From James Marek comes some madness in the month of March:
According to James, there was a scavenger hunt this year to determine which lucky individuals received tickets to the Duke’s home men’s basketball game against rival North Carolina. (Spoiler alert: Underdog North Carolina won the game.)
The original scavenger hunt had five tasks. For the purposes of this riddle, let’s say it has three tasks. For each task, whoever finishes first gets 1 point, whoever finishes second gets 2 points and so on. Your score is the total number of points you earn across all three tasks, and lower scores are better. Knowing how popular basketball is at Duke, it’s safe to say that many people are participating. But only the top-10 finishers in the scavenger hunt will get tickets.
Without knowing how anyone else did on the scavenger hunt, what is the highest score that guarantees you are in the top 10? (Being tied for 10th is acceptable.)
In order to guarantee top, one must assume the other nine participants have as low a score as possible.
This means everyone in the top 10 scored in the top 10 in each of the three tasks.
$$3 \cdot \sum \limits_{i = 1}^{10} i = 165$$5 participants can receive 16 points and 5 can receive 17 points.
| Participant | Task #1 | Task #2 | Task #3 | Total |
|---|---|---|---|---|
| $1$ | $1$ | $6$ | $9$ | $16$ |
| $2$ | $2$ | $8$ | $6$ | $16$ |
| $3$ | $3$ | $10$ | $3$ | $16$ |
| $4$ | $4$ | $2$ | $10$ | $16$ |
| $5$ | $5$ | $9$ | $2$ | $16$ |
| $6$ | $6$ | $4$ | $7$ | $17$ |
| $7$ | $7$ | $5$ | $5$ | $17$ |
| $8$ | $8$ | $1$ | $8$ | $17$ |
| $9$ | $9$ | $7$ | $1$ | $17$ |
| $10$ | $10$ | $3$ | $4$ | $17$ |
Note that
$$3 \cdot \sum \limits_{i = 1}^{11} i = 198$$11 players can each receive 18 points.
| Participant | Task #1 | Task #2 | Task #3 | Total |
|---|---|---|---|---|
| $1$ | $1$ | $11$ | $6$ | $18$ |
| $2$ | $2$ | $5$ | $11$ | $18$ |
| $3$ | $3$ | $10$ | $5$ | $18$ |
| $4$ | $4$ | $4$ | $10$ | $18$ |
| $5$ | $5$ | $9$ | $4$ | $18$ |
| $6$ | $6$ | $3$ | $9$ | $18$ |
| $7$ | $7$ | $8$ | $3$ | $18$ |
| $8$ | $8$ | $2$ | $8$ | $18$ |
| $9$ | $9$ | $7$ | $2$ | $18$ |
| $10$ | $10$ | $1$ | $7$ | $18$ |
| $11$ | $11$ | $6$ | $1$ | $18$ |
Swapping two of the numbers in the above table easily changes two of the scores to 17 and 19. 19 is clearly too large as it would be out of the top 10.
In order for 18 to be too large, every other participant in the top 10 would have to have a score less than 18.
The sum of the top 11 players would then be less than 198. This is a contradiction, as the minimum sum is 198.
Therefore, 18 is the highest score that guarantees a spot in the top 10.