1705 The Seat Numbers from Jimmy Fallon’s Twelve Days of Christmas Sweaters

Today’s Puzzle:

Jimmy Fallon’s Twelve Days of Christmas Sweaters tradition has become something I look forward to each December. The sweaters are one-of-a-kind masterpieces. I love when the sweaters are revealed. Jimmy reaches into a bright red Christmas stocking and randomly pulls out a number, the seat number of the winner of the sweater. Miraculously,  the winner of each sweater looks fabulous in it, no matter how big or small the winner is. I love this tradition, the sweaters, the winners modeling the sweaters, but I also love hearing the seat numbers. Each seat number has something special about it. (Just because it is a number!) By day three, I knew I wanted to blog about the numbers this year. The seat numbers were 295, 257, 314, 270, 419, 126, 256, 417, 433, 242, 232, and 120. I immediately knew something special about several of the numbers, but some of them I had to research. Can you figure out what is so special about each one?

Three of the seat numbers were primes. Which three?

One of those primes is both the fourth Fermat prime and the second-largest known Fermat prime. Which prime number is that?

Two of the numbers were palindromes (numbers that read the same forward and backward). Which two?

One of the seat numbers is equal to 1 × 2 × 3 × 4 × 5. Mathematicians write that as 5! Which seat number is equal to 5!?

One of the numbers is 10π rounded. Which one?

How Do Some of the Seat Numbers Shape Up?

Two of the numbers were decagonal numbers. Which two?

126 is not only a decagonal number, but it is also a pyramid formed by stacking the first six pentagonal numbers on top of each other.
1 + 5 + 12 + 22 + 35 + 51 = 126.

120 comes in THREE shapes.

One of the seat numbers is a star:

Something Special About Each Seat Number:

I’ll explain some of these reasons below.

Three of the Numbers Were the First Numbers to do Something Special:

242 is the smallest number whose square root can be simplified that is followed by three other numbers whose square root can also be simplified. Also, all four numbers have exactly six factors. Numbers with exactly six factors always have simplifiable square roots.

  • 242 = 2·11²; its six factors are 1, 2, 11, 22, 121, 242.
  • 243 = 3⁵; its six factors are 1, 3, 9, 27, 81, 243.
  • 244 = 2²·61; its six factors are 1, 2, 4, 61, 122, 244.
  • 245 = 7²·5; its six factors are 1, 5, 7, 35, 49, 245.

Square roots 242 - 245

417 is the smallest number that is the first of four consecutive integers that are divisible by a different number of primes.

419 is one less than 420, the smallest number divisible by 1, 2, 3, 4, 5, 6, and 7. As a consequence of that, 419 is the smallest number that leaves a remainder of 1, when it is divided by 2, a remainder of 2, when it is divided by 3, a remainder of 3, when it is divided by 4, a remainder of 4, when it is divided by 5, a remainder of 5, when it is divided by 6, and a remainder of 6, when it is divided by 7. This next graphic is a different way to make the same point.

What Do I Mean by Sum-Difference?

Two of the seat numbers have factor pairs that make sum-difference: the numbers in one of its factor pairs add up to a particular number and the numbers in a different factor pair subtract to the same number. Coincidentally, both of the seat numbers are related to 30, another number that makes sum-difference.

I’ve so enjoyed discovering what made all those seat numbers special, and I hope that you have enjoyed reading about them as well!

Since this is my 1705th post, I’ll write a little about that number, as well.

Factors of 1705:

  • 1705 is a composite number.
  • Prime factorization: 1705 = 5 × 11 × 31.
  • 1705 has no exponents greater than 1 in its prime factorization, so √1705 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 1705 has exactly 8 factors.
  • The factors of 1705 are outlined with their factor pair partners in the graphic below.

More About the Number 1705:

1705 is the hypotenuse of a Pythagorean triple:
1023 1364 1705, which is (3-4-5) times 341.

1703 A Wreath to Hang on Your Door

Today’s Puzzle:

A wreath is a lovely decoration to hang on your door at Christmastime. This one might have a few thorns in it, but if you are careful, they won’t bother you in the least. Just use logic to write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues create a multiplication table.

Factors of 1703:

  • 1703 is a composite number.
  • Prime factorization: 1703 = 13 × 131.
  • 1703 has no exponents greater than 1 in its prime factorization, so √1703 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 1703 has exactly 4 factors.
  • The factors of 1703 are outlined with their factor pair partners in the graphic below.

More About the Number 1703:

Did you notice a cool pattern in 1703’s prime factorization?
13·131 =1703.

1703 is the hypotenuse of a Pythagorean triple:
655-1572-1703, which is (5-12-13) times 131.

1703 is the difference of two squares in two different ways:
852² – 851² = 1703, and
72² – 59² = 1703.

1702 A Puzzle Idea from @mathequalslove Tweaked into a Subtraction Puzzle That Directs You to a Post from NebusResearch

Today’s Puzzle:

Joseph Nebus is nearly finished with all the posts in his Little 2021 Mathematics A to Z series. Every year he requests that his readers give him mathematical subjects to write about. At my suggestion, he recently wrote about subtraction, and how it is a subject that isn’t always as elementary as you might expect.  With a touch of humor, we learn that subtraction opens up whole new topics in mathematics.

I wanted to make a puzzle to commemorate his post. I gave it some thought and remembered a tweet from Sarah Carter @mathequalslove:

That puzzle originated from The Little Giant Encylopedia of Puzzles by the Diagram Group. I wondered how the puzzle would work if it were a subtraction puzzle instead of an addition puzzle, and here’s how I tweaked it:

 

There is only one solution. I hope you will try to find it! If you would like a hint, I’ll share one at the end of this post.

Factors of 1702:

  • 1702 is a composite number.
  • Prime factorization: 1702 = 2 × 23 × 37.
  • 1702 has no exponents greater than 1 in its prime factorization, so √1702 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 1702 has exactly 8 factors.
  • The factors of 1702 are outlined with their factor pair partners in the graphic below.

More About the Number 1702:

1702 is the hypotenuse of a Pythagorean triple:
552-1610-1702, which is (12-35-37) times 46.

1702² = 2896804, and
2197² = 4826809.
Do you notice what OEIS.org noticed about those two square numbers?

Puzzle Hint:

Here’s how I solved the puzzle: I let the rightmost box be x. Then using the values in the adjacent triangles and working from right to left, I wrote the values of the other boxes in terms of x.

x – 5 went in the box that is second to the right,
x – 5 + 2 = x – 3 went in the next box,
x – 3 + 5 = x + 2,
x + 2 – 6 = x – 4,
x – 4 + 5 = x + 1, and so on until I had assigned a value in terms of x for every box.

Think about it, and this hint should be enough for you to figure out where the numbers from 1 to 9 need to go.

1701 Is a Decagonal Number

Today’s Puzzle:

There is a pattern to the decagonal numbers. Can you figure out what it is?

Factors of 1701:

1701 is divisible by nine because 1 + 7 + 0 + 1 = 9.

  • 1701 is a composite number.
  • Prime factorization: 1701 = 3 × 3 × 3 × 3 × 3 × 7, which can be written 1701 = 3⁵ × 7.
  • 1701 has at least one exponent greater than 1 in its prime factorization so √1701 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1701 = (√81)(√21) = 9√21.
  • The exponents in the prime factorization are 5 and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1701 has exactly 12 factors.
  • The factors of 1701 are outlined with their factor pair partners in the graphic below.

More About the Number 1701:

1701 is the difference of two squares in SIX different ways.
851² – 850² = 1701,
285² – 282² = 1701,
125² – 118² = 1701,
99² – 90² = 1701,
51² – 30² = 1701, and
45² – 18² = 1701.

1701 is the 21st decagonal number because
21(4·21 – 3) =
21(84-3) =
21(81) = 1701.

There is decagonal number generating function:
x(7x+1)/(1-x)³ = x + 10x² + 27x³ + 52x⁴ + 85x⁵ + . . .

The 21st term of that function is 1701 x²¹.

 

1700 Time for a Horse Race!

Today’s Puzzle:

I’ve made a table of all the numbers from 1601 to 1700, their prime factorizations, and how many factors each of those numbers has.

Each number from 1601 to 1700 has 2, 3, 4, 6, 8, 10, 12, 16, 18, 24, 30 or 40 factors. Do more numbers have 2 factors, 3 factors, or a different number? Sure, you could use the table to count, but it will be more fun if we make a horse race out of it. What pony will you pick, 2, 3, 4, 6? Make your prediction of the winner, and then watch the horse race to see if you were right.

Here are all the horses lined up at the gate. Pick your pony!

And they’re off!

1601 to 1700 Horse Race

make science GIFs like this at MakeaGif
The first half of the race was VERY interesting to me. Three different horses held the lead, and the leaders in the race was even neck and neck some of the time. After you know the winner, watch the race again, following the winner from start to finish.

Factors of the Number 1700:

  • 1700 is a composite number.
  • Prime factorization: 1700 = 2 × 2 × 5 × 5 × 17, which can be written 1700 = 2² × 5² × 17.
  • 1700 has at least one exponent greater than 1 in its prime factorization so √1700 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1700 = (√100)(√17) = 10√17.
  • 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 1700 has exactly 18 factors.
  • The factors of 1700 are outlined with their factor pair partners in the graphic below.

More About the Number 1700:

1700 is the sum of two squares THREE different ways:
40² + 10² = 1700,
38² + 16² = 1700, and
32² + 26² = 1700.

Those sum of two squares mean 1700 is the hypotenuse of some Pythagorean triples, SEVEN to be exact:

  1. 260-1680-1700, which is 20 times (13-84-85),
  2. 348-1664-1700, which is 4 times (87-416-425), but it can also be calculated from 32² – 26², 2(32)(26), 32² + 26²
  3. 476-1632-1700, which is (7-24-25) times 68,
  4. 720-1540-1700, which is 20 times (36-77-85),
  5. 800-1500-1700, which is (8-15-17) times 100, but it can also be calculated from 2(40)(10), 40² – 10², 40² + 10²,
  6. 1020-1360-1700, which is (3-4-5) times 340,
  7. 1188-1216-1700, which is 4 times (297-304-425), but it can also be calculated from 38² – 16², 2(38)(16), 38² + 16².

As OEIS.org informs us, 1700 is a Catalan number. It is found in the C(13,4) position, as shown below. With the exception of the lone 1 in the 0th row of the triangle, every number in Catalan’s triangle is the sum of the number above it and the number to its left. For example, 1700 = 1260 + 440.

I hope you have enjoyed learning about the number 1700.

1695 If You Can Solve 3×3 and 5×5 Magic Squares, Then You Can Solve a 15×15 Magic Square!

Today’s Puzzle:

Completing a 15 × 15 magic square may seem daunting, but I assure you, if you can solve a 3 × 3 magic square and a 5 × 5 magic square, then you can complete a 15 × 15 magic square!

The 15 × 15 magic square below is made with twenty-five 3 × 3 magic squares. See the most famous 3 × 3 square in yellow? Do you see that the first 25 multiples of 9 are along the diagonals I’ve drawn? Do you understand the pattern that was used to make this magic square? Study it, then without looking at this one, can you make your own? Open this Excel file, 12 Factors 1683-1695, enable editing, and the sums of each row, column, and diagonal will automatically populate as you type in the numbers.

This next 15 × 15 magic square is made with nine 5 × 5 magic squares. The one in yellow is one of MANY possible 5 × 5 squares. Do you see the first nine multiples of 25 along the diagonals I’ve drawn? Once you understand this pattern, perhaps you would like to take a turn duplicating it. Try it yourself on that same Excel file, 12 Factors 1683-1695.

Factors of 1695:

  • 1695 is a composite number.
  • Prime factorization: 1695 = 3 × 5 × 113.
  • 1695 has no exponents greater than 1 in its prime factorization, so √1695 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 1695 has exactly 8 factors.
  • The factors of 1695 are outlined with their factor pair partners in the graphic below.

More About the Number 1695:

Since 1695 is the sum of three consecutive numbers, it is the magic sum of a particular 3 × 3 Magic Square. Those three consecutive numbers are in yellow in the square below.

Also, since 1695 is the sum of the five consecutive numbers shown in yellow below, it is the magic sum of this 5 × 5 magic square:

And lastly, 1695 is the sum of the 15 consecutive numbers shown in yellow below, so here is yet another 15 × 15 magic square with 1695 as the magic sum. This magic square uses the same pattern that works for all odd number magic squares. It is so satisfying to complete it yourself. Study this one and then give it a try! Open that same Excel file, 12 Factors 1683-1695, to make Excel to all the adding for you.

1695 is also the hypotenuse in FOUR Pythagorean triples:
225-1680-1695, which is 15 times (15-112-113),
828-1479-1695, which is 3 times (276-493-565)
1017-1356-1695, which is (3-4-5) times 339, and
1188-1209-1695, which is 3 times (396-403-565).

Did you notice that 565, 339, or 113 was a center number in every magic square in this post?

I hope you enjoy completing these magic squares on your own as you explore the number 1695.

1692 A Pilgrim’s Belt to Unbuckle

Today’s Puzzle:

The logic needed to unbuckle this Pilgrim’s belt puzzle has several interesting twists and turns in it. Even adults will find it a challenge. Guessing and checking will only frustrate you. Use logic to write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues work like a multiplication table.

Here’s the same puzzle if you want to print it using less ink.

Factors of 1692:

  • 1692 is a composite number.
  • Prime factorization: 1692 = 2 × 2 × 3 × 3 × 47, which can be written 1692 = 2² × 3² × 47.
  • 1692 has at least one exponent greater than 1 in its prime factorization so √1692 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1692 = (√36)(√47) = 6√47.
  • 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 1692 has exactly 18 factors.
  • The factors of 1692 are outlined with their factor pair partners in the graphic below.
  • 1568 is a composite number.

More About the Number 1692:

1692 is the difference of two squares in three different ways:
424² – 422² = 1692,
144² – 138² = 1692, and
56² – 38² = 1692.

The square of 1692 looks a little interesting:
1692² = 2862864.

Do These Records Refer to One Person, Two People, or More?

Some of my husband’s ancestors lived in Dévaványa, Hungary in the 1700s. I came across a death record for an individual who died in Dévaványa but was born in Gyoma, the place where two of my husband’s grandparents were born.

Dévaványa civil death record, number 323, 1899 October 24, Csáki Gergely, age 73 (born about 1827) in Gyoma, married to Karádi Eszter, his parents were the late Csáki István and the late Kis Rebeka.

My husband had a 4th great-grandmother named Kis Erzsébet who lived in Gyoma. Perhaps this Csáki Gergely would be related to her somehow? I decided to look for him in other records and found that he and his wife had one child born in Gyoma.

Gyoma Reformed Church christening record, page 877, Line 188, born 1859 August 6, christened 1859 August 8, a girl, Ester, born to Csáki Gergely and Karádi Ester.

That matches the information in Csáki Gergely’s death record. Great! Let me see if I can find his christening record. I’m looking for a Gergely who was born about 1827 and his parents were Csáki István and Kis Rebeka. I did find a Gergely christened on 12 September 1827 whose father was Csáki István, but his mother’s name was Fekete Rebeka. Also that Gergely died a half year later on 25 April 1828. The couple had another son, István, christened 23 Jan 1829, and one more they named Gergely christened on 12 January 1830. Could that second Gergely be the same as the one who died in 1899?

Suppose one of your relatives died and you reported the death to the authorities. Would you get his/her birth year and mother’s maiden name right? I’m not ready to declare that these two documents refer to the same person, but I’m also not going to eliminate the possibility either.

Some people in Hungary essentially had two surnames. Perhaps Rebeka sometimes listed her surname as Kis. I searched for István and Rebeka’s marriage record.

Gyoma Reformed Church marriage record, page 1009, last entry for 1826 November 15, the widower young Csáki István, age 25 (born about 1801) and Fekete György’s daughter Rebeka, age 29 (born about 1797).

There is no mention of Kis there, and young Csáki István was already a widower? I looked for Rebeka’s christening record and found she appears to have been christened on 13 Jun 1797, the daughter of Vanyai Fekete György and Szabó Ilona. (Vanyai means he was from Dévaványa.) Still no mention of the surname Kis. I found these parents’ marriage record.  Gyoma Reformed Church marriage record, 2nd entry for 29 January 1794, Dévaványa resident Fekete György, age 25 (born about 1769), and the late Bálint Szabó Márton’s daughter Ilona, age 21 (born about 1773). Again, no mention of Kis on either her father’s side or her mother’s side. It looks like I’m at a dead end.

All this means Csáki Gergely might not be related to my husband’s 4th great-grandmother after all, but my curiosity was still peaked about Csáki István being a widower at age 25. I found three records of interest.

  1. Gyoma Reformed Church marriage record page 1000, 16 November 1825, Csáki Gergely’s son István, age 24, wed B. Szabó János’s daughter, Mária, age 17 (born about 1808). This is one day less than one year before the widower Csáki István married Fekete Rebeka. And her name is B. Szabó Mária? Could she be related to Fekete Rebeka’s mother, Bálint Szabó Ilona?
  2. Gyoma Reformed Church christening record, page 73, 21 September 1826, the child christened, Imre, his parents were Csáki István and Kaptsos Mária.  Mária is the right given name, but Kaptsos and B. Szabó are not the same names.
  3. Gyoma Reformed Church death record, page78, 22 September 1826, Csáki Istvánné (Mrs. István Csáki), age 24 (born about 1802). This is just one day after the christening mentioned in the previous record. I looked up the cause of death, pokolvar, to see if it had anything to do with childbirth, but it did not. Pokol means hell, pokolvar means carbuncles. I do not know if the two words are related. Her death occurred less than two months before Fekete Rebeka’s 15 November 1826 wedding mentioned above. (It wasn’t unusual for a father of a new baby to look for a new mother soon after the death of his wife.)

Do these three records belong to one person, two people, or three? I looked for a possible christening record for Mária, and found several records for children born to János (Kaptsos or Bálint) and Ersebet Czegledi. They may be one couple or they may be two, but I arranged the information from the records in a chart here:

Note that Győri István was the godfather of three of the children. Balás Judit (not Ersébet) might have been the godmother all three times as well.

Discrepancies might have been from the participants, the informants, or even the priests. Most people were illiterate back then. I can imagine a person going to the priest when Mrs. István Csáki died. “How old was she?” asks the priest. “I don’t know,” the informant responds. “What were her parents’ names? I’ll determine her age from the christening book,” says the priest. He finds the birth of Mária in 1802, totally unaware that Mária 1802 died in 1807. “It looks like she was 24 when she died,” and that’s what he records in the book. Another possibility is that all the records are exactly correct and belong to different people so there are no discrepancies.

Although I was very interested in looking at these records, and I have an opinion of which records refer to which people, I’m not offering any of those opinions. But if you are related to any of these people, and find my research helpful, go ahead, form your own opinions, and make any connections you feel are appropriate.

 

1691 Jack 0’Lantern Time

Today’s Puzzle:

This smiling Jack O’Lantern actually has a few tricks up his sleeve. Don’t let yourself get fooled by any of them. Using logic, write the numbers from 1 to 12 in the first column as well as in the top row so that those numbers and the given clues can be a multiplication table.

Have a safe and happy Halloween!

Here’s the same puzzle with just the clues:

Factors of 1691:

  • 1691 is a composite number.
  • Prime factorization: 1691 = 19 × 89.
  • 1691 has no exponents greater than 1 in its prime factorization, so √1691 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 1691 has exactly 4 factors.
  • The factors of 1691 are outlined with their factor pair partners in the graphic below.

More About the Number 1691:

1691 is the hypotenuse of a Pythagorean triple:
741-1520-1691, which is 19 times (39-80-89).

1684 Triangular Candy Corn

Today’s Puzzle:

Candy corn is a triangular piece of Halloween candy. 1684 is a centered triangular number formed from the sum of the 32nd, the 33rd, and the 34th triangular numbers. Label the boxes next to the representations of each of those triangular numbers.

 

Factors of 1684:

  • 1684 is a composite number.
  • Prime factorization: 1684 = 2 × 2 × 421, which can be written 1684 = 2² × 421.
  • 1684 has at least one exponent greater than 1 in its prime factorization so √1684 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1684 = (√4)(√421) = 2√421.
  • 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 1684 has exactly 6 factors.
  • The factors of 1684 are outlined with their factor pair partners in the graphic below.

More About the Number 1684:

1684 is the sum of two squares:
30² + 28² = 1684.

1684 is the hypotenuse of a Pythagorean triple:
116-1680-1684, calculated from 30² – 28², 2(30)(28), 30² + 28².
It is also 4 times (29-420-421).

1680, 1681, 1682, 1683, and 1684 are the second smallest set of FIVE consecutive numbers whose square roots can be simplified.

1680 square roots

1684/2 = 842,  which is the third number in the smallest set of FIVE consecutive numbers whose square roots can be simplified.