pascal’s triangle

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).

What 1331 in Pascal’s Triangle Means

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1331 is the third row of Pascal’s triangle. What does that mean?
It means that 11³ = 1331,
and it means that 1 + 3 + 3 + 1 = 8 = 2³
Those third powers are not a coincidence.

It means that (x + y)³ = 1x³yº + 3x²y¹ + 3x¹y² + 1xºy³ or written more simply,
(x + y)³ = x³ + 3x²y¹ + 3x¹y² + y³

Likewise, (x – y)³ = 1x³yº – 3x²y¹ + 3x¹y² – 1xºy³ or simply
(x – y)³ = x³ – 3x²y¹ + 3x¹y² – y³

It also means that if you flip a coin three times, you’ll get
three heads and no tails 1 way: (HHH),
two head and one tail 3 ways: (HHT); (HTH); (THH),
one head and two tails 3 ways: (HTT); (THT); (TTH), and
no heads and three tails 1 way: (TTT).

That’s just some of what that third row being 1331 means but here are a few more facts about the number 1331.

  • 1331 is a composite number.
  • Prime factorization: 1331 = 11 × 11 × 11, which can be written 1331 = 11³
  • The exponent in the prime factorization is 3. Adding one, we get (3 + 1) = 4. Therefore 1331 has exactly 4 factors.
  • Factors of 1331: 1, 11, 121, 1331
  • Factor pairs: 1331 = 1 × 1331 or 11 × 121
  • Taking the factor pair with the largest square number factor, we get √1331 = (√121)(√11) = 11√11 ≈ 36.48287
  • 1331 is a perfect cube.

1331 is cool in some other bases, too:
It is 1000 in BASE 11, and
it’s 131 in BASE 35

Applying Divisibility Rules to 924

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Divisibility Rules and 924:

Let’s apply some basic divisibility rules to find some of the factors of 924:

  1. Like every other counting number, 924 is divisible by 1.
  2. Since 924 is even, it is divisible by 2.
  3. 9 + 2 + 4 = 15, a number divisible by 3, so 924 is divisible by 3.
  4. Its last two digits, 24, is divisible by 4, so 924 is divisible by 4.
  5. Its last digit isn’t 0 or 5, so 924 is NOT divisible by 5.
  6. 924 is even and divisible by 3, so it is also divisible by 6.
  7. Since 92-2(4) = 84, a number divisible by 7, we know that 924 is also divisible by 7.
  8. Since its last two digits are divisible by 8, and the third to the last digit, 9, is odd, 924 is NOT divisible 8.
  9. 9 + 2 + 4 = 15, a number not divisible by 9, so 924 is NOT divisible by 9.
  10. The last digit is not 0, so 924 is NOT divisible by 10.
  11. 9 – 2 + 4 = 11, so 924 is divisible by 11.

Thus, 1, 2, 3, 4, 6, 7, and 11 are all factors of 924.

Factor Cake for 924:

You can see its prime factors easily on the outside of its festive prime factor cake:

Factors of 924:

  • 924 is a composite number.
  • Prime factorization: 924 = 2 x 2 x 3 x 7 x 11, which can be written 924 = 2² x 3 x 7 x 11
  • 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) = 3 x 2 x 2 x 2 = 24. Therefore 924 has exactly 24 factors.
  • Factors of 924: 1, 2, 3, 4, 6, 7, 11, 12, 14, 21, 22, 28, 33, 42, 44, 66, 77, 84, 132, 154, 231, 308, 462, 924
  • Factor pairs: 924 = 1 x 924, 2 x 462, 3 x 308, 4 x 231, 6 x 154, 7 x 132, 11 x 84, 12 x 77, 14 x 66, 21 x 44, 22 x 42, or 28 x 33,
  • Taking the factor pair with the largest square number factor, we get √924 = (√4)(√231) = 2√231 ≈ 30.3973683.

Sum Difference Puzzle:

924 has twelve factor pairs. One of the factor pairs adds up to 65, and a different one subtracts to 65. If you can identify those factor pairs, then you can solve this puzzle!

More about the Number 924:

You may have seen one of its many possible factor trees contained in the first frame of this factor tree for 852,852 from my previous post:

924 is in the very center of the 12th row of Pascal’s triangle because 12!/(6!6!) = 924.

924 is the sum of consecutive prime numbers: 461 + 463 = 924

I like 924 written in some other bases:
770 BASE 11
336 BASE 17
220 BASE 21
SS BASE 32, S is 28
S0 BASE 33

924 has several sets of consecutive factors. Besides being divisible by the 1st, 2nd, and 3rd triangular numbers (1, 3, and 6), those consecutive factors mean the following:

  • 924 is divisible by the 6th triangular number, 21, which is 6(7)/2.
  • 924 is divisible by the 11th triangular number, 66, which is 11(12)/2.
  • 924 is divisible by the 21st triangular number, 231, which is 21(22)/2.