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The Chicago Bears score 18 points in a football game. In how many different ways can the Bears score these points? Points are scored as follows: a safety is 2 points, a field goal is 3 points, a touchdown is 6 points, and a point after touchdown (PAT) is 1 point (a PAT can't be scored unless a touchdown is scored first).

My Solution

From here on out, I abbreviated Touchdown as TD, Field Goal as FG, and point after touchdown as PAT. I created a systematic list, beginning with the maximum number of touchdowns scored and then the maximum amount of field goals with 2 touchdowns and so on until I was left with no further solutions. To make it easier for you to read, I just inputted my data into Excel and took a screenshot.

I am wondering if I accidently skipped over a possible way when making my list. Keep in mind, there are NO 2-point conversions after a TD is scored. I also assumed that you could miss a PAT, even though it is unlikely (but, so are some of these solutions below - 9 safeties, really?).

My systematic list:

enter image description here

I got a total of 14 ways.

David Mitra
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Joe
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  • Looks good. $$$$ – The Chaz 2.0 Feb 13 '12 at 23:27
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    Thank you for embedding my picture David. Mind sharing the syntax so I don't have to rely on you every time? :)

    Thanks for your quick reply, yet again Chaz. Anyone else get the same solution?

     – Joe Feb 13 '12 at 23:28
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    Joe, you should see "edited 3 mins ago" (with the correct time, of course) right below each question. Click for revisions/source – The Chaz 2.0 Feb 13 '12 at 23:30
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    You're welcome Joe. If you have enough rep, you can use one of the icons above the text input window when writing your questions to insert images. It's the sixth icon from the left. – David Mitra Feb 13 '12 at 23:31
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    Ah, lovely. Thanks for pointing it out. I see it now. Originally, I was "inspecting element" in Chrome to no avail however - just seeing the displayed code. – Joe Feb 13 '12 at 23:32
  • Yup - just got enough rep to do that. Thanks for your help - both of you. – Joe Feb 13 '12 at 23:34
  • It looks good and well-organized. That's a good thing for these problems. Breaking things down by starting with the max. number of touchdowns possible, then the max. number of field goals, etc.. – David Mitra Feb 13 '12 at 23:34
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    Great to hear. Yup, it definitely seems the route to go when it comes to most problems involving systematic lists. – Joe Feb 13 '12 at 23:35
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    If you are only interested in totals of each, looks good, nice and systematic listing. Another interpretation of the problem gives a much larger answer. Look for example at the case $2$ unconverted TD, $2$ FG. This could happen in various orders (TD, TD, FG, FG, or TD, FG, TD, FG, others). – André Nicolas Feb 13 '12 at 23:36
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    Makes sense, yeah. I'm not sure that those count as necessarily different ways, since it accounts for the same total amount of points. The question is ambiguous nonetheless. Good point of mention. – Joe Feb 13 '12 at 23:48
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    This is fine. But it's a bit annoying to have to make sure that the number of PAT is less than or equal to the number of touchdowns. It might have been easier to have columns be safeties, field goals, touchdowns-without-PAT, and touchdowns-with-PAT, worth 2, 3, 6, and 7 points respectively. Just something to keep in mind if you're faced with similar problems in the future. – Michael Lugo Feb 14 '12 at 01:22
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    Ah, valid point Michael. It did not seem too problematic when i was working through the problem (trying to figure out numbers that add up to x when y amount of TDs, FGs, etc.). However, I will note your strategy may be easier for readers to follow and for me to perform even - thanks. – Joe Feb 14 '12 at 04:47

1 Answers1

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A really simple and elegant solution is a polynomial expansion. You can consider each way a team could score (touchdowns with 2-pt conversion, touchdown with xp, touchdown, field goal, etc.) and the number of times it is possible for a team to have scored in those ways. In your example, a team could score at most 2 touchdowns with extra points, 6 field goals, or 9 safeties. This can be represented as:

$(1+x^7+x^{14})(1+x^6+x^{12}+x^{18})(1+x^3+x^6+x^9+...+x^{18})$$(1+x^2+x^4+x^6+...+x^{18}).$

The exponents denote the number of points scored, while the coefficients denote the number of possible ways to score that many points. We begin each polynomial with 1 because there is one way to score 0 points with each type of scoring play. This expansion yields 14x^18, showing us that there are exactly 14 combinations of scoring plays (excluding 2-pt conversions) that could result in 18 points for the Monsters of the Midway.

Harsh Kumar
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Birds'2019
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