Can you solve the magical maze riddle? - Alex Rosenthal
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!