Happy Lunar New Year!!!
It is 2023, the year of the rabbit and we have to say, rabbit is one of our favorites. Our MathBait founder, Shayla, was born in the year of the rabbit and she has a family member, Sammy, that is a rabbit as well. Awwweeee 😍
To celebrate the Lunar New Year, we created a little puzzle, one of our favorite paradoxes and such a fun thing to think about and explore.
The best viewer comment? The viewer who posted "*cries*". But what about this post pulled at some of our heartstrings? Let's dive into our MathBait Lunar New Year Puzzle!
The Puzzle
A rabbit is crossing a 2-mile field to find his way home (you can view the puzzle video on TikTok).
In the first hour, he is energetic! He just got started. He is able to cross half the field or travel 1 full mile.
The next hour he is getting tired, he can only cross half the remaining field. That means he travelled 1/2 a mile this time.
With his energy draining, in the next hour he crosses another half of the distance remaining, 1/4 a mile.
If this continues, every hour he is able to only half of what he did the hour before - how long until the rabbit gets home?
The Solution
There is a realistic answer and a mathematical answer to this one, which makes it ripe for debate! Drag the slider in the window below to see how far the rabbit has to go after each hour.
©MathBait Created with GeoGebra
What did you notice? Since humans are very bad at understanding large or tiny numbers, we did some conversions for you. After 13 hours of this behavior the rabbit is only about a foot away from home! After 17 hours he is only an inch away! Why not call it a day and say the rabbit does in fact make it home. To give it a nice number we can say he makes it home in 20 hours. But does he really?
You might have also noticed that at some point, the activity above does actually say he is home! At exactly 54 hours later we see the rabbit has no more to travel. So he makes it home then, right? Why are the little bars showing his distance still visible? If he got home, those should have zero height, right? In this article we will explore number bases, computers, Zeno's paradox, and ask - is infinity a real thing?
Why He Didn't Make it Home Even Though We Said He Did
Let's start with bases. This is important because we need to understand how a computer counts (unlike us they have only 2 fingers) to understand why the activity reads the rabbit as home. Next, we will dive into how much memory a computer has. Finally, we will take a quick look at infinity and ask ourselves some mind-bending questions! Let's dive in.
Bases
Bases are simply the way we all agree to count. Since humans have ten fingers, when early mathematicians were deciding on how to represent numbers; everything they saw and everything they were trying to measure around them, they decided ten was a great foundation.
It makes things easy. I count my fingers: 1, 2, 3, 4, 5, 6, 7, 8, 9. When I get to 10, I have counted all of my fingers 1 time and haven't done anything more. So I write ten as one followed by zero - 10. The first number says how many times I counted all my fingers, the second number says how many more. If I count to 14, I counted all my fingers one time and then I counted 4 more.
What about the aliens on planet Zorg? They have only 8 fingers (some say 4 on each hand but we heard an account that said they actually have 5 on their right hand and 3 on the left, but who's counting?). The number 12 to the Zorgeans is very different than 12 to us humans. Why? Well, they use the same system. The number 12, to them, says they counted all their fingers 1 time and then counted 2 more. Because they have only 8 fingers, when they count all their fingers one time, they get 8, then they count 2 more, so altogether they have actually counted to 10. This is really important stuff if you ever plan to trade with the Zorgeans. If they tell you they will trade for 12 apples, remember, you only need to give them 10!
What does this have to do with the rabbit? Everything. You see computers have only 2 fingers, on and off. This means, if a computer wants to express the number 10, they go through this process: I count all of my fingers once (that is 2), they do this 5 times to get to 10 (1, 2, 1, 2, 1, 2, 1, 2, 1, 2). Well this is where things get a bit more complicated. They don't have enough to represent this with two-digits! So we need to add another place. In human counting, each place is 10 times the previous (again because we have ten fingers). That means for instance the number 100 is saying I counted all of my ten fingers ten times. For binaries, like computers, since they have only 2 fingers, each place value is not 10 times, but 2 times the place before. So, when a computer has counted all of their 2 fingers 2 times they add another place. Two fingers, two times, is of course 4 which in binary is represented as 100. Now, to get to 10 (notice this is a significant amount of work to get to the simple number of 10) they can do this twice (4+4=8) making in computer-counting 1000, the number 8 to us. But I still need to get to 10! I need to count all my fingers one more time which results in the number 1010 in computerspeak.
For binaries, each place is two times the previous. In other words, our tens place is their 2-place, our hundreds place (10 tens) is their 4-place (2 twos) and our thousands place (10 hundreds) is their 8-place (2 fours). The binary number 101 tells us they counted both fingers 4 times (two twos) then counted one more meaning 101 in binary is 4+1=5 in human.
More like humans than we may want to admit, computers have limited memory (what was I doing last Thursday night?). This is where our model, and the rabbit's distance, breaks down.
Computers
If you are interested in doing anything in technology, this next bit is super important to know. Computers use what is called floating point representation. You probably learned this in early math and science classes as "scientific notation".
The number 2317.23 is expressed in scientific notation as 2.31723 × 10³. They call the decimal portion (in this case 0.31723) the mantissa. Since computers don't use tens, they use twos, every number on a computer has the form,
Once the computer has translated your number to binary floating point form, it stores it generally in double precision which uses 64 bytes. That means they have 64 light switches that can turn on or off to represent your number.
The first light switch is the sign (is your number positive or negative?) so we are down to 63 switches left. The next 11 switches cover the exponent and what they call the bias. The bias is to determine the sign of the exponent (if your number is less than one, like in the case of the rabbit's distance, your exponent is negative and the bias is what stores this information).
The exponent is given 11 bits of storage meaning the biggest exponent a computer can handle is 11111111111 which is 2047 to us. The rest of the space leftover (63-11=52) is used for the mantissa.
What does all this mean? Well, it boils down to the fact that a computer can't get really really close to zero. It goes through what is called an underflow error. Now it can of course represent 0, 0 is 0 on all planets and across all machines. It cannot however get very very small numbers that are close to zero but not equal to zero. Which is what we want in this case.
Key Point: The limitations of both computers and the GeoGebra software mean at a certain point it collapses everything to zero, even if the calculations aren't zero.
This impacted our model! Although it told us after 53 hours the rabbit had less than an inch to go, he wasn't mathematically home. It was simply that our program got too close to zero and gave up. Who knew computers were so lazy?
Zeno's Paradox
This conundrum (does the rabbit make it home) is commonly known as Zeno's Paradox. You can read all about it at a very high level in the Stanford Encyclopedia of Philosophy, but we will give you the shortened version.
We can see time as a series of individual moments. This is essentially how animation works. If you are interested in getting into this field, math is super important. The paradox says, if we want to run, like the rabbit, two miles, we have to at some point run halfway. And when we look at all those halfway points, well, things get trippy.
The problem with this is that while 2 miles is a finite distance, there are an infinite number of steps to get there. Since an infinite number of steps takes an infinite amount of time, even though we know we can get from here to there, the paradox says we can't, or more accurately we can, but it would take forever. Since we don't have an infinite life, it is like saying we, and the rabbit, never get home. *cries*
Infinity
This brings us to infinity. What is it, is it real? What does forever even mean in terms of time? Euclid has us believing that infinity is like a line that never ends. It goes on and on in a single direction. But others have questioned that. Isn't infinity simply something with no ending? Like a circle. There is no "start" and "stop", if you begin anywhere on a circle and follow its path, isn't it true that it never ends? Round and round we go.
Another thing to consider is: what if the bunny isn't moving at all? Did you know, when you play an animated racing game your car never moves? While it looks like you are speeding around the track, in fact, it is the track and the scenery that are moving... you're still. This gives the appearance like you are driving, but you are in fact still right where you began!
The idea that the rabbit never gets home is quite sad. But we know that we, and certainly a rabbit, are capable of making the 2 mile distance. So we are okay with saying he gets home, even if that causes a mathematical conundrum. The idea that none of us really ever move is certainly a mind-boggling and interesting one. While it may seem improbable, we experience this everyday. To us, the sun rises in the east and sets in the west. It appears to move across the sky throughout the day. We know that this is not the case. We understand that we are moving and the sun is sitting comfy. However mind-bending, it is certainly possible that what we see, or what appears to be, isn't what is happening at all.
Who knows? Maybe the rabbit never moved at all and still made it home? Happy Lunar New year and Happy year of the Rabbit!
If you like this puzzle, make sure to check out Marco the Great and the History of Numberville where Marco learns the true story of Hansel and Gretel and the never-full candy bar.
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