Mathematics

1785 A Pythagorean Triple Logic Puzzle

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Today’s Puzzle:

If you can print a copy of the puzzle from this Excel sheet, 10 Factors1773-1785, it will look like this:

Note: I have revised this puzzle since originally publishing it. I was horrified to discover that the original puzzle had two solutions. I apologize for any inconvenience I may have caused. This revised puzzle only has one solution.

Factors of 1785:

17 × 5 = 85, so 1785 is divisible by 17.

  • 1785 is a composite number.
  • Prime factorization: 1785 = 3 × 5 × 7 × 17.
  • 1785 has no exponents greater than 1 in its prime factorization, so √1785 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 1785 has exactly 16 factors.
  • The factors of 1785 are outlined with their factor pair partners in the graphic below.

More About the Number 1785:

Did you notice that 3, 5, 7, and 357 are all factors of 1785?
Or that 35 and 51 make a factor pair, and 3, 5, and 1 are also factors?

1785 is the hypotenuse of FOUR Pythagorean triples:
273-1764-1785
756-1617-1785
840-1575-1785
1071-1428-1785

1785 is the difference of two squares in EIGHT different ways:
893² – 892² = 1785,
299² – 296² = 1785,
181² – 176² = 1785, and five more ways. Can you find them?

1785 is a Palindrome in a couple of bases:
It’s 123321 in base 4, because 1(1024) + 2(256) + 3(64) + 3(16) + 2(4) + 1(1) = 1785.
And it’s 3F3 in base 22, because 3(22²) + 15(22) + 3(1) = 1785.

1784 Another Hundred Simplifiable Square Roots

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Today’s Puzzle:

What percentage of natural numbers less than or equal to 1784 have simplifiable square roots?

Here is a chart of the 601st to the 700th simplifiable square roots:

You can figure out the percentage of numbers up to 1784 that have simplifiable square roots by calculating 700×100 ÷1784.

Was the percentage higher or lower than you expected?

The green areas on the chart are for consecutive numbers with simplifiable square roots. 1680-1684 are the smallest five consecutive numbers that can make that claim. Why can they? Because every one of their prime factorizations has an exponent greater than one in it.

1680 prime factorization

Factors of 1784:

  • 1784 is a composite number.
  • Prime factorization: 1784 = 2 × 2 × 2 × 167, which can be written 1784 = 2³ × 167.
  • 1784 has at least one exponent greater than 1 in its prime factorization so √1784 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1784 = (√4)(√446) = 2√446.
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1784 has exactly 8 factors.
  • The factors of 1784 are outlined with their factor pairs in the graphic below.

More About the Number 1784:

1784 is the difference of two squares in two different ways:
447² – 445² = 1784, and
225² – 221² = 1784.

1784 is a palindrome in two bases:
It’s 494 in base20 because 4(20²)+9(20)+4(1) = 1784, and
2C2 in base27 because 2(27²)+12(27)+2(1) = 1784.

1782 Don’t Chop Down This Factor Tree!

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Today’s Puzzle:

Today is Monday, February 19. In the United States, we are celebrating Presidents’ Day, honoring most especially two important presidents who were born in February.

Exactly one week ago was February 12.

George Washington was born on February 11, 1731, Julian calendar.
Abraham Lincoln was born on February 12, 1809, Gregorian calendar.

The Julian calendar didn’t have leap days, so in 1752 a year and eleven days were added to Washington’s birthday to convert it to the Gregorian calendar.

Neither president will ever have his birthday on the third Monday of February when Presidents’ Day is observed. Too bad the second Monday of February wasn’t chosen instead. Then we could fudge a little and say that Presidents’ Day would be observed on one of their birthdays 2/7 of the time!

What days of the month are the earliest and the latest that a second Monday could be? 

When I was young I was told the story about George Washington chopping down a cherry tree. When he was confronted, he would not and could not tell a lie, and confessed his misdeed. As I got older, I learned that this was a fabricated story designed to teach children honesty of all things!

Nevertheless, some people celebrate Presidents’ Day by eating a cherry pie in remembrance of that story.

Factors of 1782:

This is my 1782nd post. Since it’s Presidents’ Day, I thought I would make a few factor trees for that number. You could think of the prime factors in red as cherries on the trees. Notice that all the prime factors are low-hanging fruit on these particular trees!

  • 1782 is a composite number.
  • Prime factorization: 1782 = 2 × 3 × 3 × 3 × 3 × 11, which can be written 1782 = 2 × 3⁴ × 11.
  • 1782 has at least one exponent greater than 1 in its prime factorization so √1782 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1782 = (√22)(√81) = 9√22. The exponents in the prime factorization are 1, 4, and 1. Adding one to each exponent and multiplying we get (1 + 1)(4 + 1)(1 + 1) = 2 × 5 × 2 = 20. Therefore 1782 has exactly 20 factors.
  • The factors of 1782 are outlined with their factor pair partners in the graphic below.

More About the Number 1782:

(5(27²) -3(27))/2 = 1782, so it is the 27th heptagonal number after 0.

Here’s another cool fact about 1782 from OEIS.org.

1780 Reflections of a Polygonal Bird

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Today’s Puzzle:

What ordered pairs were used to create this bird?

Its eye was formed from an equation of a circle:
(x – 7)²+ (y – 15)² = 3/4.

After creating the polygonal bird using ordered pairs and that circle equation, I wanted to do other things with the bird. Everything I did was like a puzzle for me to figure out.

Could I make it “fly”? Yes!

 

Could I make it reflect itself more than once over the y-axis and the x-axis? Yes! And I could make it do some sliding at the same time!

This next one was the toughest for me to do. I wanted the bird to be in motion rotating counter-clockwise around the origin. I was able to do it, but Desmos wouldn’t save the sliders exactly the way I wanted. I will need your help on this one. Click on this rotating bird link, then push play on slider a. About the time that slider goes to zero, push play on slider b. If you hit the sliders just right, it will look something like this GIF I made, but slower:

Rotating Polygonal Birds

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Factors of 1780:

Perhaps our polygonal bird would like to fly to a tree. Here’s a factor tree for 1780 that it can take a rest on.

I knew that 1780 was divisible by 4 because its last two digits are divisible by 4.

  • 1780 is a composite number.
  • Prime factorization: 1780 = 2 × 2 × 5 × 89, which can be written 1780 = 2² × 5 × 89.
  • 1780 has at least one exponent greater than 1 in its prime factorization so √1780 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1780 = (√4)(√445) = 2√445.
  • 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 1780 has exactly 12 factors.
  • The factors of 1780 are outlined with their factor pair partners in the graphic below.

More About the Number 1780:

1780 is the difference of two squares in two different ways:
446² – 444² = 1780, and
94² – 84² = 1780.

1780 is the sum of two squares in two different ways:
42² + 4² = 1780, and
36² + 22² = 1780.

1780 is the hypotenuse of four Pythagorean triples:
336-1748-1780, calculated from 2(42)(4), 42² – 4², 42² + 4²,
780-1600-1780, which is 20 times (39-80-89)
812-1584-1780, calculated from 36² – 22², 2(36)(22), 36² + 22², and
1068-1424-1780, which is (3-4-5) times 356.

1780 is KK in base 88 because
20(88) + 20(1) = 20(89) = 1780.

1779 How Many Similar Triangles Are There in This Image?

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Today’s Puzzle:

All of the triangles in the image below are similar. How many similar triangles are there in the image? Why are they similar? Hint: If I were counting them, I would list all the triangles by writing each one indicating the sides in this order every time: the smallest, the medium, and the longest side. Don’t forget to list ΔLKJ. It’s pretty tiny!

Factors of 1779:

  • 1779 is a composite number.
  • Prime factorization: 1779 = 3 × 593.
  • 1779 has no exponents greater than 1 in its prime factorization, so √1779 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 1779 has exactly 4 factors.
  • The factors of 1779 are outlined with their factor pair partners in the graphic below.

More About the Number 1779:

1779 is the hypotenuse of a Pythagorean triple:
1104-1395-1779, which is 3 times (368-465-593).

From OEIS.org we learn that 1779 = 10,016,218,555,281, and that’s the smallest 4th power that has 14 digits.

1779 is palindrome 323 in base 24 because
3(24²) + 2(24) + 3(1) = 1779.

1776 A Single Rosebud

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Today’s Puzzle:

The gift of a single red rose is a way to say, “I love you.” To me, a single red rosebud would be saying, “I love you, and my love for you is growing.” To all my faithful readers, I give you this single red rosebud:

Here is the same puzzle without any added color if you want to save on printer ink.

Factors of 1776:

Another way to show love is to plant a tree. How about we plant a factor tree? Since 1776 has twenty different factor pairs, MANY possible factor trees could be planted. I chose to base this one on the fun fact that 1776 = 4 · 444:

  • 1776 is a composite number.
  • Prime factorization: 1776 = 2 × 2 × 2 × 2 × 3 × 37, which can be written 1776 = 2⁴ × 3 × 37
  • 1776 has at least one exponent greater than 1 in its prime factorization so √1776 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1776 = (√16)(√111) = 4√111
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1776 has exactly 20 factors.
  • The factors of 1776 are outlined with their factor pair partners in the graphic below.

Did you notice all the repdigit factors of 1776 in the table?

More About the Number 1776:

1776 is the hypotenuse of a Pythagorean triple:
576-1680-1776, which is (12-35-37) times 48.

1776 looks interesting in some other bases:
It’s 5115 in base7 because 5(7³) + 1(7²) +1(7) + 5(1) = 1776, and uh oh!
OO in base73 because 24(73) + 24(1) = 24(74) = 1776.
We’ve run out of letters in the alphabet to use as numbers, but I
will note that 37(47) + 37(1) = 37(48) = 1776.

1774 A Mostly Square Heart for You to Play With

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Today’s Puzzle:

The celebrated author of Math Play, Libo Valencia, recently wrote a post on how he uses mathplay to help his nine-year-old daughter learn the multiplication table. One of the playful things they did together was find objects around the house to represent several perfect squares. For example, they happened to have some small bright yellow hexagons in their house and they used six of them to show that six times six is thirty-six. If you don’t have any bright yellow hexagons at your place, you probably have some hexagon-shaped nuts and/or bolts you could use to show 6 × 6 = 36.

All but two of the clues in today’s puzzle are perfect squares, so I’m dedicating this puzzle to Libo’s daughter. Square number thirty-six is a clue three times in the puzzle. The rules of the puzzle won’t allow 6 × 6 to be the factors for all three of them, however. I’m sure you can figure the puzzle out, anyway. Just make sure you’re having fun doing it. There is only one solution.

Factors of 1774:

  • 1774 is a prime number.
  • Prime factorization: 1774 is prime.
  • 1774 has no exponents greater than 1 in its prime factorization, so √1774 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1774 has exactly 2 factors.
  • The factors of 1774 are outlined with their factor pair partners in the graphic below.

More About the Number 1774:

1774 is a palindrome in a couple of bases:
It’s 626 in base17 because 6(17²) + 2(17) + 6(1) = 1774, and
it’s 383 in base23 because 3(23²) + 8(23) + 3(1) =1774.

1772 Is a Centered Heptagonal Number!

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Today’s Puzzle:

It’s early in 2024, so here’s a Factor Fits puzzle utilizing the factors of 20 and 24. Give it a try! There is only one solution.

Factors of 1772:

This is my 1772nd post. What are the factors of 1772?

  • 1772 is a composite number.
  • Prime factorization: 1772 = 2 × 2 × 443, which can be written 1772 = 2² × 443.
  • 1772 has at least one exponent greater than 1 in its prime factorization so √1772 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1772 = (√4)(√443) = 2√443.
  • 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 1772 has exactly 6 factors.
  • The factors of 1772 are outlined with their factor pair partners in the graphic below.

1772 is a Centered Heptagonal Number:

1772 is one more than 7 times the 22nd triangular number. For all previous centered heptagonal numbers about which I’ve written, I only mentioned their inclusion in this set of numbers. This time, I was determined to produce a graphic of the number. I used Desmos and Excel to determine all 1772 points in the graphic. It was a little time-consuming, but I got it done!

The points of the first heptagon were (1, 0), (cos2π/7, sin2π/7), (cos4π/7, sin4π/7), (cos6π/7, sin6π/7), (cos8π/7, sin8π/7), (cos10π/7, sin10π/7), (cos12π/7, sin12π/7). Here is an example of what was involved in completing one side of the other heptagons: Suppose I wanted to find five points on the line connecting (a,c) and (b,d). The five points would be
((4a+0b)/4, (4c+0d)/4), or simply (a, c),
((3a+1b)/4, (3c+1d)/4),
((2a+2b)/4, (2c+2d)/4), or simply ((a+b)/2, (c+d)/2), the midpoint,
((1a+3b)/4, (1c+3d)/4),
((0a+4b)/4, (0c+4d)/4), or simply (b, d).

I used Excel to calculate those numbers and then copied and pasted them into Desmos which graphed them beautifully. Each round took me less than ten minutes to complete. Here is the finished product:

More About the Number 1772:

1772 is the difference of two squares:
444² – 442² = 1772.

1772 is palindrome 24042 in base 5. Why?
Because 2(5⁴)+4(5³)+0(5²)+4(5¹)+2(5º) = 1772.

Facts and Factors for the Year 2024

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A Countdown to 2024:

2024 Countdown

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Here are some other countdowns:

What Kind of Factors Will the Year 2024 Bring Us?

Here’s a factor cake to celebrate 2024’s arrival:

And its factor pairs are outlined on this chart:

The sum of all the factors of a number (excluding itself) determines if a number is deficient, perfect, or abundant. Which of those describes 2024?

Powerful Facts About the Number 2024:

2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ = 2024, as illustrated below

2¹º + 10³ =2024.

2024 is the sum of eleven consecutive even square numbers:

2024 is the difference of two squares in FOUR different ways:

507² – 505² = 2024,
255² – 251² = 2024,
57² – 35² = 2024, and
45² – 1² = 2024.

One of those four equations brings us to…

Today’s Puzzle and Some Other 2024-Themed Puzzles:

Here are some other 2024-themed puzzles:

2024 in Pythagorean Triple Triangles

2024 is a leg in quite a few Pythagorean triple triangles. Here are a few:

I didn’t even include all the triangles listed on the left in the illustration because some of the points were too close together with the scale I used in Desmos.

I also didn’t include
2024² + 512070² = 512074 or
2024² + 128010² = 128026
because those triangles would have made the scale even worse.

There are more triangles, but I think this is a good enough representation.

Here’s a more complete list:

2024 is NOT the hypotenuse of any Pythagorean triple because none of its prime factors leave a remainder of 1 when divided by 4.

2024 in Pascal’s Triangle:

2024 is in the 24th row of Pascal’s triangle. Click on the image to see the numbers in the image better.

Since it is in the third column of that 24th row, 2024 is a Tetrahedral number. That means it is the sum of the first 22 triangular numbers. That fact can be illustrated as I have here in this vertically rotating Desmos 3D image. Go ahead and click on the image below to see this tetrahedron rotating more horizontally. Also, notice that I made the image with 2024 quarter-unit spheres.

As I stated before, the image is made from the first 22 triangular numbers stacked on top of each other. The sum of the first 22 triangular numbers is given below:

This tetrahedral number can also be expressed mathematically in this way:

This way:

The second way listed here:

Or this way:

2024 Consecutive Number Sums

2024 is the sum of consecutive counting numbers in three different ways:

2024 is the sum of consecutive odd numbers in four different ways: (It’s because 2024 is the difference of two squares in four different ways.)

2024 is the sum of consecutive even numbers:

8 consecutive even numbers:
246+248+250+252+254+256+258+260=2024.

11 consecutive even numbers:
174+176+178+180+182+184+186+188+190
+192+194=2024.

23 consecutive even numbers:
66+68+70+72+74+76+78+80+82+84+86+88+90+92
+94+96+98+100+102+104+106+108+110 = 2024.

2024 Magic Sums

2-0+2-4 = 0, so 2024 is divisible by 11. That means it is the magic sum of an 11 × 11 Magic Square. Here is one way that Magic Square can be completed. I followed the directions given in this post. You can see the 11 consecutive numbers listed above along the lower left to upper right diagonal.

If you would like to try completing the magic square yourself, here’s an Excel template that will automatically add the sums while you enter the numbers: 1766-1772 and 2024 Magic Squares

2024 is divisible by 8 but not by 16, so it is the Magic Sum of a 16 × 16 Magic Square. Here are two examples:

For this first one, I wrote the numbers from -1 to 14 across the top of the puzzle and continued in like manner. After the numbers were in place, I started flipping diagonals. How many diagonals did I flip?  34: The sixteen green diagonals, the sixteen blue diagonals, the pink diagonal, and the brown diagonal.

Here is how it looked when I finished:

For this second one, I wrote the numbers from -1 to 14 in the first 4 × 4 square, the numbers from 15 to 30 in the second 4 × 4 square, etc. Then I began flipping diagonals.

The Excel sheet, 1766-1772 and 2024 Magic Squares, also includes a template that will allow you to complete just a 4 × 4 magic square in the lower right corner for the numbers from -1 to 14. The whole 16 × 16 magic square will populate if you just complete that 4 × 4 magic square! But there’s a template if you want to do the whole thing from scratch as well.

More About the Number 2024:

Because 2024 is the sum of the 16 numbers from 119 to 134,
and 16 is even, it follows that
134²-133²+132²-131²+130²-129²+128²-127²+126²
-125²+124²-123²+122²-121²+120²-119²=2024.

Here is a way to make 2024 using only the digits 2, 0, 2, and 4:

(2+0+2+4)×
(2+0+2+4-(2+0)/2+4)×
(((2+0+2+4)×(-(2+0)/2+4))-(2+0+2)/4) = 2024.

The tan²(88.7266556386°) ≈ 2024, so it is the solution to this next problem:

 

1771 Pascal’s Triangle and the Twelve Days of Christmas

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A Twelve Days of Christmas Puzzle with Triangular and Tetrahedral Numbers:

I wanted a copy of Pascal’s triangle with 14 rows. I couldn’t find one, so I made my own. To fill in the missing number in a cell, simply write the sum of the two numbers above it. I would suggest filling it in together as a class so that they can see how it is done without actually having to write in all the numbers themselves. The biggest missing sum is 364.

After filling that puzzle in together as a class, I would give students this next copy of Pascal’s triangle to use.

There are many patterns in Pascal’s triangle. It can be fun to color them with that in mind. I would caution students to color lightly so that they can still read the numbers afterward. How did I color this one? If the number in a cell is not divisible by the row number, I colored it green. Of course, all the 1’s were colored green. If all the other numbers in the row were divisible by the row number, I colored all of them red. If only some of them were, I colored them yellow. Notice that the row number of every row that is red is a prime number. Composite row numbers will always have at least one entry that is not divisible by the row number.

I divided each of the numbers in this next one by 3, noted the remainder, and colored them accordingly:

  • remainder 0 – red
  • remainder 1 – green
  • remainder 2 – yellow

1771 is a Tetrahedral Number:

364 = 12·13·14/6. That means it is the 12th tetrahedral number.
If my true love gave me all the gifts listed in the Twelve Days of Christmas song, it would be a total of 364 gifts. Since I don’t have use for all those birds, if I returned one gift a day, it would take me 364 days to return them all. That’s one day less than an entire year!

1771 = 21·22·23/6. That means it is the 21st tetrahedral number.
If there were 21 days of Christmas, and the pattern given in the song held, my true love would give me 1771 gifts. Yikes, I’ll need a bigger house or maybe a bird sanctuary!

Here is one-half of the 23rd row in Pascal’s triangle showing the number 1771:

I’ve also been thinking about the next tetrahedral number after 1771 because the year, 2024, has almost arrived. Note to my true love: I don’t need or want 2024 gifts, please!

Factors of 1771:

1771 is a palindrome with an even number of digits, so 1771 is divisible by eleven.

  • 1771 is a composite number.
  • Prime factorization: 1771 = 7 × 11 × 23.
  • 1771 has no exponents greater than 1 in its prime factorization, so √1771 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 1771 has exactly 8 factors.
  • The factors of 1771 are outlined with their factor pair partners in the graphic below.

More About the Number 1771:

1771 is the difference of two squares in four ways:

886² – 885² = 1771,
130² – 123² = 1771,
86² – 75² = 1771, and
50² – 27² = 1771.

It is easy to see that 1771 is a palindrome in base 10, but it is also a palindrome in some other bases:
It’s 4H4 in base 19 because 4(19²)+17(19)+4(1)=1771,
232 in base 29 because 2(29²)+3(29)+2(1)
1T1 in base 30 because 1(30²)+29(30)+1(1)=1771, and
NN in base 76 because 23(76)+23(1).