Before the last race, I heard the runners from Nepal, Oman and Pakistan make predictions about the final standings:
"I hope the Nepalese gets bronze, for then I'm certain to get silver."
"The Omani could get gold, but in that case I'd get silver."
"The gold medal will go to the Pakistani or to me."
I didn't know which runner said what, but as I learned later, they were all right. Furthermore, knowing which country won the race allowed me to calculate the final standings, which were... ?
The statements aren't predictions on who gets which position, but rather on how they themselves fare against another player.
Statement one, stated by either the Omani or the Pakistani, is not
claiming that they would fare better than the Nepalese but that they're
weaker than the other country. (that is, if Nepal < them, then
they're certain to get silver, not gold. therefore, them < the other
competing country)
So we have two possibilites. Either the Omani said it, declaring the
Pakistani will beat him, or the Pakistani said it, declaring the Omani
would beat him.
Let's take the first case:
Fact 1. Omani says statement 1: Pakistani > Omani
Therefore, Pakistani says statement 2. The player declaring statement 2
doesn't know how his skill compares to the Omani, but is sure that if
the Omani got gold, he could beat the other competing country (he'd
definitely get the silver, not the bronze). In this case, the Nepal.
Thus:
Fact 2. Pakistani says statement 2: Pakistani > Nepalese
Which leaves Nepal declaring statement 3. The player declaring
statement 3 doesn't know how he'd fare against the Pakistani, but he's
sure he (and Pakistani) would do better than the other country, which
in this case, is Oman. Thus:
Fact 3. Nepalese says statement 3: Nepalese and Pakistani > Omani
To summarize:
Fact 1:
P > O
Fact 2:
P > N
Fact 3:
N and P > O
Therefore: P > N > O
So the Pakistani gets the gold, the Nepalese gets silver, and the Omani gets the bronze.
Let's take the second case:
Fact 1. Pakistani says statement 1: Omani > Pakistani
Fact 2. Nepalese says statement 2: Nepalese > Pakistani
Fact 3. Omani says statement 3: Omani and Pakistani > Nepalese
To summarize:
Fact 1:
O > P
Fact 2:
N > P
Fact 3:
O and P > N
Facts 2 and 3 go against each other, so this case is not possible.
This leaves only the first case. So for all their statements to be
correct, the Pakistani must get the gold, the Nepalese the silver, the
Omani the bronze, and the Chikkadog the bone! Rowlf!
rowlf!
