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%I #17 Feb 20 2022 20:25:29
%S 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,
%T 29,30,31,32,33,34,35,36,37,38,39,40
%N Possible sums of the final scores of completed American football games where both teams score.
%C One point only is an impossible score in American football. But with the safety 2 and the field goal 3, we can construct the set of integers greater than 1.
%C We can prove this by noting that if a score is even, we can build it with a series of safeties. Of course the other allowed scorings of 3, 6, and 1 after a touchdown could also be used. Now if a score is odd it is of the form 2k + 3. So for any odd number 2m + 1, we subtract 3 (or 1 field goal) from it to make it even and divide by 2 to get the number of safeties we need to add back to the field goal. Symbolically, let the odd number be 2m + 1. Then, (2m+1 - 3)/2 = m - 1 safeties are needed. Add this to 3 and you will have the number.
%C For example, say we want a score of 99. 99 = 2m + 1 and m = 49. So m - 1 = 48 safeties + 1 field goal = 99 points. From the first statement that 1 is an impossible score, it follows that there are infinitely many impossible scores in football, i.e., (1, 0), (1, 1), (1, 2), (1, 3), ...
%C Moreover, this restriction greatly influences the outcome of the ending digits for match up to the 100 square game cards that are floating around throughout the world before the Super Bowl game.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/NonRecursions.html">Non Recursions</a>
%e A safety or 2 points is the minimum possible score at the end of a game. So if two teams score one safety each, the sum of their scores will be 4. The next number, 5, is a safety for one team and a field goal for the other.
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, Feb 03 2006
%E Comments and example edited by _Jon E. Schoenfield_, Sep 20 2013
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