*Brandon Laing proposes three interesting mathematical problems and contemplates why students may not engage with these problems.*

If you select a random student on the Stellenbosch campus (or indeed almost anywhere in the world), it is almost guaranteed that they will have a negative view of mathematics. It is no secret that our beloved subject is not exactly a talking point at parties.

It is my belief that this ‘distaste’ stems from two major fronts; an individual’s academic upbringing together with the perception of the subject they absorb from their chosen media outlets. A question that every mathematics educator tackles is how one can make mathematics enjoyable and perhaps more importantly, how one can encourage the desire to explore mathematics for explorations sake? To that end I would like to discuss three mathematical problems I found making the rounds on the internet in various versions.

##### Digit Frequency

*What digit is the most frequent between the numbers 1 and 10000 (inclusive)?*

At first glance, this problem would probably put most people off from even attempting it. It ticks all the ‘wrong’ boxes for a palatable problem, namely, it has numbers and asks the user to make a claim about these (wretched) numbers. It also appears to make use of a great deal of values. However, if you give this an attempt it doesn’t matter how large the range of values is.

The problem with our digit problem boils down to the lack of will of students to experiment. In my own lectures with education students, I am often concerned with how unwilling they are to write down their ideas and explore a problem. They would simply see the large 10000 and conclude that the number is far beyond casual reason.

##### Three Students at A Hotel

*Three students rent a hotel room for the night. When they get to the hotel, they pay the $30 fee, then go up to their room. Soon the bellhop brings up their bags and gives the students back $5 because the hotel was having a special discount that weekend. The three students decided to each keep one of the $5 dollars and to give the bellhop a $2 tip. However, when they sat down to tally up their expenses for the weekend, they could not explain the following details:*

*Each one of them had originally paid $10 (towards the initial $30), then each got back $1 which meant that they each paid $9. Then they gave the bellhop a $2 tip. HOWEVER, 3 x $9 + $2 = $29*

*The students couldn’t figure out what happened to the other dollar. After all, the three paid out $30 but could only account for $29.*

I love this problem. It is so subtly cruel that it can only make you smile when you solve it. However, someone who has no interest in a mathematical riddle would most likely be put off by the amount of information and feel overwhelmed. When we set problems for our students, it is very important that we remember we are fighting a battle for their interest. Modern media lives or dies on speed of digestion after all. We must aim to make our work bite-sized and instantly engaging.

##### Foreign Country Riddle

*In a certain country ½ of 5 = 3. If the same proportion holds, what is the value of 1/3 of 10?*

At first glance, this problem is a head-turner for sure. What I find most interesting is the apparent falsity it presents. The fact that it asks us to accept that half of 5 is 3 is maddening enough to steal our interest. The problem is also short enough that we don’t lose interest before we finish reading it. Oddly enough it is probably the most mathematically involved of the three problems and yet from my students I know this one is the first one they will fully engage with. It is that glimmer of silliness that makes the problem fun and lively.

When we as mathematicians develop new ideas we also go through various phases of ‘silly’ conclusions. I’m sure all of us have attempted to formulate definitions that on first try lead to crazy outcomes or unforeseen consequences. But in those moments, we learn, we improve, and most importantly, we engage.

### Brandon Laing

Junior Lecturer, Department of Mathematics,

STELLENBOSCH UNIVERSITY