AfterMath Issue 3, 2007 - University of Western Australia · AfterMath Issue 3, 2007 4 Solving Puzzles with Mathematics: Legal Cheating? by Pantazis C.Houlis Fig. 1. The 3x3x3 cube

AfterMath Issue 3, 2007


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Contents

A few words from the editor 3 Solving Puzzles with Mathematics: Legal Cheating? by Pantazis C. Houlis 4 The Blakers Mathematics Competition 1997-2007, by Phill Schultz 7 The Blakers Mathematics Competition: Why Take Part?, by Wilson Ong 8 Useful Theorems and Methods for First-years, by Wilson Ong 10 Upcoming MS events 14


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A few words from the editor

Dear readers, Hope all your semester 1 exams went well. The Maths Society (abbreviated MS) was estab-lished back in 2002. The club aims to provide a social network for mathematics and statistics students, and AfterMath is the magazine published by the Maths Society. Our website is http://www.maths.uwa.edu.au/students/undergraduate/MS Here you’ll be able to download past issues of AfterMath and find a list of upcoming events. It has been about 3 years since the last issue of AfterMath was published as many of the active members of the club graduated from the university, and the club became dormant during those few years. But the good news is that the Maths Society is being revived again this year, and in-tends to remain a free club! The club will always welcome new members, and since member-ship is free, there’s no reason not to join! Simply subscribe to our mailing list (via the website above) to become an official member of MS, and receive notification of upcoming events. As the UWA vice-chancellor Alan Robson always likes to say, “if you leave UWA with just a de-gree, the university has failed you”. Now that the exams are over, the magazine committee has had time to put together issue 3 of AfterMath. As always, contributions to this issue are greatly appreciated. Without these contri-butions, AfterMath would not exist. Contributions to the magazine are not restricted to the magazine committee. Anyone can contribute. So if you have an idea for an interesting mathe-matical article, please do not hesitate to contact the editor: [email protected] Hope this issue will make for an interesting read over the mid-semester break. Wilson Ong Editor of AfterMath / MS Acting President


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Solving Puzzles with Mathematics: Legal Cheating?

by Pantazis C. Houlis

Fig. 1. The 3x3x3 cube

Abstract—Mathematical puzzles are always

a challenge. Some are harder than others,

and the number of possible combinations

does not necessarily reflect their difficulty

when it comes to intuition. But by using

mathematics, the most difficult part is ana-

lysing it, rather than solving it.

I. RUBIK’S CUBE: PAST, PRESENT, AND FUTURE

Mathematical puzzles always existed long ago before Erno Rubik marketed his brilliant 3x3x3 cube. A good example is the “15” slid-ing puzzle. But in the older days, the way most puzzles were made was not sophisticated enough. Moreover, many of such puzzles were based on paper notes rather than solid materials. And because of this, it was almost impossible to win the attention of those who were not involved into mathematical puzzles. In the early eighties, with the introduction of the Rubik’s Cube, the puzzle world experi-enced an amazing revolution and things were never the same again. Almost thirty years later, it is impossible to meet someone who has not played with a Rubik’s Cube.

Today, there is a new wave of puzzles coming out, some are copies of past designs. But some use some brilliant engineering concepts, and can be very impressive when seen in ac-tion. For instance, the impressive Brain-

Twist (made by Hoberman) is using in full the concept of duality for the tetrahedrons. Meanwhile, the entire puzzle community is eagerly awaiting the 6x6x6 and 7x7x7 Olym-

picubes, which will be released in the future. The architecture of the Olympicubes is simply breath-taking, enabling the construction of up to 11x11x11 cubes! And who can ignore the organic-like move-ment of one of the most ingenious puzzles of all time, the Astrolabacus? II. CONFUSING PUZZLES WHICH ARE SIMPLE AND SIMPLE PUZZLES WHICH

ARE CONFUSING Before going to the main topic, we should clear up some facts. Most of the times, it can be obvious that puzzles which look simple are simple. Interestingly, that is not always the case. The Dogic is a puzzle with the shape of an icosahedron and the number of different combinations exceed by far the ones for the Rubik’s Cube. It looks extremely intimidat-ing, but is it really that hard? If you haven’t guessed it yet, the answer is “no”, and this is because it is a puzzle which uses surface moves. An example of the other side is the Skewb

Cube. This puzzle has a very small total num-ber of combinations. Its movement and the ways it can be solved are different from Ru-bik’s Cube. But its difficulty arises from the way it functions, which can be very confus-ing.


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Overall, each person has a different way of thinking, so a puzzle which is easier for one person could be difficult for another, and vice versa...! Personally, one of my favourite puzzles is the 2x2x2 cube. The reason is, it is very deceiv-ing, and many friends and relatives of mine had claimed they can solve it at once. The end result has been very embarrassing for all of them. And even when I tell them that such a puzzle has 3.7 million combinations, they find it hard to believe it.

III. THE MYTH AND THE REALITY Out of those who had tried to solve the Ru-bik’s Cube, the big majority had failed, not because of lack of intelligence, but mainly because of lack of knowledge. Someone with analytical skills could indeed find the required moves, but it will require a fair amount of time. The biggest secret in the world of cub-ing is that anyone can solve the 3x3x3 Ru-bik’s Cube, provided that some essential se-quence of moves (algorithms) are given. It is similar to fishing. If someone is given a line and bait, it is easy to catch a fish. But a person without them, must find other ways, even if this means creating such tools from scratch. And once those algorithms are known, solving the 3x3x3 cube is as easy as memorising a few lines of text. The same principle applies to every mathematical puzzle. Those algo-rithms have been optimised throughout the years, and many speedsolving techniques had been developed, allowing competitors to solve the 3x3x3 Rubik’s Cube in just over ten sec-onds. IV. HOW TO CHEAT: A QUICK GUIDE

This guide could immediately start by stating certain keywords, such as “Group Theory”, “permutations” or “conjugates”. But let us avoid such technicalities in our attempt to

Fig. 2. The Brain-Twist

Fig. 3. The 7x7x7 cube

Fig. 4. The “15” sliding puzzle


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explain this procedure, in a way which can be understandable by all readers. When using a puzzle, there exist algorithms which can change a specific part of the puz-zle, while keeping the rest of the puzzle in-variant. For example, let us go back to the 3x3x3 cube and let us focus on its different pieces. It has three types of pieces, the cen-ters, the edges, and the corners. The centers always remain invariant, so there is no need to do anything. Regarding the edges, we only need two algorithms, one for swapping edges, and another one for fixing the orientation of the edge. The same goes for the corners, giv-ing us a total of only four required algorithms to solve the entire cube. Those four algo-rithms may be found intuitively (hard way), but they can also be found by using special-ised programs such as GAP (easy way). But the main challenge of a puzzle is actually solving it (even by using the mathematical “cheating” way), instead of using solutions made by others. Because that is when we can really understand the enjoyment puzzles can provide.

Fig. 5. The Skewb

Fig. 6. The Dogic

Fig. 7. The Astrolabacus

Pantazis has completed a Master’s Degree

in Algebraic Graph Theory, was involved

into many research projects, and now is

writing his PhD Thesis in Control Systems

Engineering. He is the founder and presi-

dent of the Mechanical and Mathematical

Puzzle Club.

http://clubs.guild.uwa.edu.au/clubs/mmpc/


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The Blakers Mathematics Competition 1997-2007

by Phill Schultz

Larry Blakers (1917-1995) was the first mathematics Honours student at UWA, and the first Australian born Professor of Mathe-matics. His academic career was interrupted by the Second World War, which broke out while he was a PhD student at Princeton Uni-versity. He enlisted in the Royal Canadian Air Force and served throughout the war as a Fly-ing Officer. It was during his sojourn in Can-ada that he met and married Terri, who still lives in Edward Street. Returning to Princeton in 1945, he earned his PhD in 1948 for re-search in Algebraic Topology. His thesis con-tains a well known theorem which bears his name. In 1952 he succeeded Charles Weatherburn in the sole Chair in Mathematics at UWA. Larry's interests then turned to organisation and education. He was largely responsible for the foundations of the Australian Mathemati-cal Society, the Mathematical Association of WA and the National Mathematics Summer School at ANU, all of which continue to flourish. He was always interested in fostering mathematical talent in high school and Uni-versity students. On his retirement as Head in 1982, the Department commissioned his por-trait which now hangs in the Common Room. When Larry died in 1995, he bequeathed $7000 to the Department which in accordance with his lifelong interests was used to support various educational activities. The most im-portant of these is the Blakers Mathematics

Competition, an annual competition for under-graduates now celebrating its tenth year. Originally open to UWA undergrads in all Faculties, it has now expanded to all WA

Universities and there is lively inter-Varsity rivalry for the major prizes. In fact, it is the only such academic competition, in any disci-pline, in WA. In standard, it is a step below the Sydney Uni-versity Mathematics Competition

(http://www.maths.usyd.edu.au/u/SUMS/) being aimed at the top 10% of students in Sci-ence and Engineering, but it does not require any mathematical background beyond 1st and 2nd year calculus, linear algebra and probabil-ity. Since the problems always include several in Applied Math, a general background in Physics or Engineering is also useful. The problems are composed by a committee of three, currently myself, Malcolm Hood, a re-tired lecturer in Applied Math, and Sam Mueller, lecturer in Statistics. Prizes are offered for the best entries from 1st, 2nd and 3rd year students, and for elegant so-lutions to the more difficult problems. The 2007 Competition is open until 4pm, Friday July 27. The Problems can be accessed by clicking on the Blakers Mathematics Competi-

tion logo on the Mathematics Department home page, http://www.maths.uwa.edu.au/.

http://www.maths.uwa.edu.au/students/undergraduate/blakers-mathematics-competition/index_html


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The Blakers Mathematics Competition: Why Take Part?

by Wilson Ong

Since 1997, the Blakers Mathematics Compe-

tition (BMC) has been held every year at UWA. (For the origin and history of the com-petition, see the article entitled “The Blakers Mathematics Competition 1997-2007” by Phill Schultz). It is free to enter and partici-pants do not have to solve all 10 problems to win prizes. The competition remains open for approximately 2 months, which allows par-ticipants to solve the problems at their own pace. I participated in the competition for the first time in 2006, during my freshman year. It was a great opportunity to exercise creative think-ing, and also be rewarded for it (the school of mathematics and statistics offer up to $300 in prizes). The more difficult problems in the BMC require extensive creative thinking to arrive at a solution. As such, participating in the competition exposes one to new ways of thinking, that most likely was not taught in high school, or even lectures. The BMC prob-lems are very different to the routine ques-tions typically asked in assignments, tests, and exams. Many people, when seeing the BMC ques-tions for the first time, may get discouraged due to the level of difficulty. However, every paper consists of some easy problems, and some more challenging ones. Those who are new to these competitions should start with an easy problem. In the 2007 paper, question 3 would be a good place to start. Admittedly some of the more challenging problems are, well, pretty challenging! Even so, I have found that solutions to most of the BMC prob-lems do not require the use of any sophisti-cated mathematical methods learned at uni-versity. In fact, many of the BMC problems

have more than one solution, and often the most elegant and beautiful solutions do not make use of advanced mathematics, but merely elementary concepts such as symme-try. With enough thought and patience, the solution to the problem will reveal itself. As for students who are able to solve all, or close to all the problems, there is a rewarding feeling of self-satisfaction and insight which helps build confidence in tackling more diffi-cult problems in the future. At the presenta-tion of prizes, you also have the opportunity to discuss and compare your solutions with other fellow participants. So over the mid-semester break, why not put the Blakers Mathematics Competition on your to do list? "The value of a problem is not so much com-

ing up with the answer as in the ideas and at-

tempted ideas it forces on the would be

solver." - I.N. Herstein "The value of an education ... is not the learn-

ing of many facts but the training of the mind

to think something that cannot be learned

from textbooks."

- Albert Einstein On the next page is a sample BMC problem (2006 paper, question 2). This problem has more than one solution. Some may make use of calculus, such as the method of Lagrange multipliers. But the solution shown here is short, simple, and only makes use of inequali-ties—a good example of how simple some solutions can be. Nothing is needed apart from lower high school maths!


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Solution by Wilson Ong—July 2006


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Useful Theorems and Methods for First-years

by Wilson Ong

This article aims to give a very brief introduc-tion to some theorems and methods which I have found somewhat useful in first-year mathematics, in particular MATH1010: Cal-culus and Linear Algebra, and MATH1020: Calculus, Statistics, and Probability. The ones that will be covered in this article include: For MATH1010: 1) Pappus’ theorem for surface areas and

volumes of surfaces/solids of revolution 2) Finding the inverse of a 3x3 matrix using

the minors matrix 3) 3x3 determinant formula for the cross

product For MATH1020: 4) Descarte’s rule of sign 5) Synthetic division for linear factors 6) Transition matrices for probability There are generalisations for some of these methods. As mentioned, this article only gives a very brief introduction to the methods. To find out more, such as proofs of some of the theorems, I suggest you do your own research (although the proof to (both parts of) Pappus’ theorem is simple and straightforward pro-vided you know how to locate a centroid). 1) Pappus’ theorem for surface areas and

volumes of surfaces/solids of revolution

These are two very simple and somewhat in-tuitive equations. Pappus’ theorem for volumes states that the volume V of the solid of revolution generated by rotating a plane region R about an axis is given by the distance traveled by the centroid of R times the area of R, provided that the axis

does not pass through R, and is coplanar with R. i.e. If r is the distance from the axis to the centroid of R, then where A is the area of region R. Example:

The volume of the torus shown below is given by since the centroid of a circle is located in its centre, and its area is given by π r 2.

Alternatively (1) may be derived by directly applying the definition/formula for the vol-ume of the solid of revolution. But this re-quires many more steps, even more if one does not recognize that the area of a semi-circle of radius a is given by the integral Alternatively (2) can also be deduced by using

rAV π2=

))(2(2

min ormajor rrV ππ= (1)

2

222 adxxa

a

a

π=−∫− (2)


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a trigonometric substitution for x. Pappus’ theorem for surface areas There is an analogous equation relating the centroid and surface area of surfaces of revo-lution: where A is the surface area of the surface of revolution, and S is the arc length of the plane curve which is revolved to generate the sur-face. As with Pappus’ theorem for volumes, the axis must not intersect the curve, and must be coplanar with the curve. Example:

The torus is the previous example will have surface area given by 2) Finding the inverse of a 3x3 matrix us-

ing the minors matrix

Typically in MATH1010, students are taught how to find the inverse of a 3x3 invertible ma-trix using row operations. However an alter-native (and perhaps faster) method for finding the inverse exists and goes as follows: (a) Find the determinant (b) Find the minors matrix (c) Find the cofactors matrix from the minors

matrix (d) The inverse is the transpose the cofactors

matrix multiplied by the reciprocal of the determinant

The process is best illustrated with an exam-ple: Suppose we want to invert A=

rSA π2=

)2)(2( min ormajor rrA ππ=

(a) The determinant is given by

This formula is easily remembered by noting that the expressions in the brackets are actu-ally the determinants of the sub 2x2 matrix after eliminating the row and column corre-sponding to the coefficient of the brackets. (b) Call the minors matrix M. Then mij is found by eliminating the ith row and jth col-umn in A, then finding the determinant of the remaining sub 2x2 matrix. So m11= m12= etc... And we eventually have M= (c) We find the cofactors matrix by simply multiplying mij by (-1)

i+j. i.e. Pointwise multi-ply M by the matrix Thus the cofactors matrix is (d) So A-1 = =

432

211

131

1

)1231(1)2241(3)2341(1

)(

)()(

2231322113

23313321122332332211

−=

×−×+×−×−×−×=

−+

−−−=

aaaaa

aaaaaaaaaaA

2341 ×−×2241 ×−×

−

−

−

215

329

102

−

−−

−

111

111

111

−−

−

−

215

329

102

−

−

−−

231

120

592

1

1

−

−−

−

−

231

120

592


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We see that the matrix we have found is in-deed the inverse since A.A-1= =I3x3 Note: The transpose of the cofactors matrix is called the Adjoint matrix. Important remark: When trying to invert a 3x3 matrix “by hand”, the minors-cofactors proc-ess can often be faster and may require less calculations than using row operations. The example above only shows the 3x3 case for a general process for inverting an n by n matrix. However it may be much more efficient to use the row operations method for matrices of size n>3. 3) 3x3 determinant formula for the cross

product

There is a bonus for remembering how to cal-culate the determinant of a 3x3 matrix! The cross product of any 2 vectors a=(a1, a2, a3), and b=(b1, b2, b3) in R

3 is given by a x b = 4) Descarte’s rule of sign

A very simple rule which tells you how many real roots a polynomial may have. The rule is as follows: Let P(x)=anx

n+…+a1x+a0 where ai is real, i=0,1,…,n. Then the number of real positive solutions to

the equation P(x)=0 is equal to the number of times that the signs of 2 consecutive coeffi-cients (ai and ai-1, i=1,2,…,n) strictly change (i.e. ai.ai-1 < 0); or less than that by a multiple of 2. Note: Descarte’s rule of sign counts roots with multiplicity>1 as separate roots. Example:

The polynomial P(x) = x5 – x4 + 3x3 + 9x2 – x + 5

may have 4, 2 or 0 real positive roots because the coefficients change sign 4 times (between +x5 and -x4, -x4 and +3x3, +9x2 and -x, -x and +5). Also the graph of P(-x) is that of P(x) re-flected about the vertical axis, so applying Descarte’s rule of sign to

P(-x) = – x5 – x4 – 3x3 + 9x2 + x + 5 we find that P(x) must have exactly 1 real negative root (since we cannot have a nega-tive number of roots!). 5) Synthetic division for linear factors

Suppose you want to divide a linear factor into a polynomial of degree 1 or higher. (Often in MATH1020 you may need to do this to simplify an integrand). This can be done “by hand” using long division. However a faster algorithm known as synthetic division might save you a bit of pen ink (and time!). The synthetic division algorithm is best shown with an example: Divide x − 2 into x3 − 5x2 + 3 x − 7 (Note: The coefficient of x in the linear factor must be 1 before applying the algorithm.) We first write out the coefficients of the divi-dend in a row, and the root of the linear factor in a box (to distinguish it from the coeffi-cients): 1 -5 3 -7

432

211

131

−−

−

−

231

120

592

321

321

bbb

aaa

kji

2


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Drop the first number down to the 3rd row: 1 -5 3 -7 1 (a) Multiply the rightmost number in the 3rd

row by the boxed number, and put the re-sult in the 2nd row of the next column:

1 -5 3 -7 2 1 (b) Add up the 2 numbers in that column, and

put the result in 3rd row of that column: 1 -5 3 -7 2 1 -3 Now repeat steps (a), and (b) until the 3rd row is filled: 1 -5 3 -7 2 -6 1 -3 1 -5 3 -7 2 -6 1 -3 -3 1 -5 3 -7 2 -6 -6 1 -3 -3 1 -5 3 -7 2 -6 -6 1 -3 -3 -13 The coefficients of the quotient and remainder are shown in the 3rd row. So If the coefficient of the linear factor is not 1,

2

2

2

2

2

2

2

2

1333

2

735 223

−−−−=

−

−+−

xxx

x

xxx

we can re-write ax+b as before us-ing synthetic division. 6) Transition matrices for probability

A typical elementary conditional probability problem might be: “If it rains on any day, there is a probability

of 1/3 that it will rain again the next day. If it

does not rain, there is a probability of 1/4 that

it will rain the next day. If it does not rain to-

day (say, Monday), what is the probability it

will rain 3 days later (Thursday)?”

This problem can easily be solved by using a probability tree:

etc… By considering all the paths which lead to rain on Thursday yields P(rains on Thurs)= We can arrive at the same answer by using a probability transition matrix:

=P

Then the probabilities of rain n days after any

+a

bxa

Rain No Rain

No Rain

1/4 3/4

1/4 3/4 1/3 2/3

Rain No Rain Rain No Rain

Tues

Mon

Wed

576

157

4

1.

4

3.

4

3

3

1.

4

1.

4

3

4

1.

3

2.

4

1

3

1.

3

1.

4

1=+++

Initial State

Final State

4/33/2

4/13/1

Rain No Rain

Rain

No Rain


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given day is given by P n. So the probabilities for Thursday are given by P cubed:

=P3

As before, if it did not rain today, the prob-ability of it raining three days later is 157/576.

Initial State

Final State

576/419216/157

576/157216/59

Rain No Rain

Rain

No Rain

Why this works: For each additional day in question, imagine adding 2 more branches at the top of the tree (thereby increasing the height of the tree by 1) with the node of each new branch being the entire old tree. Then consider how matrix multiplication is done. Probability transition matrices can be used in similar problems to this one, even if the tree is non-binary. The ideas mentioned here follow from a more general concept known as Markov chains.

Upcoming MS events

No social events were organized for semester 1, 2007 due to the small number of members. But now that the club has a considerably larger membership base, we can expect a reasonable turn-out at events. So the events/meetings we have scheduled for semester 2, 2007 include: - A general BBQ - Meet the other members of MS. - The Annual General Meeting (AGM) - elections for committee positions (if you are inter ested in running for a position, please contact the MS Acting President. You don’t have to be a member of the guild!) - AfterMath Writers Meeting - to discuss ideas and articles for issue 4 of AfterMath. - A Numb3rs Screening - Organised by Shreya. More information about the events will be sent out via the MS mailing list closer to the time. Please feel free to bring all your friends to these events as we are always keen to increase our membership base!

Jokes

High school teacher: “Can anyone in the class tell me what Linear Algebra is?” Student: “It is doing algebra on linear expressions. For example a(bx+c)=abx+ac” Theorem: A cat has nine tails. Proof: No cat has eight tails. Since one cat has one more tail than no cat, it must have nine tails. “Divide fourteen sugar cubes into three cups of coffee so that each cup has an odd number of sugar cubes in it.” “That’s easy: one, one, and twelve.” “But twelve isn't odd!” “It's an odd number of cubes to put in a cup of coffee…”

Documents

AfterMath Issue 3, 2007 - University of Western Australia · AfterMath Issue 3, 2007 4 Solving Puzzles with Mathematics: Legal Cheating? by Pantazis C.Houlis Fig. 1. The 3x3x3 cube