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.