1694 Football Game Day

Today’s Puzzle:

Today all over the United States family and friends will gather to watch or play a game of football. If you would like to change things up a little, here’s a game ball for you to practice multiplication and division facts. Just write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues form a multiplication table. Some clues might be tricky, but enough of them aren’t that I am confident you can score with this football!

Here’s the same puzzle without any distracting color:

Factors of 1694:

  • 1694 is a composite number.
  • Prime factorization: 1694 = 2 × 7 × 11 × 11, which can be written 1694 = 2 × 7 × 11².
  • 1694 has at least one exponent greater than 1 in its prime factorization so √1694 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1694 = (√121)(√14) = 11√14.
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1694 has exactly 12 factors.
  • The factors of 1694 are outlined with their factor pair partners in the graphic below.

More About the Number 1694:

From OEIS.org we learn that 1694³ = 4,861,163,384, a number that uses each of the digits 1, 3, 4, 5, and 8 exactly twice.

1665 Why I Am Rooting for England over Denmark Today

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.

1171 The Best Team in the Best Conference

The NCAA college football season has not had a single game, yet you can find out which team is in first place through twenty-fifth place now or anytime during the season by looking here. How are these football standings determined? By FIFTEEN people voting. Sure, it’s only one of several polls, but the four teams who play for the national championship are determined by a computer that uses polls like that one. Can you believe that there are people who find that rather unsatisfying? Your team could finish the season with the exact same record as one of those four teams but not be allowed to compete for the championship.

What do college football teams have to play for then? Almost every team is in a conference. They can play hoping to win their conference. Those teams who have a winning record can also be selected to play in one of 38 bowl games in December or early January. Winning a bowl game allows a team to finish the season with a win and is an honor to the school. Other than that, 35 of those bowl games mean absolutely nothing.

Perhaps this is a bit simplistic, but why can’t each conference send their best teams to play in bowl games against teams from a different conference. The conference that wins the most bowl games would be deemed the best conference.  The team that won that conference’s championship would be the best team in the best conference and the national champion. Every bowl game would then be important. Each eligible team would still only have to play one bowl game. More people would watch EVERY bowl game which would cause them all to make more money. The sports stations would also make more money as they keep their viewers updated with the win/loss records for every conference week after week.

Of all the things that are happening in the world today, this issue is far from being the most important, but thinking about it, like sports or this football-shaped mystery level puzzle, is a nice diversion.

Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Here’s the same puzzle but without all the color.

Now I’ll write a few things about the number 1171:

  • 1171 is a prime number.
  • Prime factorization: 1171 is prime.
  • The exponent of prime number 1171 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1171 has exactly 2 factors.
  • Factors of 1171: 1, 1171
  • Factor pairs: 1171 = 1 × 1171
  • 1171 has no square factors that allow its square root to be simplified. √1171 ≈ 34.21988

How do we know that 1171 is a prime number? If 1171 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1171 ≈ 34.2. Since 1171 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1171 is a prime number.

1171 is the sum of seven consecutive prime numbers:
151 + 157 + 163 + 167 + 173 + 179 + 181 = 1171

1171 is a palindrome in three bases:
It’s 14141 in BASE 5 because 5⁴ + 4(5³) + 5² + 4(5) + 1 = 1171,
1J1 in BASE 26 (J is 19 base 10) because 26² + 19(26) + 1 = 1171,
and 191 in BASE 30 because 30² + 9(30) + 1 = 1171