In this post we present few examples of how our intuition fails us when it comes to certain mathematical problems…
1.Circumference increase vs Radius increase
The equatorial radius of the Earth is 6,378.1 km.
Assuming that the earth is a perfect sphere:
- Tightly wrap a string around the equator of the Earth.
- Cut that string at any point and increase its length by 1 meter.
- Uniformly lift above the earth’s surface, removing the slack caused by the addition in the length of the string.
Guess what will be the distance that the string is lifted above the surface of the Earth?
- 0.16 cm
- 1.6 cm
- 16 cm
- 160 cm
The correct answer is “c”:
The distance that the string is lifted above the surface of the earth will be approximately 16 cm.
The first time I saw this, I was quite astonished, wondering what difference would 1 meter make, considering that the earth is huge.
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Now, lets do the math
For any circle:
Lets call the original radius r
and let R be the radius of the circle with circumference increased by 1 m.
The larger circumference is equal the original circumference plus 1 m:
which describes the value of the “lift” of the increased by 1 m circumference above the surface of the original sphere.
This value (for 1 m increase in circumference) is 0.1592 m (approximately 16 cm).
What is even more incredible is that this number remains the same for a circle of any size (this number does not depend on the radius)!!!
This is an astonishing conclusion!
It means that adding 1 m to any circumference, e.g. circumference of the sun, Jupiter or any other planet of the solar system, would raise the circumference above the surface by the same number: 16 cm.
2. Origin of the Chess Story
There’s a famous legend about the origin of chess that goes like this. When the inventor of the game showed it to the emperor of India, the emperor was so impressed by the new game, that he said to the man
“Name your reward!“
The man responded,
“Oh emperor, my wishes are simple. I only wish for this. Give me one grain of rice for the first square of the chessboard, two grains for the next square, four for the next, eight for the next and so on for all 64 squares, with each square having double the number of grains as the square before.“
The emperor agreed, amazed that the man had asked for such a small reward – or so he thought. After a week, his treasurer came back and informed him that the reward would add up to an astronomical sum, far greater than all the rice that could conceivably be produced in many many centuries!
We are all like the emperor in some ways – we find it hard to grasp how fast functions like “doubling” makes numbers grow – these functions are called “exponential functions” and are actually found everywhere around us – in compound interest, inflation, moldy bread and populations of rabbits.
3. Water lilies
Another puzzle involving exponential functions goes like this :
A particular lake has water lilies growing on it. On the first day, there is one water lily. Each day, the number of water lilies doubles. After 30 days, the water lilies cover half the lake. How long before they also cover the other half of the lake, so the whole lake is full?
The answer is both obvious and very surprising – the lilies that took 30 days to cover half the lake take only one more day to cover the other half. They fill the lake on day 31.
4. Salary for a job
I’ve prepared, here, some worksheets for kids to help them learn about this kind of amazing math. The worksheet goes with a story – a modern version, if you like, of the rice and chessboard legend. It goes like this.
You are offered a job, which lasts for 7 weeks. You get to choose your salary.
- Either, you get $100 for the first day, $200 for the second day, $300 for the third day. Each day you are paid $100 more than the day before.
- Or, you get 1 cent for the first day, 2 cents for the second day, 4 cents for the third day. Each day you are paid double what you were paid the day before.
Which do you choose?
Most people unfamiliar with this kind of dilemma will choose the first option.
- For the first choice, the person earns $2800 in week 1, then $7700, $12600, $17500, $22400, $27300 then $32200, for a grand total of $122,500
- For the doubling scheme, the person earns $1.27 in week 1, then $162.56, $20807.68, $2,663,383.04, $340,913,029.12, $43,636,867,727.36 then $5,585,519,069,102.08 for a total of $5,629,499,534,213.11.
- For the Fibonacci scheme, the totals are $0.53, $15.42, $447.71, $12,999.01, $377,419.00, $10,958,150.01 then $318,163,769.29, for a total of $329,512,800.97.