Today is the annual Sly Wizard Tournament
featuring competitors from the wizarding world’s
three greatest schools, and you’ve been entrusted
with an enormous responsibility. You are to administer the tournament. First, you and you alone will determine
how many events there will be and their scoring system. Then the three wizards will
enter your maze and compete in your chosen
events in utmost secrecy; only you and they will see what happens. The competition begins and... Wow, that was one for the record books. The winner was... wait, you have no idea. The last thing you remember was
a dark wizard showing up and casting a forgetting curse. The competitors seem even more confused—
each is convinced they won. This is bad. There can be no do-overs, and the last failure to declare a winner
set off the first Great Wizarding War. You’ve got to figure this out— and fast. But for the life of you,
you can’t remember a thing. You know you had to follow a few rules: there had to be three or more events,
each with a single winner and loser. Every event used the same scoring system, where first place received more points
than second, and second more than third. All points were positive integers. Maybe there’s a record somewhere...
oh, of course, your scorecard. Well, that leaves something to be desired. All that you wrote down was
that in Calchemy, Newt-niz won,
Leib-ton took second, and that was the only time
Magnificent Marigold’s Magical Macademy got third all day. Oh, and some final scores: one school got 22 points,
and both of the others got 9. Why are you so bad at taking notes?! No time for self-recrimination. The wizarding world is waiting. Who won the tournament? Pause here to figure it out yourself.
Answer in 3 Answer in 2 Answer in 1 At first, it may seem like there are
an overwhelming number of possible scoring systems, so let's see if we can narrow our options. We can start by looking for clues
in the total scores. Every event was scored the same way, so the sum of all points
in the entire tournament must be a multiple of one event’s total. In other words, if there were three events
that scored 3, 2, 1, which adds up to 6,
the total points for the day would be 18, which is three events times 6 points. Our total is 40. So we can make a table of possibilities:
one event totaling 40 points, two totaling 20, four totaling 10,
and so on. We know there were at least three events,
which eliminates these options, and we can also get rid of events
with fewer than 6 points, because the smallest possible total
is 3 plus 2 plus 1. That leaves two prospects. Let’s try to narrow those further
by breaking down the possible points earned in each event: if first place received 7 points, the teams that had totals of 9
couldn’t have won an event because their total score
would be 10 or more. That means the team with 22
would have to have won all four. But then their total would be 28,
so we can eliminate that option. And with four events, these numbers
can't be made to add up to 22. Finally, if first place got 5 points, the highest possible score
with four events would be 20, getting rid of these two as well. In fact, five events scoring 4 each
could only reach 20 as well. That leaves us with just one possibility: five events each scored 5, 2, 1. There’s exactly one way to make
those scores add up to 22: the winner finished first
four times and second once. The scores of 9 mean one team won once
and lost four times, and the other lost once
and took second four times. That must be Marigold’s Macademy, whose only third place finish
was Calchemy based on your note. And Leib-ton’s second place finish
in Calchemy means they scored 22 and won the Sly Wizard Tournament. You have just enough evidence to prove it,
keep your job, and avert war. Phew!
