Month: February 2024

1789 An Easy Way to Solve This Subtraction Distraction Puzzle

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Today’s Puzzle:

To begin, I want to find a set of twelve consecutive numbers that will make the puzzle work. I want all of those numbers to be positive and relatively small. Thus, I would want to put a 12 in one of the last two boxes. Since the last triangle is -9, I would put the 12 in the last box. (If it were +9, I would put the 12 in the next to the last box.) Then I would do the following calculations based on the numbers in the triangles from right to left:

  • 12
  • 12 – 9 = 3
  • 3 + 4 = 7
  • 7 – 3 = 4
  • 4 + 4 = 8
  • 8 – 2 = 6
  • 6 – 1 = 5
  • 5 + 8 = 13
  • 13 – 4 = 9

And I would put the answers in the boxes from right to left:

The empty triangle makes me have to stop. Now I know I have to make some adjustments because one of the boxes has a 13 in it, but how much do I need to adjust each of those numbers? To answer that question, I will note what numbers from 1 to 12 are missing. I am missing 1, 2, 10, 11. The 13 I have means I can’t have the 1. I next access which of those missing numbers will yield -1. I note that 10 – 11 = -1, and write those numbers above the -1 triangle.

That leaves only the number 2 to place, but 10 + 4 ≠ 2, but 14. I place the 14 instead of the 2.

Now I have the twelve consecutive numbers from 3 to 14 in the boxes. If I subtract 2 from each of those twelve numbers, I will have all the numbers from 1 to 12. Also, it is easy to see that the number missing from the empty triangle is 2 whether I use 11 – 9 or 9 – 7.

Factors of 1789:

  • 1789 is a prime number.
  • Prime factorization: 1789 is prime.
  • 1789 has no exponents greater than 1 in its prime factorization, so √1789 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1789 has exactly 2 factors.
  • The factors of 1789 are outlined with their factor pair partners in the graphic below.

How do we know that 1789 is a prime number? If 1789 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1789. Since 1789 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1789 is a prime number.

More About the Number 1789:

1789 is the sum of two squares:
42² + 5² = 1789.

1789 is the hypotenuse of a Pythagorean triple:
420-1739-1789 calculated from 2(42)(5), 42² – 5², 42² + 5².

Here’s another way we know that 1789 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 42² + 5² = 1789 with 42 and 5 having no common prime factors, 1789 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1789. Since 1789 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1789 is a prime number.

1789 is in the twin prime 1787, 1789.

1789 is in the prime triplet 1783, 1787, 1789.

The first four multiples of 1789 are 1789, 3578, 5367, and 7156. Each of those multiples contains a 7. OEIS.org informs us that 1789 is the smallest number that can make that claim.

1789 is 414 in base 21 because 4(21²) + 1(21) + 4(1) = 1789.

 

1788 Subtraction Distraction

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Today’s Puzzle:

Factors of 1788:

A factor tree for 1788 isn’t very big because one of its prime factors has 3 digits.

  • 1788 is a composite number.
  • Prime factorization: 1788 = 2 × 2 × 3 × 149, which can be written 1788 = 2² × 3 × 149.
  • 1788 has at least one exponent greater than 1 in its prime factorization so √1788 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1788 = (√4)(√447) = 2√447.
  • 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 1788 has exactly 12 factors.
  • The factors of 1788 are outlined with their factor pair partners in the graphic below.

More About the Number 1788:

1788 is the hypotenuse of a Pythagorean triple:
612-1680-1788 which is 12 times (51-140-149).

1788 is the difference of two squares in two different ways:
448² – 446² = 1788, and
152² – 146² = 1788.

Two more square facts about 1788:
227² – 226² + 225² – 224² + 223² – 222² + 221² – 220² = 1788.

86² – 85² + 84² – 83² + 82² – 81² + 80² – 79² + 78² – 77² + 76² – 75² + 74² – 73² + 72² – 71² + 70² – 69² + 68² – 67² + 66² – 65² + 64² – 63² = 1788.

1787 The11-Digit Palindromes of Base 2

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Today’s Puzzle:

1787 is an 11-digit palindrome in base 2. I wondered how many 11-digit palindromes there are, what they are, and what numbers they represent in base ten. I decided to try to make you wonder about all that as well. Try it out yourself before you read how I solved this puzzle.

The only digits in base 2, are 0 and 1. The first digit of any number must be 1 or else the number will not have eleven digits. The last digit also must be one for the number to be a palindrome. In fact, all five last digits will be determined by the first five digits. Thus, we only need to find all possible combinations of 0 and 1 that can occur in the second through sixth positions. There are 2⁵ ways to write 0 and 1 in those 5 positions. That means we know right away that there are 32 different 11-digit palindromes in base 2. I opened Excel and wrote those 32 different 11-digit numbers beginning with 00000 and ending with 11111. I put a 1 in front of them and had Excel copy the appropriate numbers into the last 5 spots. That gave me all the 11-digit palindromes. Then I had Excel multiply the values in each cell with the powers of 2 that head up each column to give the base 10 representations. This chart was the final product.

Did you notice that the first base 10 number in the chart is the number just after 2¹º and the last number is the number right before 2¹¹?

Factors of 1787:

  • 1787 is a prime number.
  • Prime factorization: 1787 is prime.
  • 1787 has no exponents greater than 1 in its prime factorization, so √1787 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1787 has exactly 2 factors.
  • The factors of 1787 are outlined with their factor pair partners in the graphic below.

How do we know that 1787 is a prime number? If 1787 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1787. Since 1787 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1787 is a prime number.

More About the Number 1787:

1787 and 1789 are twin primes.
1783, 1787, and 1789 are a prime triplet.

1787 is a palindrome in some other bases, too!
It’s 919 in base 14 because 9(14²) + 1(14) + 9(1) = 1787,
595 in base 18 because 5(18²) + 9(18) + 5(1) = 1787, and
191 in base 38 because 1(38²) + 9(38) + 1(1) = 1787.

1786 is a Centered Triangular Number

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Today’s Puzzle:

A formula for the nth triangular number is n(n+1)/2. Centered triangular numbers are the sum of three consecutive triangular numbers. What would be a formula for finding centered triangular numbers? What value of n in your formula would produce the number 1786?

Factors of 1786:

  • 1786 is a composite number.
  • Prime factorization: 1786 = 2 × 19 × 47.
  • 1786 has no exponents greater than 1 in its prime factorization, so √1786 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 1786 has exactly 8 factors.
  • The factors of 1786 are outlined with their factor pair partners in the graphic below.

More About the Number 1786:

From OEIS.org we learn that 1786³ = 5,696,975,656. Notice that all those digits are 5 or greater.

1786 is 1G1 in base35,
because 1(35²) + 16(35) + 1(1) = 1786.

 

1785 A Pythagorean Triple Logic Puzzle

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Today’s Puzzle:

If you can print a copy of the puzzle from this Excel sheet, 10 Factors1773-1785, it will look like this:

Note: I have revised this puzzle since originally publishing it. I was horrified to discover that the original puzzle had two solutions. I apologize for any inconvenience I may have caused. This revised puzzle only has one solution.

Factors of 1785:

17 × 5 = 85, so 1785 is divisible by 17.

  • 1785 is a composite number.
  • Prime factorization: 1785 = 3 × 5 × 7 × 17.
  • 1785 has no exponents greater than 1 in its prime factorization, so √1785 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1785 has exactly 16 factors.
  • The factors of 1785 are outlined with their factor pair partners in the graphic below.

More About the Number 1785:

Did you notice that 3, 5, 7, and 357 are all factors of 1785?
Or that 35 and 51 make a factor pair, and 3, 5, and 1 are also factors?

1785 is the hypotenuse of FOUR Pythagorean triples:
273-1764-1785
756-1617-1785
840-1575-1785
1071-1428-1785

1785 is the difference of two squares in EIGHT different ways:
893² – 892² = 1785,
299² – 296² = 1785,
181² – 176² = 1785, and five more ways. Can you find them?

1785 is a Palindrome in a couple of bases:
It’s 123321 in base 4, because 1(1024) + 2(256) + 3(64) + 3(16) + 2(4) + 1(1) = 1785.
And it’s 3F3 in base 22, because 3(22²) + 15(22) + 3(1) = 1785.

1784 Another Hundred Simplifiable Square Roots

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Today’s Puzzle:

What percentage of natural numbers less than or equal to 1784 have simplifiable square roots?

Here is a chart of the 601st to the 700th simplifiable square roots:

You can figure out the percentage of numbers up to 1784 that have simplifiable square roots by calculating 700×100 ÷1784.

Was the percentage higher or lower than you expected?

The green areas on the chart are for consecutive numbers with simplifiable square roots. 1680-1684 are the smallest five consecutive numbers that can make that claim. Why can they? Because every one of their prime factorizations has an exponent greater than one in it.

1680 prime factorization

Factors of 1784:

  • 1784 is a composite number.
  • Prime factorization: 1784 = 2 × 2 × 2 × 167, which can be written 1784 = 2³ × 167.
  • 1784 has at least one exponent greater than 1 in its prime factorization so √1784 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1784 = (√4)(√446) = 2√446.
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1784 has exactly 8 factors.
  • The factors of 1784 are outlined with their factor pairs in the graphic below.

More About the Number 1784:

1784 is the difference of two squares in two different ways:
447² – 445² = 1784, and
225² – 221² = 1784.

1784 is a palindrome in two bases:
It’s 494 in base20 because 4(20²)+9(20)+4(1) = 1784, and
2C2 in base27 because 2(27²)+12(27)+2(1) = 1784.

1783 Another Mystery Puzzle

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Today’s Puzzle:

Can you find the factors for this mystery-level puzzle? There is only one solution.

Factors of 1783:

  • 1783 is a prime number.
  • Prime factorization: 1783 is prime.
  • 1783 has no exponents greater than 1 in its prime factorization, so √1783 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1783 has exactly 2 factors.
  • The factors of 1783 are outlined with their factor pair partners in the graphic below.

How do we know that 1783 is a prime number? If 1783 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1783. Since 1783 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1783 is a prime number.

More About the Number 1783:

1783 is palindrome 1L1 in base 33 because
1(33²) + 21(33) + 1(1) = 1783.

1783 is the sum of two consecutive numbers:
891 + 892 = 1783.

1783 is the difference of the squares of those same two consecutive numbers:
892² – 891² = 1783.
Of course, every other odd number can make a similar claim.

1782 Don’t Chop Down This Factor Tree!

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Today’s Puzzle:

Today is Monday, February 19. In the United States, we are celebrating Presidents’ Day, honoring most especially two important presidents who were born in February.

Exactly one week ago was February 12.

George Washington was born on February 11, 1731, Julian calendar.
Abraham Lincoln was born on February 12, 1809, Gregorian calendar.

The Julian calendar didn’t have leap days, so in 1752 a year and eleven days were added to Washington’s birthday to convert it to the Gregorian calendar.

Neither president will ever have his birthday on the third Monday of February when Presidents’ Day is observed. Too bad the second Monday of February wasn’t chosen instead. Then we could fudge a little and say that Presidents’ Day would be observed on one of their birthdays 2/7 of the time!

What days of the month are the earliest and the latest that a second Monday could be? 

When I was young I was told the story about George Washington chopping down a cherry tree. When he was confronted, he would not and could not tell a lie, and confessed his misdeed. As I got older, I learned that this was a fabricated story designed to teach children honesty of all things!

Nevertheless, some people celebrate Presidents’ Day by eating a cherry pie in remembrance of that story.

Factors of 1782:

This is my 1782nd post. Since it’s Presidents’ Day, I thought I would make a few factor trees for that number. You could think of the prime factors in red as cherries on the trees. Notice that all the prime factors are low-hanging fruit on these particular trees!

  • 1782 is a composite number.
  • Prime factorization: 1782 = 2 × 3 × 3 × 3 × 3 × 11, which can be written 1782 = 2 × 3⁴ × 11.
  • 1782 has at least one exponent greater than 1 in its prime factorization so √1782 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1782 = (√22)(√81) = 9√22. The exponents in the prime factorization are 1, 4, and 1. Adding one to each exponent and multiplying we get (1 + 1)(4 + 1)(1 + 1) = 2 × 5 × 2 = 20. Therefore 1782 has exactly 20 factors.
  • The factors of 1782 are outlined with their factor pair partners in the graphic below.

More About the Number 1782:

(5(27²) -3(27))/2 = 1782, so it is the 27th heptagonal number after 0.

Here’s another cool fact about 1782 from OEIS.org.

1781 A Mystery Puzzle for You to Solve

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Today’s Puzzle:

Is this mystery-level puzzle difficult or easy to solve? I’m not telling. You’ll have to try it for yourself to find out. As always, there is only one solution.

Factors of 1781:

1781 ÷ 4 leaves a remainder of 1, and 41² + 10² = 1781. Could 1781 be a prime number? It will be unless it has a prime number hypotenuse less than √1781 as a divisor. In other words, is it divisible by 5, 13, 17, 29, 37, or 41?

1781 obviously isn’t divisible by 5, and since it’s 41² + 10², it isn’t divisible by 41 either. That means we only have to check if it is divisible by 13, 17, 29, and 37.

So is it prime or composite?

  • 1781 is a composite number.
  • Prime factorization: 1781 = 13 × 137.
  • 1781 has no exponents greater than 1 in its prime factorization, so √1781 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 1781 has exactly 4 factors.
  • The factors of 1781 are outlined with their factor pair partners in the graphic below.

More About the Number 1781:

Not only does 41² + 10² = 1781, but
34² + 25² = 1781.

That 34² lets us know right away that 1781 is not divisible by 17, but any number that is the sum of two squares in more than one way is never a prime number.

1781 is the hypotenuse of FOUR Pythagorean triples:

531-1700-1781, calculated from 34² – 25², 2(34)(25), 34² + 25²,
685-1644-1781, which is (5-12-13) times 137,
820-1581-1781, calculated from 2(41)(10), 41² – 10², 41² + 10², and
1144-1365-1781, which is 13 times (88-105-137).

1781 is also the difference of two squares in two different ways:
891² – 890² = 1781, and
75² – 62² = 1781.

1780 Reflections of a Polygonal Bird

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Today’s Puzzle:

What ordered pairs were used to create this bird?

Its eye was formed from an equation of a circle:
(x – 7)²+ (y – 15)² = 3/4.

After creating the polygonal bird using ordered pairs and that circle equation, I wanted to do other things with the bird. Everything I did was like a puzzle for me to figure out.

Could I make it “fly”? Yes!

 

Could I make it reflect itself more than once over the y-axis and the x-axis? Yes! And I could make it do some sliding at the same time!

This next one was the toughest for me to do. I wanted the bird to be in motion rotating counter-clockwise around the origin. I was able to do it, but Desmos wouldn’t save the sliders exactly the way I wanted. I will need your help on this one. Click on this rotating bird link, then push play on slider a. About the time that slider goes to zero, push play on slider b. If you hit the sliders just right, it will look something like this GIF I made, but slower:

Rotating Polygonal Birds

make science GIFs like this at MakeaGif

 

Factors of 1780:

Perhaps our polygonal bird would like to fly to a tree. Here’s a factor tree for 1780 that it can take a rest on.

I knew that 1780 was divisible by 4 because its last two digits are divisible by 4.

  • 1780 is a composite number.
  • Prime factorization: 1780 = 2 × 2 × 5 × 89, which can be written 1780 = 2² × 5 × 89.
  • 1780 has at least one exponent greater than 1 in its prime factorization so √1780 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1780 = (√4)(√445) = 2√445.
  • 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 1780 has exactly 12 factors.
  • The factors of 1780 are outlined with their factor pair partners in the graphic below.

More About the Number 1780:

1780 is the difference of two squares in two different ways:
446² – 444² = 1780, and
94² – 84² = 1780.

1780 is the sum of two squares in two different ways:
42² + 4² = 1780, and
36² + 22² = 1780.

1780 is the hypotenuse of four Pythagorean triples:
336-1748-1780, calculated from 2(42)(4), 42² – 4², 42² + 4²,
780-1600-1780, which is 20 times (39-80-89)
812-1584-1780, calculated from 36² – 22², 2(36)(22), 36² + 22², and
1068-1424-1780, which is (3-4-5) times 356.

1780 is KK in base 88 because
20(88) + 20(1) = 20(89) = 1780.