is (12-35-37) times 42.

1548 Puzzling Gerrymandering Questions

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Today’s Puzzle:

If you had two types of candy and four kids, this is a good way to divide the candy. But if you had two types of voters and four congressional districts, is it a fair way to determine those congressional districts?

Gerrymandering happens when congressional district boundaries are drawn to give an advantage to one political party over another. The green party might be given an advantage in the drawing above, but does simply dividing the graphic into four quadrants give an advantage to the yellow party?

One article complains that one of the worst examples of gerrymandering is a congressional district shaped like a duck, but it is unclear if that duck is keeping like-minded people together or keeping them apart.

or

It is unclear to us if like-minded people were grouped together, but we can be certain that it was very clear to those who drew the boundaries.

Each of the graphics above had 12 yellow sections and 36 green sections. If you think it is only fair to let like-minded people elect someone who thinks like them,  how should congressional boundaries be drawn if the 12 yellow sections and the 36 green sections look like this?

No matter how you draw the boundaries, green will be in the majority in each congressional district, and most likely the majority will choose each district’s representative. But if you believe that gerrymandering is justified to benefit like-minded people, should those people NOT be represented by a like-minded representative simply because they don’t live next to each other?

These are good questions to puzzle over. Denise Gaskins has created the Gerrymandering Project to help you manipulate a 10 by 10 map of your creation in a variety of ways. This project will help every voter and future voter understand the mathematics and the politics of drawing boundaries on a larger scale. Check it out!

Factors of 1548:

  • 1548 is a composite number.
  • Prime factorization: 1548 = 2 × 2 × 3 × 3 × 43, which can be written 1548 = 2² × 3² × 43.
  • 1548 has at least one exponent greater than 1 in its prime factorization so √1548 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1548 = (√36)(√43) = 6√43.
  • 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 1548 has exactly 18 factors.
  • The factors of 1548 are outlined with their factor pair partners in the graphic below.

Factor Tree for 1548:

The last two digits of 1548 are a multiple of 4, so 1548 is divisible by 4.
1 + 5 + 4 + 8 = 18, a multiple of 9, so 1548 is divisible by 9.

Here’s how I used those two facts to make an autumn factor tree for 1548:

More about the Number 1548:

1548 is the difference of two squares in three different ways:
388² – 386² = 1548,
132² – 126² = 1548, and
52² – 34²  = 1548.

 

1547 is a Hexagonal Pyramidal Number

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Today’s Puzzle:

1547 is the 13th hexagonal pyramidal number. Looking at the graphic that I made below of 1547 tiny squares, can you determine what the prime factorization of 1547 is?

Factors of 1547:

  • 1547 is a composite number.
  • Prime factorization: 1547 = 7 × 13 × 17.
  • 1547 has no exponents greater than 1 in its prime factorization, so √1547 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 1547 has exactly 8 factors.
  • The factors of 1547 are outlined with their factor pair partners in the graphic below.

More about the Number 1547:

1547 is the hypotenuse of FOUR Pythagorean triples:
147-1540-1547, which is 7 times (21-220-221),
595-1428-1547, which is (5-12-13) times 119,
728-1365-1547, which is (8-15-17) times 91, and
980-1197-1547, which is 7 times (140-171-221).

1545 The Complete Message Made with Puzzle Letters

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The Puzzles:

The last six Find the Factors Puzzles spelled out this message: I voted. I like the quote from Larry J. Sabato, “Every election is determined by the people who show up.” My vote will matter, and so will yours.

You can also print these puzzles from this excel file: Taxman & 1537-1544

American politics has become rather ugly, but voting is still important.

I am thrilled that I was able to vote for one of these candidates for governor of Utah, and I am very happy that one of them will become my governor even if it isn’t the same person I voted for.

Factors of 1545:

  • 1545 is a composite number.
  • Prime factorization: 1545 = 3 × 5 × 103.
  • 1545 has no exponents greater than 1 in its prime factorization, so √1545 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 1545 has exactly 8 factors.
  • The factors of 1545 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1545:

1545 is the hypotenuse of a Pythagorean triple:
927-1236-1545, which is (3-4-5) times 309.

1541 is the 23rd Octagonal Number

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Today’s Puzzle:

Into which geometric shape can you arrange 1541 tiny squares?

The answer is an octagon as illustrated below.

Can you divide that octagon into 6 triangles, 5 of them representing the 22nd triangular number and 1 of them representing the 23rd?

It is possible because 5(22·23)/2 + (23·24)/2 = 1541.

We can simplify the left side of the equal sign:
5(22·23)/2 + (23·24)/2 =
5(11·23) + (23·12) =
23(5·11) + 23(12) =
23(55 + 12) =
23(67) = 1541

Factors of 1541:

  • 1541 is a composite number.
  • Prime factorization: 1541 = 23 × 67.
  • 1541 has no exponents greater than 1 in its prime factorization, so √1541 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 1541 has exactly 4 factors.
  • The factors of 1541 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1541:

1541 is the difference of two squares in two different ways:
771² – 770² = 1541 (Not coincidently, 770 + 771 = 1541), and
45² – 22² = 1541.

How Much of My $1540 Will The Taxman Get?

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The Taxman Game

You might think that the picture below is just the numbers from 1 to 55 arranged in a 5×11 rectangle.

It’s really PLAY MONEY, one bill in each denomination from 1 to 55. If you add up the total value of all this play money, you will get 1540 because the sum of the numbers from 1 to 55 is 1540.

Most of that 1540 in play money can be yours to keep if you outsmart THE TAXMAN.

You can take any bill that you want, but The Taxman has to get paid every time you do. The Taxman will get ALL the available factors of the bill you take. If there isn’t at least one bill on hand that is a factor of your chosen bill, then you can’t take that bill. When none of the bills left are factors of any of the other bills that are left, The Taxman gets ALL the remaining bills. The value of those bills could add up quickly, and that’s how The Taxman might outsmart you.

You can print this excel file to play the Taxman game: Taxman & 1537-1544 It has all the play money bills from 1 to 100, and all the factors are typed on the top of the bills!

The factors at the top of each bill make it possible to play the game even if you don’t know how to multiply or divide yet. My granddaughter learned to play the game sometime around her eighth birthday. Before tackling 55 bills, she learned what to do for a smaller number of bills. I told her:
If 1 is the only bill, The Taxman wins.
If 1, 2 are the only bills, you take the 2, The Taxman gets the 1, and you win.
If 1, 2, 3 are the only bills, the best you can do is a draw (1 + 2 = 3),
But if there are at least four bills, it is possible to win every time.

She played the game over and over adding one more bill to the game each time she played. She enjoyed it very much and played it over 20 times until bedtime required her to stop. Each time she played we talked about how she would arrange the cards in a rectangle. For example, if she used cards 1 – 6, she would look at the factors of 6 and decide on a 2 × 3 rectangle or a 1 × 6 rectangle.

Knowing I wanted to write this post about the 55th triangular number (1540), I had her jump to using 55 bills the next day, but I helped her with it a little. We didn’t want to just win the game, we wanted to make it so The Taxman would get as little money as possible. Here is the final order of what we took and what The Taxman got.

Laying it out like that made it easy to have do-overs when needed.

A Way to Keep Score When Playing Taxman:

When it was time to score the game, instead of adding up all the numbers to see if we took more than The Taxman, we found smaller sums that equaled as many of our choices as we could as illustrated below. For example,  47 + 6 = 53, so there was no need to add 47 + 6 to The Taxman’s total or to add 53 to ours.

I found it enjoyable finding all those sums, but it isn’t necessarily easy for a child to do. After arranging all those sums on the table, we were able to see that we scored
35 + 39 + 38 + 12 + 30 + 50 + 24 + 48 + 42 = 318 more than The Taxman did.

We can use algebra to figure out exactly how much of our 1540 in play money we get to keep.
If The Taxman got X of the play money, we kept X + 318 of it, and
X + X + 318 = 1540,
2X + 318 = 1540,
2X + 318 – 318 = 1540 – 318,
2X = 1222,
X = 611, that’s what The Taxman got.
We kept 611 + 318 = 929 of our play money.

What percentage of the play money did The Taxman get?
611/1540 ≈ 0.40 which is 40%. Wow, that’s a lot!

Factors of 1540:

  • 1540 is a composite number.
  • Prime factorization: 1540 = 2 × 2 × 5 × 7 × 11, which can be written 1540 = 2² × 5 × 7 × 11.
  • 1540 has at least one exponent greater than 1 in its prime factorization so √1540 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1540 = (√4)(√385) = 2√385.
  • 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 1540 has exactly 24 factors.
  • The factors of 1540 are outlined with their factor pair partners in the graphic below.

Factor Cake for 1540:

1540 has lots of prime factors with 11 being the greatest so it makes a lovely factor cake:

What Kind of Shape Is 1540 in?

I know that 1540 = 55×56÷2 so, 1540 is the 55th triangular number. Guess what? Since 55 is also a triangular number (55 = 10×11÷2), we could have arranged the play money bills in a triangle like this instead of a 5 × 11 rectangle.

1540 is not only the 55th triangular number, but it is also the sum of the first 20 triangular numbers:

1+3+6+10+15+21+28+36+45+55+66+78+91+105+120+136+153+171+190+210 = 1540.

If you stack those twenty triangular numbers from smallest to biggest, you will get something shaped like a tetrahedron. That’s why we say that 1540 is the 20th tetrahedral number. We can also use a formula for the 20th tetrahedral number:
20(21)(22)/6 = 1540.

1540 is also the 28th hexagonal number. (All hexagonal numbers are also triangular numbers. That’s why it’s the 2(28)-1 = 55th triangular number.) It is the 28th hexagonal number because 2(28²) – 28 = 1540.

1540 is the 20th decagonal number because 4(20²) – 3(20) = 1540.

I didn’t make a visual, but 1540 is also the19th centered nonagonal number.
That’s because
3 × 19 = 57 and (57 – 2)(57 – 1)/2 = 1540.

1540 is also the hypotenuse of a Pythagorean triple triangle:
924-1232-1540 which is (3-4-5) times 308.

1540 is quite the shape-shifter number, isn’t it!

1534 Dem Bones

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Today’s Puzzle:

You’ve probably heard someone sing about Dem Bones before. It is a song that describes how bones in the body are connected to each other. Ezekiel 37:1-14 was the inspiration for the lyrics, and the song is the inspiration for today’s mystery level puzzle:

Factors of 1534:

  • 1534 is a composite number.
  • Prime factorization: 1534 = 2 × 13 × 59.
  • 1534 has no exponents greater than 1 in its prime factorization, so √1534 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 1534 has exactly 8 factors.
  • The factors of 1534 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1534:

1534 is the hypotenuse of a Pythagorean triple:
590-1416-1534 which is (5-12-13) times 118.

1534 is 4(7³) + 3(7²) + 2(7¹) + 1(7º). That means it is 4321 in base 7.