Author: ivasallay

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.

1778 Happy Valentine’s Day!

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Today’s Puzzle:

I ♥ U. Here’s a Valentine’s Day puzzle for you to enjoy. It might be a little tricky so remember to use logic to find all the factors! There are some other mathy Valentine’s Day activities at the end of the post.

Factors of 1778:

 

  • 1778 is a composite number.
  • Prime factorization: 1778 = 2 × 7 × 127.
  • 1778 has no exponents greater than 1 in its prime factorization, so √1778 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 1778 has exactly 8 factors.
  • The factors of 1778 are outlined with their factor pair partners in the graphic below.

More About the Number 1778:

1778 is the sum of four consecutive numbers:
443 + 444 + 445 + 446 = 1778.

1778 is the sum of seven consecutive numbers:
251 + 252 + 253 + 254 + 255 + 256 + 257 = 1778.

1778 is not the difference of two squares, but it is this:
446² – 445² + 444² – 443² =  1778.

1778 is palindrome, A6A in base13, because
10(13²) + 6(13) + 10(1) = 1778.

Other Mathy Valentine’s Day Activities:

 

1777 A Different Heart

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Today’s Puzzle:

Every year I make some heart-shaped puzzles, but this heart is different: I haven’t used this design before. Can it win you over? Some of the clues are tricky, so make sure you use logic to find the one and only solution.

Factors of 1777:

  • 1777 is a prime number.
  • Prime factorization: 1777 is prime.
  • 1777 has no exponents greater than 1 in its prime factorization, so √1777 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1777 has exactly 2 factors.
  • The factors of 1777 are outlined with their factor pair partners in the graphic below.

How do we know that 1777 is a prime number? If 1777 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1777. Since 1777 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1777 is a prime number.

More About the Number 1777:

1777 is the sum of two squares:
39² + 16² = 1777.

1777 is the hypotenuse of a primitive Pythagorean triple:
1248-1265-1777, calculated from 2(39)(16), 39² – 16², 39² + 16²

Here’s another way we know that 1777 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 39² + 16² = 1777 with 39 and 16 having no common prime factors, 1777 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1777. Since 1777 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1777 is a prime number.

1777 looks interesting in some other bases:
It’s 12121 in base 6 because 1(6⁴) + 2(6³) + 1(6²) + 2(6¹) + 1(6º) = 1777,
2L2 in base 25, because 2(25²) + 21(25) + 2(1) = 1777, and
1B1in base 37, because 1(37²) + 11(37) + 1(1) = 1777.

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.

1775 and Cupid’s Arrow

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Today’s Puzzle:

Will Cupid’s Arrow hit you right in your heart this year? Who knows? Solving this puzzle might help! It’s a level 3 puzzle so begin with the clues in the top row, then work your way down the puzzle row by row until you have found all the factors.

Factors of 1775:

  • 1775 is a composite number.
  • Prime factorization: 1775 = 5 × 5 × 71, which can be written 1775 = 5² × 71.
  • 1775 has at least one exponent greater than 1 in its prime factorization so √1775 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1775 = (√25)(√71) = 5√71.
  • 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 1775 has exactly 6 factors.
  • The factors of 1775 are outlined with their factor pair partners in the graphic below.

More About the Number 1775:

1775 is the difference of two squares in three different ways:
888² – 887² = 1775,
180² – 175² = 1775, and
48² – 23² = 1775.

1775 is the hypotenuse of two Pythagorean triples:
497-1704-1775, calculated from (7-24-25) times 71, and
1065 1420 1775, calculated from (3-4-5) times 355.

From OEIS.org, we learn that 1775 is one of the numbers in this Fibonacci-like series:
1, 7, 8, 15, 23, 38, 61, 99, 160, 259, 419, 678, 1097, 1775, . . .
Did you notice that 1+7=8, 7+8=15, and so forth? That’s why it’s called a Fibonacci-like series.

1775 is the repdigit PP in base 70. P is the 25th number in base 70. Thus,
25(70) + 25(1) = 25(71) = 1775.

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.

1773 You Will L♥ve This Multiplication Table Puzzle!

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Today’s Puzzle:

It’s almost Valentine’s Day! Enjoy this heart-shaped multiplication table puzzle! You only need to know one set of ten math facts to complete this puzzle, but which set is it? The two’s? the three’s? the four’s? or something different? You CAN figure it out, so give it a try! There is only one solution.

Factors of 1773:

Does 1+7+7+3 = a number divisible by 3? I’ve played enough cribbage to know instantly that 1+7+7=15. Add the remaining 3 to the 15, and we get 18, a number divisible by both 3 and 9, so 1773 is divisible by both 3 and 9.

  • 1773 is a composite number.
  • Prime factorization: 1773 = 3 × 3 × 197, which can be written 1773 = 3² × 197.
  • 1773 has at least one exponent greater than 1 in its prime factorization so √1773 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1773 = (√9)(√197) = 3√197.
  • 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 1773 has exactly 6 factors.
  • The factors of 1773 are outlined with their factor pair partners in the graphic below.

More About the Number 1773:

1773 is the sum of two squares:
42² + 3² =1773.

1773 is the hypotenuse of one Pythagorean triple:
252-1755-1773, calculated from 2(42)(3), 42² – 3², 42² + 3².
It is also 9(28-195-197).

1773 is palindrome 909 in base 14 because
9(14²) + 0(14) + 9(1) = 1773.

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

make science GIFs like this at MakeaGif
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).