Monday, January 22, 2007

Football puzzle

I'm still tired and cranky from last night's game. So here's a football puzzle for you. How many different ways can you score 20 points in a football game?

Assume the following possibilities for scoring:

  • If the other team doesn't show up, the game is forfeit, and you score 1 point (and the game is over.)
  • On defense, if you tackle the other team's quarterback in their endzone, you get 2 points for a "safety".
  • On offense, you can kick the ball over the defense's goalposts for a 3 point field goal.
  • On offense, you can carry the ball across the defense's goal line for a 6 point touchdown.
  • On offense, immediately after a touchdown, you can kick the ball through the defense's goalposts for an additional 1 point ("point after touchdown"). We can combine this with the touchdown and write it as a (6+1) or 7 point score.
  • On offense, immediately after a touchdown, you can make another play to carry the ball across the defense's goal line for an additional 2 points ("two point conversion"). We can combine this with the touchdown and write it as a (6+2) or 8 point score.

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  1. lots of ways.

    3 TDs, 2 extra points and a missed extra point.
    10 safeties (which will never happen...)
    6 FGs and a safety


    Mike are you a Patriots fan? I live in the Boston area myself, although I'm an Eagles fan.

  2. Yes, I'm a Pats fan. Which is why I'm so cranky today. :-( I think the Colts made just one more play than the Pats did and good for them (sort of... It's hard to be gracious when I'm upset, but I'm trying).

  3. You could get
    2 TDs with the 2 pt conversions and 2 safeties.

    And by the way, GO BEARS!!!!

  4. And, for the record, i came up with 23 different scenarios to make 20 points. I will not list them here, unless requested.


  5. Mike....what was crazy about that game was the penalties. They killed the Pats. 12 men in the huddle? Offensive pass interference? They had 4 penalties that came at critical times. Without 2 or 3 of them, Pats win in a blowout.

  6. 1. 2222222222
    2. 222222233
    3. 22223333
    4. 2333333
    5. 6662
    6. 662222
    7. 66332
    8. 62222222
    9. 6332222

    I agree

  7. Let's build our way up to 20 points (btw, I'm going to make it a little bit harder and say the order the team scores in counts):

    There is only one way to get 0 points;

    There is one way to get 1 point.

    There is one way to get 2 points

    There is one way to get 3 points;

    There is one way to get 4 points (two safeties).

    When we get to 5 points, we see that this can in two ways, by a safety and a field goal in either order.

    For 6 points, we could get 3 safeties, or two field goals, or a touchdown.

    We can get 7 points by 7, or 2+2+3 or 2+3+2 or 3+2+2.

    We can get 8 points by 2+2+2+2, or (2+3+3, 3+2+3, 3+3+2), or ( 6+2, 2+6 ) or 8.

    Let's get formulaic from this point on:S(N) = S(N-2) + S(N-3) + S(N-6) + S(N-7) + S(N-8)

    That is, to get a total score of N, your last score was a 2, 3, 6, 6+1 or 6+2. And the number of ways of getting a score of N is the sum of the number of ways of getting a score that is N-2, N-3, N-6, N-7 or N-8.

    Our table begins like this:
    N S(N)
    0 1
    1 1
    2 1
    3 1
    4 1
    5 2
    6 3
    7 4
    8 7

    Now we can compute new values automatically:
    N S(N)
    8 7
    9 10
    10 14
    11 21
    12 30
    13 44
    14 65
    15 95
    16 140
    17 205
    18 300
    19 440
    20 644
    21 944
    22 1384
    23 2028
    24 2973
    25 4357
    26 6385
    27 9358
    28 13714
    29 20099
    30 29457

    Note that 20 points can be gotten in 644 ways!

    As for the Pats loss, it's a shame, but that's the way it goes. I believe mistakes were made and bad calls were made (or not made) by the refs. But that statement applies to both teams, so that in the end, hurray for Indy. I hope the Bears kick their butts! ;-)

  8. I don't think your formula is correct, because you cannot score just one point by itself, it has to be combined with a TD.

  9. Mike, does it really matter in which order you score the points? I would not think so, so a field goal scored before a touch-down is the same as a field goal after a touch down. Both equals 9 points.

  10. Sean, the formula takes care of that. The only time you can score 1 point by itself is if the other team forfeits.

    Mr Don, I agree it doesn't matter. I was just trying to make my answer to cover all possibilities.

  11. I'm not big on football but I would have to say there's several different ways to score 20 points but no one said it would be easy.


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