Re: Baseball Pennant, by George Summers

James Dow Allen wrote:

Fumbling through my library I stumbled on
_New Puzzles in Logical Deduction_
by George J. Summers. I riffled through
the book, deciding that the puzzles were
challenging, but more tedious than interesting.

I don't normally write in books, especially
in ink, but at Puzzle #48 (out of 50) I
found I'd left myself some notes: The
puzzle was "Hard" yet two of the 6 clues
were "Unnecessary"!

I'll give the puzzle and the 4 needed clues
(along with the 2 "unnecessary" clues rot13).
(I've rewritten the puzzle slightly: it had
a useless complication that needs to disappear
when a redundant clue disappears.)

(George Summers doesn't mention that baseball
games do not have tied final scores. I don't
know if he overlooked that non-Americans might
not know that, or just noticed that absence of
tied scores can be deduced from the clues.)

#48 The Baseball Pennant

(Summers describes an (irrelevant) playoff scenario.)

1. Each pair of four teams plays a game; runs
scored by each team in its games are as follows
(in some order):
Sexton 1 - 3 - 7
Treble 1 - 4 - 6
Ulster 2 - 3 - 6
Verdue 2 - 4 - 5

2. Each team won a different number of games.

3. Gur fpber sbe rnpu tnzr jnf qvssrerag sebz
gung bs nal bgure tnzr.

4. The greatest difference in runs scored by
two teams at one game was 3 runs; this difference
occurred only once when the team that lost the
greatest number of games lost by 3 runs.

5. Two teams scored the same number of runs during
the first round and two teams scored the same
number of runs during the second round. (A "round"
consists of all the teams playing games simultaneously.)

6. Ehaf fpberq va gur guveq ebhaq vapyhqr gjb
qvssrerag bqq ahzoref naq gjb qvssrerag ahzoref.

Determine the scores in each of the six games.

There were six games. The teams won 3, 2, 1, and 0 games, in some
order. Since there were 6 wins, there were also 6 losses, and so
there was no room for draws.

Sexton had at least one win (with 7 runs). Sexton and Treble each
had at least one loss (with 1 run); one of these losses was to either
Ulster or Verdue (whichever won 3 games) with 2 runs (the only way to
win with 2 runs).

The loss by 3 points (by the team that lost 3 games) could have been

1 (T) to 4 (V) -> Treble lost its 6-run game to Sexton
2 (U) to 5 (V) -> Ulster lost its 6-run game to Sexton
3 (U) to 6 (T) -> Ulster lost its 6-run game to Sexton
4 (T) to 7 (S) -> Treble won its 6-run game (contradiction)
4 (V) to 7 (S) -> Verdue lost its 5-run game to Treble or Ulster

This leads to the following possibilities (one per column):

S T : 7 6 S T : S T : S T : S T :
U V : U V : 2 5 U V : 2 W U V : W 2 U V : 6 5

S U : S U : 7 6 S U : 7 6 S U : W S U : W
T V : 1 4 T V : W T V : W T V : 6 5 T V : W 2

S V : S V : W S V : W S V : 7 4 S V : 7 4
T U : 4 6 T U : W 3 T U : 6 3 T U : W T U : W

First possibility:

S T : 7 6
U V : [23] [25]

S U : [13] [23]
T V : 1 4

S V : [13] [25]
T U : 4 6

Neither (ST UV) nor (SV TU) can contribute toward clue 5, contradiction.

Second possibility:

S T : [13] [146]
U V : 2 5

S U : 7 6
T V : [14] [24]

S V : [13] [24]
T U : [46] 3

Neither (ST UV) nor (SU TV) can contribute toward clue 5, contradiction.

Third possibility:

S T : [13] [14]
U V : 2 [45]

S U : 7 6
T V : [14] [245 W]

S V : [13] [245]
T U : 6 3

Clue 5 requires the following, which works:

S T : 3 4
U V : 2 4

S U : 7 6
T V : 1 2

S V : 3 5
T U : 6 3

Fourth possibility:

S T : [13] [14]
U V : 3 2

S U : 1 2
T V : 6 5

S V : 7 4
T U : [14] [26 W]

Clue 5 requires the following, which works:

S T : 3 1
U V : 3 2

S U : 1 2
T V : 6 5

S V : 7 4
T U : 4 6

Fifth possibility:

S T : 3 4
U V : 6 5

S U : 1 2
T V : 4 2

S V : 7 4
T U : 1 [23]

Clue 5 can't be satisfied, contradiction.

The rot13'd clues don't eliminate either of the two solutions.
.

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