The Next Superbowl

Copyright 2002 by Paul Niquette.  All rights reserved.

Superbowl Sunday is the perfect day to drive your car on the freeway or ride your bike in the city, to push a stroller in the park or take a jog on the beach, to wait on line for the chairlift or practice aerobatics in the sky, to peruse an exhibit in the museum or shop for bargains at the swap meet.  Unless... 
...you happen to share a partisan interest in the athletic contest, which will be viewed by a worldwide audience that would fill a thousand stadiums -- or unless you otherwise happen to have a stake in the outcome.

For the next Superbowl, you are offered an opportunity to join a football pool.  The winner will be determined by the least significant digits of the final score and will receive $100.  You are invited to purchase as many squares in the following matrix as you please for $1 each.
 


0
1
2
3
4
5
6
7
8
9
0
                   
1
                   
2
                   
3
                   
4
                   
5
                   
6
                   
7
                   
8
                   
9
                   

By the way, the team in blue is favored to win.
Good luck.



 
Which squares will you purchase?
     


 
Football Pool
Sports pools generally allow participants to pick squares in a payoff matrix like the one in this puzzle.  A pool is quite distinguished from other games of chance that feature participant choices, inasmuch as each player purchases an exclusive right to the selected outcome.  Thus, the winnings are neither parimutuelly determined, as in a horserace, nor shared among multiple winners, as in a sports book.  This aspect also eliminates financial exposure of the pool's sponsor, who is generally a friend or coworker not a bookie.

The specific outcome of the athletic contest is not in play.  Accordingly, sophisticated knowledge of the teams will afford no advantage in picking squares.  Knowledge about the game and its scoring can make a difference, though.

  • In a high-scoring game, like basketball, you might reasonably expect all possible values for least significant digits to be equally likely in the final score.  You would have one chance in a hundred of winning for each square you purchase, resulting purely from "the luck of the draw."
  • Most games, however, end with low scores (baseball, hockey, soccer), and for those, a full-sized payoff matrix based on least significant digits would not be advised.
  • Football manifests scores high enough to distribute all ten possible values for their least significant digits uniformly.  Not quite so...
Football applies rather idiosynchratic scoring increments -- units of 1, 2, 3, and 6 points.  Two-point conversions are extremely rare, as are missed, one-point conversions.  Accordingly, most scoring opportunities produce either 7 or 3 points.  If those two occur exclusively, you can easily show that the only possible football scores reachable in play for any given team would be...
0, 3, 6, 7, 9, 10, 12, 13, 14, ...
...from which you will conclude that a team scoring 20 points or below would never produce a cypher 1 for a least significant digit, and a team scoring less that 18 points would never pay off for purchasers of squares in the corresponding 8-column or 8-row.   With final scores less than 31 points, the cypher 1 would be possible only once (21), the cyphers 4, 5, and 8 twice, cyphers 3, 6, 7, and 9 thrice, and the cypher 0 is four times more likely than 1.

Perhaps with that in mind, the designers of some football pools are known to keep the rows and columns blank until all 100 squares have been sold.  The numerical values are then applied to the full matrix, using some randomizing method (numbered slips drawn from a hat, say).  At the time of your purchase, you would have one chance in a hundred of winning for each square you purchase.  Later, as you sit down to watch the game, you may or may not be pleased with what your selections turned out to be, but as a sophisticated solver, you will understand that they resulted purely from the "luck of the draw."

The payoff matrix in The Next Superbowl puzzle above does have its rows and columns already labeled numerically.  Hmm, let's get to work. {Return}