tetrahedral

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).

1140 is the 18th Tetrahedral Number

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1140 is the 18th tetrahedral number because it satisfies this formula:
(18)(18+1)(18+2)/6 = 1140

It is the 18th tetrahedral number because it is the sum of the first 18 triangular numbers:

Since 18 is an even number, 1140 is the sum of the first 9 EVEN squares.

If the 1140 tiny squares in that graphic were cubes, they could be stacked into a tower with either a triangular base or a square base. Then we would see the beauty of this tetrahedral number.

We can see the number 1140 as well as ALL the previous tetrahedral numbers on this portion of Pascal’s Triangle. (They are the green squares.):

1140 has its place as the 3rd number (as well as the 17th number) on the 20th row of Pascal’s triangle because of this next formula:

Here are some other facts about the number 1140:

  • 1140 is a composite number.
  • Prime factorization: 1140 = 2 × 2 × 3 × 5 × 19, which can be written 1140 = 2² × 3 × 5 × 19
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1140 has exactly 24 factors.
  • Factors of 1140: 1, 2, 3, 4, 5, 6, 10, 12, 15, 19, 20, 30, 38, 57, 60, 76, 95, 114, 190, 228, 285, 380, 570, 1140
  • Factor pairs: 1140 = 1 × 1140, 2 × 570, 3 × 380, 4 × 285, 5 × 228, 6 × 190, 10 × 114, 12 × 95, 15 × 76, 19 × 60, 20 × 57, or 30 × 38,
  • Taking the factor pair with the largest square number factor, we get √1140 = (√4)(√285) = 2√285 ≈ 33.76389

Here are some factor trees that use 11 of 1140’s factor pairs:

1140 is the sum of consecutive prime numbers two different ways:
179 + 181 + 191 + 193 + 197 + 199 = 1140,
569 + 571 = 1140

1140 is the hypotenuse of a Pythagorean triple:
684-912-1140 which is (3-4-5) times 228

1140 looks interesting when it is written in a couple other bases:
It’s palindrome 474 in BASE 16 because 4(16²) + 7(16) + 4(1) = 1140,
and it’s 330 in BASE 19 because 3(19²) + 3(19) = 3(19² + 19) = 3(19)(20) = 1140

969 is the 17th Tetrahedral Number and the 17th Nonagonal Number

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A tetrahedron is a pyramid whose base and sides are all triangles.

The nth tetrahedral number is the sum of the first n triangular numbers. So if you made a pyramid of the first n triangular numbers, you would get the nth triangular pyramidal number, also known as the nth tetrahedral number.

969 is the 17th tetrahedral number.

That image might look a little like a Christmas tree lot where you could select a tree in several different sizes. If we had tiny cubes instead of squares, we could stack them on top of each other to make a tetrahedron. That is the visual reason why 969 is a tetrahedron.

Look at the graphic below of a portion of Pascal’s triangle. You can easily see the first 19 counting numbers. The first 18 triangular numbers are highlighted in red, and the first 17 tetrahedral numbers are highlighted in green. The 16th tetrahedral number, 816, plus the 17th triangular number, 153, equals 969.

Because of its spot on Pascal’s triangle, I know that (17·18·19)/(1·2·3) = 969. That is the algebraic reason 969 is a tetrahedral number.

969 is also the 17th nonagonal number because 17(7·17 – 5)/2 = 969. I am not going to try to illustrate a 9-sided figure, but I’m sure it would be a cool image if I could.

All of this means that 969 is the 17th tetrahedral number AND the 17th nonagonal number. 1 is the smallest number to be both a tetrahedral number and a nonagonal number. 969 is the next smallest number to be both. Amazingly, it is the 17th of both, too!

969 obviously is a palindrome in base 10.

In base 20, it is 289. I find that quite curious because 17² = 289, and 17 is a factor of 969. Why would we write this number as 289 in base 20? Because 2(20²) + 8(20) + 9(1) = 969

Because 17 is one of its factors, 969 is the hypotenuse of a Pythagorean triple:
456-855-969 which is 57 times (8-15-17)

  • 969 is a composite number.
  • Prime factorization: 969 = 3 × 17 × 19
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 969 has exactly 8 factors.
  • Factors of 969: 1, 3, 17, 19, 51, 57, 323, 969
  • Factor pairs: 969 = 1 × 969, 3 × 323, 17 × 57, or 19 × 51
  • 969 has no square factors that allow its square root to be simplified. √969 ≈ 31.12876