For the first day of Math Week, we will be focusing on algebra, the study of abstract systems involving variables (our favorite \(x\)), operations and more.

Submission is now open! Submission for Day 1 puzzles has closed :/
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Submit!

If you are able to solve any of the puzzles, submit your solution by Tuesday March 4, 11:25am to win a treat.

Puzzle 1 - Music Festival

As you know, Lisgar is famous for its marvelous music program. Our school ensembles, led by the hardworking music teachers, have taken part in many music festivals. This year the Lisgar Symphony Orchestra performed in the Capital Region Music Festival on February 22, 2025. The festival was held in Earl of March SS and the musicians had to get there by school bus. The school bus can be modelled as follows:

  • Each row has \(2\) benches, one on the left and the other on the right. In the middle is the aisle. Each bench seats \(2\) people.
  • The front \(13\) rows are violinists. They are the closest to the door.
  • The middle \(8\) rows are violists.
  • At the back is \(12\) rows of cellists and bassists.
  • The bus is full.

Side note: The wind section decided to take OC Transpo and were not on the bus.

Our bus arrived at Earl of March at 3pm. Ms. Sommers asked the students to get off one by one according to the following rules:

  1. Front rows get off first. A row is allowed to get off only if everyone in every row in front of it has left.
  2. Anyone sitting next to the windows cannot get off unless the other person who sits next to the aisle and shares the bench with them has got off.
  3. As an exception to rule 1: since cellos and basses are big instruments, Ms. Sommers asked the cello/bass players to get off first to get their instruments ready. The violinists and violists have to wait until all of them have left.

The first person gets off at exactly three o’clock, and it takes 10 seconds before the next player can get off (e.g. the second player gets off at \(3:00:10\)).

I am a violist sitting next to the window in the third front-most row of the violist section. I have to practice before the performance! Can you tell me what’s the earliest possible time that I can get off?

Puzzle 2 - Average Marks

In the 2024-2025 school year, there are \(6\) MCR3U (Grade 11 Functions, University Preparation) classes open at Lisgar. Since so many people (including some grade 10s) wanted to take MCR3U, every class is at its full capacity: \(30\) students. I am in one of these MCR3U classes.

When the report card is out, I found out that the course average (that is the average mark of all students taking MCR3U) is \(82\). I feel bad for my class, since I heard from my classmate (who literally asked everyone in our class for their mark) that the average mark for just our class is \(77\), lower than the course average.

Can you help me figure out what the course average is if we don’t count my class?

Side note: The marks are not real data.

Puzzle 3 - A Very Algebra-ish Problem

I spent my whole day looking through AOPS Intermediate Algebra to get an idea for the third puzzle of the algebra day. Unfortunately, all problems are too algebra-ish and no story can be made from them.

Therefore, I decided to give you this very algebra-ish problem this year and try harder for the 2026 Lisgar Math Week:

The following equation is true for \(A = 20\) and \(B = 21\):

$$\frac{x - A}{B} + \frac{x - B}{A} = \frac{B}{x - A} + \frac{A}{x - B}$$

Since you are reading puzzle 3, I’ll assume that you all have amazing mathematical intuition and can figure out quickly that \(x = 0\) is a solution to this equation. However, it does have more solutions. Find the product of all possible values of \(x\) other than \(0\).