Three players A, B and C are nominated and three sportswriters respectively from periodicals D, E and F designate a player as first, second or, third place. However, the point awards for the first-, second- and third-place votes are not known precisely. All we know is that a first place vote is worth x points, a second-place vote is worth y points and a third-place vote is worth z points with x > y > z.

When the points have been counted up, Player A has emerged with 20 points, Player B with 10 points and Player C with 9 points.

If Player A received a second place vote from Periodical D, who received a second place vote from Periodical E?

DECLARE SUB award (perm!)
DIM SHARED prz(3)
CLS
FOR x = 5 TO 10
 prz(1) = x
 FOR y = INT((13 - x) / 2 + 1) TO x - 1
  prz(2) = y
  z = 13 - x - y
  prz(3) = z
  IF z >= 0 THEN
   FOR d = 1 TO 6 ' Choose which permutation
   FOR e = 1 TO 6
   FOR f = 1 TO 6
     REDIM SHARED player(3)
     award d
     award e
     award f
     p1 = player(1): p2 = player(2): p3 = player(3)
     IF player(1) < player(2) THEN SWAP player(1), player(2)
     IF player(2) < player(3) THEN SWAP player(2), player(3)
     IF player(1) < player(2) THEN SWAP player(1), player(2)
     IF player(1) = 20 AND player(2) = 10 AND player(3) = 9 THEN
       PRINT prz(1); prz(2); prz(3), d; e; f, p1; p2; p3
     END IF
   NEXT
   NEXT
   NEXT
  END IF
 NEXT
NEXT

SUB award (perm)
SELECT CASE perm
  CASE 1
   player(1) = player(1) + prz(1)
   player(2) = player(2) + prz(2)
   player(3) = player(3) + prz(3)
  CASE 2
   player(1) = player(1) + prz(1)
   player(2) = player(2) + prz(3)
   player(3) = player(3) + prz(2)
  CASE 3
   player(1) = player(1) + prz(2)
   player(2) = player(2) + prz(1)
   player(3) = player(3) + prz(3)
  CASE 4
   player(1) = player(1) + prz(2)
   player(2) = player(2) + prz(3)
   player(3) = player(3) + prz(1)
  CASE 5
   player(1) = player(1) + prz(3)
   player(2) = player(2) + prz(1)
   player(3) = player(3) + prz(2)
  CASE 6
   player(1) = player(1) + prz(3)
   player(2) = player(2) + prz(2)
   player(3) = player(3) + prz(1)
END SELECT
END SUB

reports

               which        players'
possible    permutation      scores
 points       D  E  F        1  2  3
 
8  4  1       1  1  4       20  9  10
8  4  1       1  4  1       20  9  10
8  4  1       1  5  5       10  20  9
8  4  1       2  2  3       20  10  9
8  4  1       2  3  2       20  10  9
8  4  1       2  6  6       10  9  20
8  4  1       3  2  2       20  10  9
8  4  1       3  3  6       9  20  10
8  4  1       3  6  3       9  20  10
8  4  1       4  1  1       20  9  10
8  4  1       4  4  5       9  10  20
8  4  1       4  5  4       9  10  20
8  4  1       5  1  5       10  20  9
8  4  1       5  4  4       9  10  20
8  4  1       5  5  1       10  20  9
8  4  1       6  2  6       10  9  20
8  4  1       6  3  3       9  20  10
8  4  1       6  6  2       10  9  20
10  3  0      1  1  6       20  9  10
10  3  0      1  6  1       20  9  10
10  3  0      1  6  6       10  9  20
10  3  0      2  2  5       20  10  9
10  3  0      2  5  2       20  10  9
10  3  0      2  5  5       10  20  9
10  3  0      3  3  4       9  20  10
10  3  0      3  4  3       9  20  10
10  3  0      3  4  4       9  10  20
10  3  0      4  3  3       9  20  10
10  3  0      4  3  4       9  10  20
10  3  0      4  4  3       9  10  20
10  3  0      5  2  2       20  10  9
10  3  0      5  2  5       10  20  9
10  3  0      5  5  2       10  20  9
10  3  0      6  1  1       20  9  10
10  3  0      6  1  6       10  9  20
10  3  0      6  6  1       10  9  20

which indicates that the two sets of (x, y, z) that work are (8, 4, 1) and (10, 3, 0) in the sense of providing possible sets of scores of 20, 10 and 9.

In the case of (10, 3, 0), two of the magazines agreed on the same order and the third magazine had the exact reverse order. Thus the score of 20 was from two first-place positions and one last place; the score of 10 was from one first-place position and two last; and the score of 9 was from three second-place evaluations. However, we know that player A had the 20 points and also was a recipient of a second-place vote. That situation doesn't happen in the (10, 3, 0) case.

In the (8, 4, 1) case, again two periodicals agreed on the same order, but in this case the first-placer from two magazines was second-placer in the third periodical; the second-placer of two was the third-placer in the third; and the last placer of two periodicals was first according to the third periodical. So the results came from 8+8+4=20, 4+4+1=9 and 1+1+8=10.

Player A was the one who had the 20 points, including a second-place from periodical D. The other two second-place positions were obtained by the person who had a total of 9 points, that is, Player C.

So indeed, it's Player C who received a second-place vote from Periodical E (as well as from Periodical F).

Posted by Charlie on 2013-06-07 17:33:31