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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.

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1  
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
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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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