Chukky problems

CHUKKY PROBLEMS

These problems are geared to show an individual the beauty, the power, the implication and the

ramification of science and mathematics. The problems are not calculation intensive. Rather the questions

are aimed at highlighting a general misconception among people about some scientific concepts; instigate

arguments concerning the result of experiments; illustrate how simple understanding of scientific laws

can solve complex problems; show the brainchild behind some famous mathematical proofs. Some of the

questions also employ more curious scenarios which help to highlight ones level of understanding of a

scientific concept.

Like every classical problem there is a gift for this challenge. I am awarding a free copy of the renowned

engineering and science utility software MATHLAB 2009 which normally costs over 500 dollars for

college students. All you need to do is answer just one question that applies to you category. The

categories are listed in front of the questions. You have – elementary school, Junior High School, Senior

High School, College, and All Levels. As one question pertinent to your category.

Some of the questions may be found online through search engines but it would be of no good to yourself

if you consult such sources. Challenge yourself with the problem, and then after much research, and

challenge, you may consult search engines if you give up. But make a concerted effort to solve the

problems yourself. The whole idea is to educate and not necessarily challenge. So feel free to consult for

help if need be.

If you have any questions, objections, suggestions or ideas about the listed problems below please feel

free to contact me at pjz081@my.utsa.edu. I would be more than excited to respond. Also if you have any

questions which you feel are challenging or inspiring like the questions below please do contact me and

we can compile and update them into the questions below. Yeah am Mechanical Engineering major so it

is pretty much math and physics but please we welcome all sides of science. As long as your questions

can be very challenging please don’t hesitate to drop by my email box. Thank you. Have fun and enjoy.

“Science, an interesting epiphany of creation.”

Answers to the questions should be posted on Scholars Assembly wall in order for other members of the

group to learn from your genius and knowledge. If you are not on Scholars Assembly we employ you to

join us but if you do not want to you can paste your answer on your Facebook wall mentioning my name

in it to draw my attention. Since it is mathematics and typing answers would be difficult to write on a

Facebook post or to write even in MS word, you may right it in your hand writing and upload it on

4shared and then share the link with us. If you have difficulty doing that, just send me the answer as an

email and with your permission I shall publish your answers.

I am still pondering on what gift to award our young scientist in secondary and elementary school who

answer any of these problems. I would be so privileged to have come across such a genius in my life time.

If you have any suggestion for what a good academic gift can be please let me know by emailing me.

Your humble servant,

Chukky

Question 1 – Elementary School/ Junior High School

Question Origin - From a Facebook post by Tommy Lash

Did you know that the L.C.M of two numbers {x and y} multiplied by the H.C.F of the two

numbers {x and y} is equal to the product of x and y. Can you tell me why this is so?

That is can you show me why this happens. Note as an elementary school student, I do

not expect a formal mathematical proof from you. Just tell me in your own words why

this happens?

Question 2 – Elementary School/ Junior High School

Question Origin – From the tricks my lesson teacher in elementary school taught me. My

classmates were always awed at how fast I did this kind of calculations in my head.lol. I

had a trick.

Did you know that 11 x 14 = 1(1+4)4 = 154; 11 x 16 = 1(1+6)6 = 176;

Like the first question above, I do not need a formal proof I just want to know why this

happens. This rule breaks down when it comes to numbers ending with “9” what

adjustment is needed to make to correct it.

Question 3 – Senior High School

Question Origin – Friend said, “I thought 0/0 is the same as 1/0”

They are clearly not the same. One is called undefined the other is called indeterminate. Only

by checking the dictionary meaning of these words one can see what the difference is.

But now the question is this, use two examples to illustrate why they are different.

Question 4 – Senior High School/College

Question Origin-Entry Exam Indian Institute of Technology / Putnam Competition

Solve the following integral- Integrate [x4(1 – x)4]/(1 + x2)from zero to 1? Write your answer

in terms of π. Plot a graph of the above equation with a computer or graphing calculator?

Shade the area between zero and 1. After you have done all this you should make a very

remarkable discovery? What discovery did you make? It took Archimedes years to make

the same discovery; it is going to take you only minutes to make this discovery because

of your knowledge for calculus which was absent in the time of Archimedes. Aren’t you

special? lol

Question 5 – College

Question Origin – From a mistake in the Engineering textbook Engineering Mathematics by

KA Stroud. Yeah, if you get this one, you can write KA Stroud a letter telling him about

the mistake of his Maclurin series.

1/(1-x) = 1 + x + x2 + x3 + x4 ……… Inserting x=2 gives 1/(1-2)= -1 but if you put it into the

Maclurin series you get a huge number that never converges. What is the problem with

this series as it is so defined. What is missing?

Question 6 – Senior High School/College

Question Origin – Observation.

The next time you enter an examination without a calculator and you want to find the sine of

a small angle (in degrees) to approximately 3 significant figures, can you show that sin

(x) = 0.0175x for x in degrees. When you proof this you would find out that 0.0175 is

only a 3.s.f approximation of a constant. What is that constant?

Question 7 – Junior High School

Question Origin: Famous “Mathemagic Puzzle” by witches and wizard in ancient Greece

Why is it that you cannot square a circle, that is draw a circle that has the same area as a

square. Or why is it that you cannot half the volume of a cube? Lol the day you do any of

this please let me know so we can go claim the prize for doing the impossible. lol.

Question 8 – College

Question Origin - Mathematics Teacher, November 1987

S = 1- 1/2 + 1/3 - 1/4 + 1/5 – 1/6 + 1/7 – 1/8 + 1/9 – 1/10 +1/11 – 1/12 ……….

Multiply both sides by 2

2S = 2 – 1 + 2/3 - 1/2 + 2/5 – 1/3 + 2/7 – 1/4 + 2/9 – 1/5 + 2/11 – 1/6………..

Now add the 1st term above to the 2nd, the 3rd to the 6th , the 5th to the 10th, the 7th to the 14th

and so on following the pattern of addition given above:

Now if you added as above you shall get the following:

2S = 1- 1/2 + 1/3 - 1/4 + 1/5 – 1/6 + 1/7 – 1/8 + 1/9 – 1/10 +1/11 – 1/12……….

Therefore 2S = S

2 = 1 (This reminds me of love saying that 1 + 1 = 1 not 2.Even the fractions

above are in love. lol) So how is this possible. How can 2 = 1 as seen above.

Question 9– College (Engineering /Physics)

Question Origin – Laplace Equations/ Potential Theory

Why are the following conditions so crucial to the existence of the fundamental forces of

nature like gravity, electromagnetic forces, and even nuclear forces,hence life itself:

Divergence of Vector = Zero; Curl of Vector = Zero. Hint remember that these forces have a

field associated with them and whenever you hear field remember flux and whenever you

hear flux remember the Prince of Mathematics – Carl Gauss, his flux theory.

Question 10 – College (Engineering/Physics)

Question Origin: Ampere’s Law

There is a big debate as to whether magnetic force is conservative or not. Well, though this

debate exists, how is it that Ampere’s law is obeyed. It requires the satisfaction of the

Laplace equation for Ampere’s law to be valid and a condition of the Laplace equation is

the conservative nature of the analyzed vector. So what how is it that there is a debate

about the conservative nature of the magnetic field. Please do note that the Amperes

equation in this case is the pristine form of Ampere’s equation with not Maxwellian

correction that is no inclusion of displacement current. To simplify the question, assumes

Ampere’s law for a hypothetically infinitely wire carrying a current through it. Use an

analysis of a loop around this wire to answer the question.


Question Origin - Friction

They say friction is a force that acts in the opposite direction. OK, fine. Let’s imagine we are

pulling a box across a rough surface. As we push the box we feel friction pulling back

against us right. If I stop pushing, then friction continues to act against the moving truck

until it stops right? Now why doesn’t the box stop and start moving in the opposite

direction since friction acts in the opposite direction. Or am I wrong can this happen?

Have you ever rolled a ball across the ground and it stops and starts rolling back. Runfor

your life if this shouldever happen. lol. Does this remind you about a famous law in

physics?


Question Origin – Friction

How does a car move? That is, how does a tyre rolling on the ground move forward? What is

the force that pushes a tire forward as it rolls on a ground? Remember a free rolling tyre.


Question Origin – The nature of Force

1. This question would help you answer the second. When you do work to an object in a

conservative field, where is the energy stored? That is where is the work that you did on the

object stored? In a non-conservative field like imagine pulling a box on a floor in the

presence of friction, where does the work you did on the box go to? What is the difference in

the two cases? Are there really non conservative force? Conservation means that energy done

some where is transformed elsewhere, so is it right to say for example, friction is non-

conservative? This highlight the quote by the great 21st century physicist Richard Feynman

who said no force is non-conservative. All forces are conservative. So can you figure out

what is wrong with classifying forces as conservative and others as not conservative? A hint:

Should it not be called mechanically conservative/non-conservative?

2. Isn’t the Work-Energy Theorem a misnomer? If I move a particle from one point to

another in a potential field (be it electrical or gravitational) with a uniform speed, I did

work but the Kinetic energy did not change. But I thought the Work Energy Theorem

said Work = change in Kinetic energy. Here there was no change in kinetic energy

(uniform velocity) but we did work.

3. If I get two toy cars and I attach very powerful magnets of opposite polarity into them and

then I drive them into each other. Since the magnets are powerful, does the collision

cause a deformation on the cart? If you say yes doesn’t that mean that when objects are

pulled by the earth’s gravitational fieldthat they undergo deformation? If you still concur

yes again does that not then mean that energy is lost when objects fall from one point to

another in a gravitational field in the absence of any external influence other than gravity

like air, wind or other forces? If this does not make sense to you and you say no then

explain why we would not have a deformation. Remember this, for a reaction to occur,

entropy must not decrease (2nd law of thermodynamics). Argue for which side is right.

Does the answer to this question remind you of the LHC (the Large Hadron Collider). Do

you now see what they are trying to do in that never-done-before-experiment, the LHC?

That’s the experiment that found neutrino’s moving faster than the speed of light(as they

claim)

Question 14 – College (Engineering/ Physics)

Question Origin: Rotational Mechanics is different from Translational Mechanics but ninety

percent of people do not realize this and hence you have errors in physics textbook like,

the reason why the earth momentum is conserved is that the sun is too large to move so

only the earth moves but that a false answer that seems to be true. So to expose the

falsehood of such an answer, an answer given by many physics textbooks and articles,

look at the scenario below:

A boy ties a tennis ball with a thin string to winding device. He initially sets the string into a

given angular momentum. The experiment takes place in space so no friction and angular

momentum is constant. Then the winding device which is fixed to a support is turned on

and begins to wind the string in thereby reducing the length of the string. This problem is

usually solved by conservation of angular momentum. But here is the question. If you say

that momentum is conserved then how is it that this objected changed angular velocity?

Hint: Does it take an external torque to change angular velocity? Remember,the rate of

change of linear velocity is acceleration (in non-relativistic speeds, that means high speed

so we avoid special relativity effects) but is the rate of change of angular velocity angular

acceleration (non-relativistic as well)?

Question 15 – College (Mathematics)

Question Origin – Function Approximation

Remember how Maclurin and Taylor impressed the world by differentiating power series

particularly power polynomials to derive polynomial approximations for transcendental

functions. Now Fourier says I can even do better and yes he did. Maclurin series can’t work for

functions whose derivatives are undefined but Fourier’s series can. This is due to the fact that it

uses integration as opposed to differentiation to derive its function approximation. Show how

Fourier series is an integral analogue of its differential counterpart Taylor/Maclurin Series? Can

you also explain why Fourier series is an edge over Taylor/Maclurin Series when it comes to

approximating discontinuous functions? Hint: remember – indefinite integrals.

Question 16 – College (Mathematics)

Question Origin: A friend wanted a straight forward proof.

Can you show that e is a limit? That is, derive the limit “e”. Show that it is a limit and that this

limit e is associated with the natural antilogarithm.

Question 17 (Secondary School)

Question Origin: 25 x 25 = (2x3)(5 x 5) = 625 ; 35 x 35 = (3 x 4)(5 x 5) = 1225; 45 x 45 = (4 x

5)(5 x 5) = 2025

Surprised? It does not end. Watch this 37 x 33 = (3 x 4)(7 x 3) ; 48 x 42 = (4 x 5)(8 x 2) = 2016

The question is this; prove the above phenomenon is valid? Use your proof to explain or show

why this occurs?

Question 17 – Open to everyone scientist or non-scientist, to every level

Question Origin - From a puzzle from the book “Compendium of Conundrum”

A group of birds are in a huge air tight cage, and you put the birds on a weight scale. After you

pass some current on the ground so that the birds would get electrocuted if their feet touch the

ground so they end up flying in the cage (what a cruel experiment!). Now when they are flying in

the cage, would the reading on the weight scale change compared to when the birds were all

standing on the cage. Does it change or not make an argument. Hint; remember the cage is air

tight .

Question 18 – Open to all levels

Question Origin - A joke by a physics professor about a dummy who says that 64/16 = 4/1

because he cancels 6 in the numerator and denominator of the fraction.

While this is clearly not the way to do division, lol, can you list other scenarios where this funny

incidence occurs involving numbers less than 100. You can skip cases like 80/10 = 8/1 (cancel

the zeros) or 70/10 = 7/1 (cancel the zero) skip those cases which are very obvious. Hint: A

computer scientist would find the answer to this problem in no time with a program. Perhaps you

could ask for their help here. Its an easy program anyway.

Question 19 – Open to all level involves simple logic

Question Origin – Legends from famous ancient mathematics challenge.

You are asked to mathematically proof that a statement or axiom is mathematically invalid. What

is the fastest approach to making such a proof? As in you just need to show something and you

proof at once that that theory, or axiom is invalid.

Question 20 – Senior High School

Question origin – A funny surprising 2009 WAEC question

Using planks formula E = hf and Einstein’s mass-energy equivalence formula E = mc2 can you

derive De – Broglie’s wavelength equation wavelength = h/p where h in the above is planks

constant and p stands for momentum of quantized entity. I would like to remind you that this

connection between these equations was a coincidence and De Broglie did not just derive his

equation by simply manipulating Einstein’s and Plank’s formulas.

Goodluck.

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