%I A114143
%S A114143 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 A114143 29,30,31,32,33,34,35,36,37,38,39,40
%N A114143 The possible sums of the final scores of completed Chicago Bears football 
               games where both teams score.
%C A114143 1 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. 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 scorings 
               allowed 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 needed. Add this to 3 and you will have the number. 
               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 there is an infinite number 
               of impossible scores in football. Ie., 1,0 1,1 1,2 1,3 ... Moreover, 
               this restriction greatly influences the outcome of the ending digits 
               for match up to the 100 square game cards rhat are floating around 
               throughout the World before the Super Bowl game.
%H A114143 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               NonRecursions.html">Non Recursions</a>
%e A114143 A safety or 2 points is the minimum possible score at the end of a game. 
               So if
%e A114143 two teams score one safety each, the sum of their scores will be 4. The 
               next
%e A114143 number 5 is a safety for one team and a field goal for the other.
%Y A114143 Sequence in context: A143836 A089166 A030543 this_sequence A020705 A081313 
               A081333
%Y A114143 Adjacent sequences: A114140 A114141 A114142 this_sequence A114144 A114145 
               A114146
%K A114143 easy,nonn
%O A114143 0,1
%A A114143 Cino Hilliard (hillcino368(AT)gmail.com), Feb 03 2006