1711 is a Triangular Number So The Taxman Wants His Share

Today’s Puzzle:

Imagine this puzzle is made up of 58 envelopes, each containing the amount of money printed in bold type on its front. You can’t take any of the envelopes without the Taxman also taking a share. The Taxman will take EVERY available envelope that has a factor of the number you take on it. When you have taken all the cash you can, the Taxman gets ALL the leftover cash, and the game is over. You want the Taxman to get as little money as possible.

How much money is at stake? (58 × 59)/2 = 1711. That means 1711 is a triangular number, the sum of all the numbers from 1 to 58. Thus, the total amount of money you will be splitting with the Taxman is 1711.

To play this game, you can print the cards from this excel file: Taxman & 1537-1544. The factors of a number is printed in small type at the top of its card.

Here below I show the order I selected the cards when I played. For example, I took 53, and all the Taxman got was 1. I took 49, and the only card left for the Taxman to take was 7, and so forth. The last card I could take was 42, and the Taxman got 21, but since I couldn’t take anymore cards, the Taxman also got 31, 34, 37, 41, 43, and 47.

To win the game, you must get over half of the 1711 cash, but of course, you will want the Taxman to get much less than nearly half the money.

I didn’t want to make a long addition problem to find out how much I kept, so I came up with an easier strategy: I pushed the Taxman’s share to the side, and looked for ways to make 100 from my cards. (It is almost the 100th day of school, after all.) I found 9 ways to make 100, so I clearly kept more than half of the 1711.

I added that final 53 + 50 + 39 + 36 in my head by thinking that if I took 3 away from 53 and gave it to the 36, it would be the same as 50 + 50 + 39 + 39, a fairly easy sum.

Factors of 1711:

  • 1711 is a composite number.
  • Prime factorization: 1711 = 29 × 59.
  • 1711 has no exponents greater than 1 in its prime factorization, so √1711 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 1711 has exactly 4 factors.
  • The factors of 1711 are outlined with their factor pair partners in the graphic below.

1711 is a Shape-Shifting Number:

  • 1711 is the 58th Triangular Number because (58·59)/2 = 1711.
  • It is the 20th Centered Nonagonal Number because it is one more than nine times the 19th triangular number: 9(19·20)/2 + 1 =1711. AND
  • It is the 19th Centered Decagonal Number because it is one more than 10 times the 18th triangular number: 10(18·19)/2 +1 = 1711.

That last figure I’ve illustrated below with ten triangles circled around the center dot:

Now get this: Not only is 1711 the 19th Centered Decagonal Number, but similar-looking 17111 is the 59th Centered Decagonal Number! (A mere coincidence, but the 59th is even cooler because 59·29 = 1711.)

More About the Number 1711:

1711 is the hypotenuse of a Pythagorean triple:
1180-1239-1711, which is (20-21-29) times 59.

1711 is the difference of two squares in two different ways:
856² – 855² = 1711, and
44² – 15² = 1711.

1711 is a fun number to explore!

Find the Factor Pairs of 1710. Which One Sums to 181? Which One Subtracts to 181?

Today’s Puzzle:

1710 has TWELVE factor pairs. Two of those factor pairs are special. If you add up the factors in one of them, you will get 181, but get this, if you subtract the factors in the other one, you will also get 181. Find the two factor pairs that do that, and you will have essentially solved this puzzle:

This Sum-Difference puzzle would be helpful if you had to factor any of these trinomials:
45x² – 181x + 38,
95x² – 181x – 18,
90x² + 181x + 19,
38x² + 181x – 45,

The trinomials are in the form ax² + bx + c, where b = 181, and a·c = ±1710. Whether the 1710 is positive or negative makes all the difference in the world when factoring them!

Factors of 1710:

  • 1710 is a composite number.
  • Prime factorization: 1710 = 2 × 3 × 3 × 5 × 19, which can be written 1710 = 2 × 3² × 5 × 19.
  • 1710 has at least one exponent greater than 1 in its prime factorization so √1710 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1710 = (√9)(√190) = 3√190.
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1710 has exactly 24 factors.
  • The factors of 1710 are outlined with their factor pair partners in the graphic below.

The following chart lists 1710’s factor pairs with their sums and differences:

More About the Number 1710:

1710 is the hypotenuse of a Pythagorean triple:
1026-1368-1710, which is (3-4-5) times 342.

1709 Sometimes “Guess and Check” Is a Good Strategy

Today’s Puzzle:

Most of the puzzles I publish are logic puzzles, and I encourage you to find the logic of the puzzle and not guess and check.  However, guess and check is a legitimate strategy in mathematics, and it is a legitimate strategy to solve this particular puzzle.

Since one of the clues is -9, we know that the two boxes under it must be [1, 10], [2, 11], or [3, 12].

Suppose you assume it’s 1 – 10 = -9. If you fill out the rest of the boxes you would get:

You know that isn’t right because zero is not a number from 1 to 12. No problem. Simply add one to each of the numbers you wrote in, and the puzzle will be solved with only numbers from 1 to 12.

Suppose you assumed it’s 3 -12 = -9. The rest of the boxes would look like this:

Again, 13 is not included in the numbers from 1 to 12, but you can fix it by subtracting 1 from each of the numbers you wrote in. Easy Peasy.

Factors of 1709:

  • 1709 is a prime number.
  • Prime factorization: 1709 is prime.
  • 1709 has no exponents greater than 1 in its prime factorization, so √1709 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1709 has exactly 2 factors.
  • The factors of 1709 are outlined with their factor pair partners in the graphic below.

How do we know that 1709 is a prime number? If 1709 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1709. Since 1709 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1709 is a prime number.

More About the Number 1709:

1709 is the sum of two squares:
35² + 22² = 1709.

1709 is the hypotenuse of a Pythagorean triple:
741-1540-1709, calculated from 35² – 22², 2(35)(22), 35² + 22².

Here’s another way we know that 1709 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 22² = 1709 with 35 and 22 having no common prime factors, 1709 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1709. Since 1709 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1709 is a prime number.

1708 Happy Birthday, Jo Morgan!

Today’s Puzzle:

A few days ago I published a new kind of factoring puzzle. Jo Morgan of Resourceaholic.com, keeps an eye out for new mathematical resources on Twitter. She was one of the first to notice and like my puzzle. Because of her, lots of other people noticed the puzzle, too. Today is Jo’s birthday, and I decided to make a similar puzzle for her to enjoy. You might find it slightly more difficult than the earlier puzzle, but use logic from the beginning, and you will be able to fit in all the factors.

Factors of 1708:

  • 1708 is a composite number.
  • Prime factorization: 1708 = 2 × 2 × 7 × 61, which can be written 1708 = 2² × 7 × 61.
  • 1708 has at least one exponent greater than 1 in its prime factorization so √1708 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1708 = (√4)(√427) = 2√427.
  • 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 1708 has exactly 12 factors.
  • The factors of 1708 are outlined with their factor pair partners in the graphic below.

More About the Number 1708:

1708 is the hypotenuse of a Pythagorean triple:
308-1680-1708, which is 28 times (11-60-61).

1707 Subtraction Distraction

Today’s Puzzle:

Even though this Subtraction Distraction puzzle has more boxes than the one I published a couple of weeks ago, it is actually an easier puzzle. Can you write the numbers 1 to 12 in the boxes so that each triangle is its adjacent left box minus its adjacent right box?

Factors of 1707:

1 + 7 + 0 + 7 = 15, a multiple of 3, so 1707 is divisible by 3.

  • 1707 is a composite number.
  • Prime factorization: 1707 = 3 × 569.
  • 1707 has no exponents greater than 1 in its prime factorization, so √1707 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 1707 has exactly 4 factors.
  • The factors of 1707 are outlined with their factor pair partners in the graphic below.

More About the Number 1707:

1707 is the hypotenuse of a Pythagorean triple:
693-1560-1707, which is 3 times (231-520-569).

1706 Can You Make the Factors Fit?

Today’s Puzzle:

It’s 2022. Happy New Year! I wanted to make a puzzle that has 20 and 22 as clues and thought about what I could do.

I continue to be inspired by an old addition puzzle Sarah Carter @mathequalslove shared on Twitter:

I decided to tweak that puzzle into a multiplication puzzle. I ran into a problem, however. Having products in every triangle made the puzzle way too easy. How do I fix that? I removed some of the product clues. Can you use logic and factoring to know where each factor from 1 to 12 belongs? Can you determine the missing products? I hope you have lots of fun finding the puzzle’s only solution! And I hope you make the factors fit instead of having a fit trying!

Here’s something I haven’t told you before: I made lots of multiplication-table puzzles years before I started blogging. I wanted to give the puzzles a good name. At first, I called them “Turn the Tables on Multiplication” or “Turn the Tables” for short. I thought that title was clever but a little bit unwieldy. For a short time, I called the puzzles “Factor Fits.” It was a play on words because all the factors fit, but they might give you fits as you try to find them. I finally settled on “Find the Factors.” That title doubled as instructions for the puzzles. I still liked the name “Factor Fits,” and this puzzle lets me give new life to that name.

Factors of 1706:

  • 1706 is a composite number.
  • Prime factorization: 1706 = 2 × 853.
  • 1706 has no exponents greater than 1 in its prime factorization, so √1706 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 1706 has exactly 4 factors.
  • The factors of 1706 are outlined with their factor pair partners in the graphic below.

More About the Number 1706:

1706 is the sum of two squares:
41² + 5² = 1706.

1706 is the hypotenuse of a Pythagorean triple:
410-1656-1706, calculated from 2(41)(5), 41² – 5², 41² + 5².
It is also 2 times (205-828-853).

2022 Facts and Factors

Today’s Puzzle:

Solve these two multiplication problems and see how mirror-like they are!

Only 50 numbers less than 10000 can make a similar claim to fame:

Countdown to 2022:

Early in 2021, I found a countdown equation for 2022 that also involves its factors. A couple of weeks ago, I found another one. I can’t decide which one I like the best. Even though you can only focus on one equation at a time, the countdown will show both of them concurrently:

2022 Countdown Equations

make science GIFs like this at MakeaGif
Here is a still of the last frame:

 

Factors of 2022:

You might be asked to find the factors of 2022 several times in the coming year.
It’s an even number, so it’s divisible by 2.
It’s also easy to remember that it is divisible by 3 because 2 + 2 + 2 = 6, a multiple of 3.
This graphic may help you remember that it is divisible by 337:
  • 2022 is a composite number.
  • Prime factorization: 2022 = 2 × 3 × 337.
  • 2022 has no exponents greater than 1 in its prime factorization, so √2022 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 2022 has exactly 8 factors.
  • The factors of 2022 are outlined with their factor pair partners in the graphic below.

More About the Number 2022:

2022 is in the hypotenuse of a Pythagorean triple:
1050-1728-2022, which is 6 times (175-288-337).

It is also the short leg in three other Pythagorean triples:
2022-2696-3370, which is (3-4-5) times 674,
2022-340704-340710, which is 6 times (337-56784-56785), and
2022-1022120-1022122, which is 2 times (1011-511060-511061)

2022 uses only 2’s and 0’s in base 10 and in base 3:
2022₁₀ = 2202220₃.

2022 is the sum of positive consecutive numbers in three different ways:

2022 Magic Squares:

All of the above are the facts that I came up with. Lots of people on Twitter have found other facts about 2022. Be sure to check them out!

Tweets Celebrating 2022’s Mathematical Properties:

Here are some tweets about 2022 that I’ve seen on Twitter. I’ll add more as I see them. They are not listed in order of difficulty, but more or less, in the order that I saw them.

 


I’ll end with this thread that you’ll just have to click on to appreciate. It’s packed with tons of mathematical reasons 2022 will a fabulous year!

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.

1704 Christmas Factor Tree

Today’s Puzzle:

If you know the factors of the clues in this Christmas tree, and you use logic, it is possible to write each number from 1 to 12 in both the first column and the top row to make a multiplication table. It’s a level six puzzle, so it won’t be easy, even for adults, but can YOU do it?

Factors of 1704:

If you were expecting to see a factor tree for the number 1704, here is one of several possibilities:

  • 1704 is a composite number.
  • Prime factorization: 1704 = 2 × 2 × 2 × 3 × 71, which can be written 1704 = 2³ × 3 × 71.
  • 1704 has at least one exponent greater than 1 in its prime factorization so √1704 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1704 = (√4)(√426) = 2√426.
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1704 has exactly 16 factors.
  • The factors of 1704 are outlined with their factor pair partners in the graphic below.


More About the Number 1704:

1704 is the difference of two squares in FOUR different ways:
427² – 425² = 1704,
215² – 211² = 1704,
145² – 139² = 1704, and
77² – 65² = 1704.

Why was Six afraid of Seven? Because Seven ate Nine.
1704 is 789 in a different base:
1704₁₀ = 789₁₅ because 7(15²) + 8(15¹) + 9(15º) = 1704.

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.