## IB AA HL P3 TZ2 - May 2021

Get the paper

### Full Solutions

Check this video made with Manim Click to Play

## IB AA HL P3 TZ1 - May 2021

Get the paper

### Full Solutions

Check this video made with Manim Click to Play

## Friday Conversations with Curio

### Musings on Infinity...

Watermelon and Black Crows

Each Friday I hope to relate one of the many interesting conversations I have had with Curio, my partner in infinity. Perhaps first a few words to describe him. I can’t accurately describe his face, as it is always somewhat obscured, or perhaps it is my memory that is obscured. When I call him to mind he resembles an amalgam of the Little Prince, Bart Simpson, myself as a ten year old, and another favorite ten year old of mine. He is not troubled by infinity, and yet he doesn’t ignore it as most of us adults do. He loves infinity. It is not the only thing we talk about, though what we will talk about next is not known, it is occasionally about infinity.

On this day, we met at an aquarium in the oldest desert in the world. There were very few fish so we had time to chat. I asked him to focus on his breath and when I snapped my fingers to tell me what was the first thing that entered his mind.

Since starting meditation practice several years ago I have occasionally been asked or asked myself to notice the first thing that comes into my awareness starting now. The purpose is to illustrate that we don’t have the control over what enters our consciousness as we think we do. One of the first times I did this exercise I thought of watermelon. Why did watermelon come to mind? I might have recently eaten some. Maybe I noticed one in a shop or on the side of the road. But really, I have no idea why watermelon came to mind above or ahead of anything else.

Since then, when I try this exercise, more often than not I think of watermelon. For this there is a reason. It is because of the memory of the earlier experiment. But again it is not as if I choose to think of watermelon. The memory is just one of many variables that determine what will enter my mind.

As I waited for Curio and thought of watermelon I also thought about the possibility that he would think the same thought as me. In math we use the word probability giving a sense of measurability or definitiveness. Though it may be unlikely, it wouldn’t be impossible for him to think of watermelon. Perhaps he had just eaten or seen some, or feeling hot and thirsty in the desert, imagined some.

But if I delve deeper into what I think of when I think of watermelon I realise it is hard to pinpoint. It is a combination of things. I see something. Sometimes it is the uncut large green oval; sometimes it is the red flesh bordered by the pale rind grading to green; sometimes it’s both, layered on top of each other, or side by side, they float and move. I hear the sound of the word being spoken. I might even conjure a smell. I might see the letters “w a t e r m e l o n" in my imagination. All of these combined, or only some of these, in different order and length, or coincident, I realise that each thought of watermelon is slightly different.

So what is the probability that Curio thinks exactly the same watermelon thought in every detail as I do?

If we had both just seen a cow skydiving then the possibility that we both share the same general thought is relatively high. Otherwise, if we are coming from two completely different experiences then the probability must be fairly low. In any case, even skydiving cows, the probability of sharing the same thoughts in exact detail seems to be nearly zero. Could it be zero? This would imply that the denominator in this probability would be infinite. In other words, the total number of possible thoughts is infinite. This is what we chatted about.

“Focus on your breath… when I snap my fingers tell me the first thing that enters your mind.”

I snapped. Curio said, “a bird, black, flying, rather large, like a crow.”

I told him about the watermelon. “Is it possible that you and I could have had the exact same thought?”

“Yes”

“In every detail?”

“Probably not.”

“Meaning probability of zero?”

Here he paused. This was not the first time we had talked about infinity. We had talked about a number line from 0 to 1, and choosing a random point on the line. There are an infinite number of points between 0 and 1. The probability of choosing 0.5 for example is given by \[ \frac{1}{\infty} \] which is zero right? Here we have a brief discussion about the difference between “is zero” and “approaching zero” which I will save for another time, but we both agree there is no number that the probability could be other than zero. Furthermore the probability of choosing one of two different points say 0.25 and 0.5, is \[ \frac{2}{\infty} \] but no different from one point. Choosing one out of a million or out of a quadrillion would not be different. The probability could only be zero.

So a point is chosen at random, and the probability of having named that point beforehand is zero. The probability of choosing any particular point is zero, yet a point is chosen. Most people drop it here to avoid getting dizzy.

Back to the watermelon and the crow… If we think about the fine details of any given thought it seems impossible to find a hard limit on the number of possibilities. The natural reaction to a hard limit is to say I can add one more to that, just like counting - the place where we all probably first encounter infinity.

Thoughts are different from number lines though. They are not necessarily random (can they even be perfectly random? Again, for another day…). They are determined by the variable events that preceded them. And this is not calculable. We should be careful not to use zero in place of not calculable.

We went to look at the fish in the aquarium and silently I wondered if he had the same thought as me.

## Chutes and Ladders

### Probability

Here is a probability question from a student.

Question: What is the probability, given that you are 4 spaces away from winning, that you will win in exactly 3 moves?

Check this video made with Manim to see how we can arrive at the answer Click to Play

Now what we'd like to do is try to gereralise. What if your were *n* spaces away from winning?
What if you had a *d* sided die? What if the goal was to win in specifically *k* moves?
This is what keeps Mathematicians up at night!

Working on it...

### Some thoughts...

## Our Motto

### One World

The world is interconnected. Knowledge can be transferred around the globe. A student with a need on one continent can be directed towards a solution by a tutor on another continent. We are accustomed to this, but it is truly remarkable.

This interconnectedness shrinks the world, and we realise that we need global solutions for the problems that we face now and will face in the future.

Learning Mathematics transcends the things that can divide us (politics, ethnicity, religion, etc). We speak the same language when we solve Math problems together. The fundamental truths of Mathematics is the same in your world and my world, and everyone's world. When we engage in this study, we realise that it is One World.

### Infinite Solutions

The concept of infinity is one of the things that makes Math so fun and fascinating. Our minds have limits when it comes to contemplating the infinite, but Math has taken us remarkably far in the journey.

What do we mean by infinite soultions? Surely that's hyperbolic.

But if you think about the available pathways a mind can go when faced with a problem, we see that the possibilities are more than we can imagine, and we get a glimpse of the infinite, even within our own single mind, not to mention 7+ billion minds. Are the ways to think about something confined by some upper limit? Why would that be?

When it comes to thinking about a problem, it is reasonable to say that the minute details of that thought, the content and the order, have never precisely happened before. Your thoughts are unique, and limitless in possibilities. Flashes of brlliance are not uncommon

### One Step at a Time

And now for practicality. There are problem solving techniques that work. We can break complex problems down into a series of simpler ones. A mountain is not climbed in one leap. As a student takes each step, progress is made, and a flash of brilliance might come at any moment. It's this combination of infinite possibilities and incremental progress that makes the study of Mathematics so exciting.