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.