A single-meeting round-robin chess tournament commenced with five players Adrian, Beau, Craig, Damian, and Elliott playing against each other exactly once.
At the end of the tournament Adrian came 1st, Beau came 2nd, Craig came 3rd, Damian came 4th, and Elliott came 5th.
Beau and Elliott shared their impressions as follows:
- Beau: "I am the only one who finished without a single loss."
- Elliott: "I am the only one who did not win a single game."
The rules were:
- A winner got a full point.
- For a draw each opponent got half a point, and:
- The ranking is decided only by looking at the points.
Reconstruct the tournament table from the clues mentioned above.
From 1, B has zero losses but is in second place, everyone
else has at least one loss. First place
A then must have at least one loss, but still has the highest score, which cannot
be greater than 3. <o:p></o:p>
Test case 1 – A wins with 3. <o:p></o:p>
A has 1 loss and 3 wins, B has 1 win and three ties –B ‘s
win must be A’s loss. Now we can say that
C thru E have all lost to A and tied with B.
C cannot have a score greater than 2, assume 2.<o:p></o:p>
C then needs 1.5 points remaining to be assigned, which must
be a win and a tie. D needs 1 more point
assigned, so it must be a win. In this
scenario, C can’t suffer a loss to D or E (needs 1.5 points), so with a little
thought C wins over D and ties with E (C=2 points total), D needs one more
point to get 1.5 (E cannot have a win), so D wins over E.<o:p></o:p>
Summary table (A-B-C-D-E order, X =no game – can’t play
against yourself):<o:p></o:p>
A X, loss, win, win,
win (3 points)
B win, X, tie, tie, tie (no losses, 2.5
points))
C loss, tie, X, win, tie (2 points)
D loss, tie, loss, X, win (1.5 points)
E loss, tie, tie, loss, X (1 point, no
wins)<o:p></o:p>
I believe this fits the problem, but I have not proved it to
be a unique answer. <o:p></o:p>
If A wins the tournament with 3 points, the loss MUST be to
B, or 1 is violated, and A and B would tie for points. I believe you cannot rearrange C’s or D’s game scores in this case without more inconsistencies. <o:p></o:p>
What remains is the cases of A winning the tournament with
only 2.5 or 2 points. Here I have run
out of time and will have to address later.<o:p></o:p>
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Posted by Kenny M
on 2022-07-14 09:02:07 |
Solution, not proven unique
