Math Investigations

### What is Math Investigations?

Math Investigations is an optional program open to students in MATH 121 and MATH 122. The class meets for 50 minutes every week, for 7 weeks per semester, to work on questions in small groups under the guidance of two undergrad TAs. The classes are less formal than typical tutorials, and explore ideas outside of the material covered in lecture. The program gives students the opportunity to work together to apply math that they've learned to more challenging problems, and develop some new problem solving techniques. It is suitable for students of all backgrounds and strengths. Above all, Math Investigations strives to give students conceptually-stimulating math problems in a friendly and personal setting.

### When and Where is it?

Winter Term:

Note: It is not necessary to have come to Math Investigations last term to join this term.

There are three sections that meet throughout each week; students may choose the tutorial time each term that best fits their schedule. To enroll, show up at any of these, in Week 2, and sign the attendance sheet.

• Wednesdays 11:30-12:20
• Thursdays 10:30-11:20
• Fridays 1:30-2:20

All tutorials will be held in JEF 202 which is located behind the Math Help Centre on the second floor.

### Is attendance mandatory?

The classes will begin in week 3 of the semester, and attendance is required for those who would like to receive credit for the class. There are exceptions, of course, for illness, sports events, and other extenuating circumstances: for each missed class, students are required to submit a make-up assignment as outlined below.

### What sort of questions will we be working on?

A typical question that might take two weeks:

Question: I take a stick of length 1 and break it at a random point. I then take the longer of the two pieces and break it at a random point. I now have three pieces. What is the probability that these can form a triangle?

Solution: With some assumptions made about the meaning of "random", the probability is (2 ln(2) - 1).

### What's the workload like?

The workload is flexible. During the term, students must attend their sessions (50 mins per week) and participate. The problems are engaging and often demanding. Although this is not required, some students spend considerable time investigating them and learn a great deal. The focus here is on your learning. At the end of each term, everyone submits a journal. The first journal will be a writeup of some problems we've done, and the second one is more of a reflection on the class. Instructions for the first semester journal are given below.

### How will Math Investigations fit into the marking scheme?

Students who fulfill the attendance requirements for the program will be awarded a mark for the program based on participation and their journals. This mark will then be added to their Math 121/122 marks.

Ultimately, the impact the program has on students' marks tends to be small, and those concerned about their grades in Math 121 or 122 are encouraged to prioritize spending their time mastering the main class material. This is a program for those interested in going beyond the course material, and is not tailored to those students who need extra help in that material.

Participation can only increase a student's mark.

### Do I have to be a math genius to join?

No! In the past, students with a wide range of ability and background have benefited from these classes. Students work in groups, and the TAs help ensure that everyone is kept on the same page.

### If I am good at math, will I be bored?

No! Many strong math students have given enthusiastic reviews of the program. The problems are challenging and unconventional, and draw on a range of disciplines and lateral-thinking skills.

### My question isn't answered here!

E-mail us at math.investigations@gmail.com.

### Puzzlers!

Three men go to stay at a motel, and the man at the desk charges them \$30.00 for a room. They split the cost ten dollars each. Later the manager tells the desk man that he overcharged the men, that the actual cost should have been \$25.00. The manager gives the bellboy \$5.00 and tells him to give it to the men.

The bellboy, however, decides to cheat the men and pockets \$2.00, giving each of the men only one dollar.

Now each man has paid \$9.00 to stay in the room and 3 x \$9.00 = \$27.00. The bellboy has pocketed \$2.00. \$27.00 + \$2.00 = \$29.00 - so where is the missing \$1.00?

Given an 8x8 chess board, can you tile the board with dominoes such that two opposing corners are not covered?

Given a 3x3x3 block of cheese, can a mouse, starting from one corner of the cheese cube, eat all the cheese and finish at the center of the cube if the mouse can only move to adjacent cubes (i.e. the mouse cannot eat in a diagonal direction)?

You have a row of 100 closed lockers, and you have a 100 students who each have been assigned different numbers from 1 through 100. Students begin on the left of the row of lockers, and start walking past the lockers to the right. They start counting, adding one to the count for each locker passed. For the student assigned the number 'n', when they reach a multiple of n in their count, they open the locker if it is closed, and close the locker if it is open. After all 100 students walk past the lockers once, how many lockers are open?

You have two jugs with total volumes of 8 and 5 liters. Assuming an infinite supply of water, what quantities of water can you obtain exactly using just the two jugs?

Given nine dots in 2-D space arranged in a 3x3 grid, can you draw four straight lines without lifting your pencil from the page that intersect all nine points?

What's the total number of unique ways to rearrange 10 glazed timbits and 3 chocolate timbits? What's the total number of ways you can split 10 timbits among four people? (splittings where some people receive no timbits are valid)

Start on a 2D square grid will one filled square, and fill in all adjacent squares. Count the total number of filled in squares and repeat the process of filling in all adjacent squares (see attached image). What's the sequence that describes the total number of filled in squares?

Let's say you part your hair on the right side. You look in the mirror, and your image has its hair parted on the left side, so the image is left-to-right mixed up. But it's not top-to-bottom mixed up, because the top of the head of the image is there at the top, and the feet are down at the bottom. The question is: how does the mirror know to get the left and right mixed up, but not the up and down?