Puzzle

1793 Are You Easily Distracted?

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Today’s Puzzle:

Don’t let the x’s in the puzzle distract you! This puzzle can actually be solved quite easily! Just follow the previous suggestion of putting a 12 in one of the last two boxes, fill in the rest of the boxes (don’t worry if any of the numbers are greater than 12), identify the largest number, and adjust all of the numbers so that that largest number becomes the new 12.

Factors of 1793:

Solve this problem: 1 – 7 + 9 – 3 =

If the answer is 0 or any other multiple of 11, then 1793 is a multiple of 11.

  • 1793 is a composite number.
  • Prime factorization: 1793 = 11 × 163.
  • 1793 has no exponents greater than 1 in its prime factorization, so √1793 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 1793 has exactly 4 factors.
  • The factors of 1793 are outlined with their factor pair partners in the graphic below.

More About the Number 1793:

1793 is a palindrome in base 32:
1O1 1(32²) + 24(32) + 1(1) = 1024 + 768 + 1 = 1793.
(O is the 15th letter of the alphabet, and 15 + 9 = 24, so O would be 24 if we all had 32 fingers.)

OEIS.org informs us that 1793 is a Fibonacci-inspired Pentanacci number.

 

Why Is 1792 a Friedman Number?

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Today’s Puzzle:

I’ve mentioned before that putting a 12 in one of the last two boxes will let you avoid negative numbers as you explore the relative relationship of the clues. For this puzzle, I would suggest that you put the 12 in the third from the last box. Why? Because the last triangle on the bottom has an 8 in it, and we will need to use either 12 – 8 = 4, and 4 – 2 = 2 for the last three boxes or 11 – 8 = 3, and 3 – 2 = 1.

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After you make your way to the empty triangle on the left of the puzzle, you will notice that you are missing the numbers 1 and 8. There isn’t any way to get a 5 by subtracting those two numbers, but if you realize that 13 – 5 = 8, you should know what adjustments you need to make to solve the puzzle.

Factors of 1792:

If the last digit of a number is 2 or 6, and the next-to-the-last digit is odd, then the whole number is divisible by 4.

If the last digit of a number is 0, 4, or 8, and the next-to-the-last digit is even, then the whole number is also divisible by 4.

1792 will allow us to apply those two divisibility observations several times as we make this factor tree:

  • 1792 is a composite number.
  • Prime factorization: 1792 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7, which can be written 1792 = 2⁸ × 7.
  • 1792 has at least one exponent greater than 1 in its prime factorization so √1792 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1792 = (√256)(√7) = 16√7.
  • The exponents in the prime factorization are 8 and 1. Adding one to each exponent and multiplying we get (8 + 1)(1 + 1) = 9 × 2 = 18. Therefore 1792 has exactly 18 factors.
  • The factors of 1792 are outlined with their factor pair partners in the graphic below.

More About the Number 1792:

1792 is a Friedman number because 7·2⁹⁻¹ = 1792.

Notice that the digits 1, 7, 9, and 2 and only those digits are used on both sides of the equal sign, and they are used the same number of times. 1792 is only the 26th Friedman number.

1792 is the difference of two squares in SEVEN different ways:
449² – 447² = 1792,
226² – 222² = 1792,
116² – 108² = 1792,
71² – 57² = 1792,
64² – 48² = 1792,
46² – 18² = 1792, and
44² – 12² = 1792.

1791 What a Distraction This Puzzle Is!

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Today’s Puzzle:

If you followed my advice from other posts and put a 12 in one of the last two boxes, you’ll be able to place five other numbers before hitting the roadblock that is the empty triangle.

Now we see that the highest known value is 15. The following numbers less than 15 are missing 1, 2, 3, 4, 5, 6, 8, 10, and 13. Since we have a 15, and our largest number can’t be greater than 12, let’s eliminate the smallest (15 – 12 = 3) three numbers from the list. We now have 4, 5, 6, 8, 10, and 13.

What can you do now? I suggest that you put an x such that -11 < x < 11 in the empty triangle and continue writing in values for the squares.

Regardless if x is a positive number or a negative number, the smallest number in a box will be either 7 or else 5 + x.

Since there isn’t a 6 + x or an 8, we know that one of those circled positions must be 1 and the other must be 2. If we assume the 7 should have been 2, we can lower the six numbers on the right of the puzzle by 5.

Then assuming that 5 + x must be 1 and filling in the puzzle we would get:

Uh oh! We can’t have two 9’s, 6’s, or 10’s, so those were NOT good assumptions.

I assure you that if switch the positions of the 1 and the 2, you will be able to complete the puzzle and place each number up to 12 in a box:

Factors of 1791:

  • 1791 is a composite number.
  • Prime factorization: 1791 = 3 × 3 × 199, which can be written 1791 = 3² × 199.
  • 1791 has at least one exponent greater than 1 in its prime factorization so √1791 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1791 = (√9)(√199) = 3√199.
  • 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 1791 has exactly 6 factors.
  • The factors of 1791 are outlined with their factor pair partners in the graphic below.

More About the Number 1791:

1791 is the difference of two squares in three different ways:
896² – 895² = 1791,
300² – 297² = 1791, and
104² – 95² = 1791.

1791 is A7A in base 13 because 10(13²) + 7(13) + 10(1) = 1791, and
636 in base 17 because 6(17²) + 3(17) + 6(1) = 1791.

1790 How Can You Solve This Subtraction Distraction?

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Today’s Puzzle:

How would I solve this puzzle? I would want to find a set of 12 consecutive numbers that were all positive and relatively small. Since there is an empty triangle near the right side of the puzzle, I would begin with the triangle on the bottom with a 4 in it and write 12 in a box above it. Then I would think and write 12 – 4 = 8 for the other box above the 4. My thinking would look like this:

  • 12
  • 12 – 4 = 8 (going to the right of the 12)
  • 12 – 6 = 6 (going to the left of the 12)
  • 6 + 7 = 13
  • 13 – 6 = 7
  • 7 – 2 = 5
  • 5 + 5 = 10
  • 10 + 5 = 15
  • 15 – 6 = 9

So that the puzzle looks like this:

I would note that I’m missing the following numbers: 1, 2, 3, 4, 11, and 14, and would figure out which of those missing numbers fit in the last three squares. Because I have a 15, I would note that 15 – 12 = 3 and would subtract 3 from each square to get numbers from 1 to 12. Figuring out what belongs in the empty triangle won’t be difficult either.

Factors of 1790:

  • 1790 is a composite number.
  • Prime factorization: 1790 = 2 × 5 × 179.
  • 1790 has no exponents greater than 1 in its prime factorization, so √1790 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 1790 has exactly 8 factors.
  • The factors of 1790 are outlined with their factor pair partners in the graphic below.

More About the Number 1790:

1790 is the hypotenuse of a Pythagorean triple:
1074-1432-1790 which is (3-4-5) times 358.

1789 is 414 in base 21, but
1790 is 4I4 in base 19 because 4(19²) + 18(19) + 4(1) = 1790.

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.

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.

 

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.