916 Witch’s Hat

Today’s puzzle looks a little like a witch’s hat. Solving it could be bewitching. It contains a lot of perfect squares, and it’s only a level 2. Give it a try! Just make sure you don’t write a number greater than ten in the top row or the first column. Use all the numbers from one to ten in both places. There is only one solution.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

916 looks the same upside-down as it does right-side up, so it is a Strobogrammatic number.

It looks like someone put a spell on that number!

Not only that, in BASE 26, our 916 looks like 196. There is a whole lot of magic going on here!

30² + 4² = 916, so 916 is the hypotenuse of a Pythagorean triple:
240-884-916, which can be calculated from 2(30)(4), 30² – 4², 30² + 4².

  • 916 is a composite number.
  • Prime factorization: 916 = 2 × 2 × 229, which can be written 916 = 2² × 229
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 916 has exactly 6 factors.
  • Factors of 916: 1, 2, 4, 229, 458, 916
  • Factor pairs: 916 = 1 × 916, 2 × 458, or 4 × 229
  • Taking the factor pair with the largest square number factor, we get √916 = (√4)(√229) = 2√229 ≈ 30.2654919

 

 

Advertisements

48 and What’s wrong with this?

Here is a four piece puzzle arranged two different ways. There are 24 blue squares and 24 yellow squares. 24 + 24 = 48. Read on to see why the factors of 48 are significant to this puzzle.

Those triangles are some fabulous mathematical slight of hand.  Something is wrong with the puzzle, but what?

Something is fishy with these two triangle puzzles, too:

And with these two triangles made with six puzzle pieces!

If I had time, I could make an infinite number of these types of puzzles. Why do we get a solid triangle sometimes, but one with a hole in it the other times?

Believe it or not, the answer to the triangle puzzles has something to do with factoring.

Here’s that first puzzle again, but smaller:

The first triangle doesn’t have a hole it while the second one does because

Who would have thought that comparing fractions could turn into a type of magic trick?!!

Both triangles have 48 blue or yellow squares, but the second triangle wants you to think it should have 49.

(Psst….6/7 is also the slope of the green triangle and 7/8 is the slope of the red triangle. Since the two triangles don’t have identical slopes, the big “triangles” formed in the puzzles aren’t really triangles at all.)

I made all the puzzles in this post by comparing fractions. In each case, I simply used two consecutive composite numbers and then factored them and came up with the fractions I wanted to use. The numerators and the denominators of the fractions became the sides of the triangles.

Now, what do you think of this video of a never-ending chocolate bar?

Let me tell you a little bit about the number 48:

48 is a composite number. 48 = 1 x 48, 2 x 24, 3 x 16, 4 x 12, or 6 x 8. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Prime factorization: 48 = 2 x 2 x 2 x 2 x 3, which can also be written 48 = 2⁴ x 3.

Sometimes 48 is a clue in the FIND THE FACTORS puzzles. Even though it has many other factors, we use only 6 x 8 for the FIND THE FACTORS 1-10 puzzles and 6 x 8 or 4 x 12 for the FIND THE FACTORS 1-12 puzzles.

 

factor tree