### Today’s Puzzle:

England is playing Denmark in the Euro 2020 Semi-Finals today. Why am I rooting for England when I don’t really follow soccer, as we call it in America? It ISN’T because 5 of my great-grandparents claim English ancestry, and just only one of my great-grandparents was born in Denmark. England wins 5 to 1. No, that isn’t the reason why I’m excited today and created this soccer/football puzzle by freehand in paint:

How many pentagon and hexagon transformations are there on the game ball?

The reason I am rooting for England is that an ingenious mathematics teacher, AC@eymaths, created and shared an exciting and even better transformation puzzle on Twitter: pdf of a transformation puzzle worksheet.

The transformation looked like a wonderful idea, still, I wasn’t sure what all the fuss was about or the meaning of “Miss, is it coming home?” The next day I asked about it:

I googled and found this music video from Three Lions.

I love the enthusiasm shown even while expressing these lyrics:

“Everyone seems to know the score, they’ve seen it all before

They just know, they’re so sure

That England’s gonna throw it away, gonna blow it away

But I know they can play.”

I know that exact feeling! I’ve had high hopes for a team that only disappointed me. I love how everyone in England is in the moment and feeling enthusiastic no matter what! I have watched the music video over and over again. It’s also wonderful that so many mathematics teachers at several different levels are embracing their students’ excitement:

### Factors of 1665:

Since this is my 1665th post, I’ll share factoring information about the number 1665.

Obviously, 1665 ends with a 5, so it is also divisible by 5.

6, 6, and 1 + 5 use up all the digits and give us three 6’s (three of the same multiple of 3), so 1665 is divisible by 9 and, of course, by 3.

The prime factors of 1665 work together to give us several repdigits as factors, too: 111, 333, and 555.

- 1665 is a composite number.
- Prime factorization: 1665 = 3 × 3 × 5 × 37, which can be written 1665 = 3² × 5 × 37.
- 1665 has at least one exponent greater than 1 in its prime factorization so √1665 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1665 = (√9)(√185) = 3√185.
- 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 1665 has exactly 12 factors.
- The factors of 1665 are outlined with their factor pair partners in the graphic below.

### More About the Number 1665:

1665 is the sum of two squares in TWO different ways:

39² + 12² = 1665, and

33² + 24² = 1665.

1665 is the hypotenuse of FOUR Pythagorean triples:

513-1584-1665, calculated from 33² – 24², 2(33)(24), 33² + 24²,

but is also **9** times (57-176-**185**),

540-1575-1665, which is (12-35-**37**) times **45**.

936-1377-1665, calculated from 2(39)(12), 39² – 12², 39² + 12²,

but is also **9** times (104-153-**185**), and

999-1332-1665, which is (3-4-**5**) times **333**.