BR Maths Corner-1

[Eric Demopheles]

Here's a puzzle discussed at Terry Tao's blog:

http://terrytao.wordpress.com/2008/02/0 ... rs-puzzle/

Solutions and detailed arguments are in the comments.

[Eric Demopheles]

I found it in the interwebs. The solution is quite simple. But it's fun to try nevertheless. It is important to do the puzzles in the order. Puzzle 1 helps in obfuscating the mind as preparation for the second.

Puzzle 1: Grandmother made a cake whose base was a square of size 20 by 20 cm and the height was 10 cm. She wanted to divide the cake fairly among her 8 grandchildren. How should she cut the cake?

Puzzle 2: Grandma made a cake whose base was a square of size 20 by 20 cm and the height was 10 cm. She put chocolate icing on top of the cake and on the sides, but not on the bottom. She wanted to divide the cake fairly among her 8 grandchildren so that each child would get an equal amount of the cake and the icing. How should she cut the cake?

[Nandu]

Puzzle 2: Grandma made a cake whose base was a square of size 20 by 20 cm and the height was 10 cm. She put chocolate icing on top of the cake and on the sides, but not on the bottom. She wanted to divide the cake fairly among her 8 grandchildren so that each child would get an equal amount of the cake and the icing. How should she cut the cake?

cut along the two aces and the two diagonals

[Nandu]

Here's a puzzle discussed at Terry Tao's blog:

http://terrytao.wordpress.com/2008/02/0 ... rs-puzzle/

Solutions and detailed arguments are in the comments.

I haven't looked at the solution, and I don't know how to tackle this, but I just wanted to point out that the puzzle belongs to the general class of induction puzzles. http://en.wikipedia.org/wiki/Induction_puzzles

[Eric Demopheles]

Puzzle 2: Grandma made a cake whose base was a square of size 20 by 20 cm and the height was 10 cm. She put chocolate icing on top of the cake and on the sides, but not on the bottom. She wanted to divide the cake fairly among her 8 grandchildren so that each child would get an equal amount of the cake and the icing. How should she cut the cake?

cut along the two aces and the two diagonals

Darn! 8 grandchildren was not a good number! I should have put 9 or 11, or some such number of grandchildren. But, say, want to try again for those numbers of grandchildren?

[Nandu]

Darn! 8 grandchildren was not a good number! I should have put 9 or 11, or some such number of grandchildren. But, say, want to try again for those numbers of grandchildren?

Mark the center. Then just pick nine points on the perimeter that are equidistant from each other. i.e. pick any point, then pick a point 4 x 20 / 9 cm from it along the boundary, and so forth until you come all the way around. It doesn't matter where you pick that first point. Now, from the center, cut a straight line to each of these points, and you will get nine pieces, all with same amount of cake and icing. Proof: We only need to consider the square cross section and show each piece has same area and same share of the outer perimeter. The latter should be obvious from the way we picked the points. For area, notice that area of triangle is half base times altitude, and altitude is always 10cm for us, irrespective of the shape of the piece, and the base is the same as the part of the piece that is on the perimeter of the square.

[Nandu]

Here's a puzzle discussed at Terry Tao's blog:

http://terrytao.wordpress.com/2008/02/0 ... rs-puzzle/

Solutions and detailed arguments are in the comments.

I haven't looked at the solution, and I don't know how to tackle this, but I just wanted to point out that the puzzle belongs to the general class of induction puzzles. http://en.wikipedia.org/wiki/Induction_puzzles

Hmm... turns out the solution is not much different from the Josephine puzzle. The interesting part is why it took the foreigner saying something to trigger it, but when you think about it, substituting small numbers instead of n=100, it becomes obvious.

[Eric Demopheles]

Good work, Nandu.

Next is a lesson in plausible deniability, a notion much discussed in this board. I found the story from the wikipedia page for "Supertask". Might give the strategists some ideas.

A man called Prometheus angers Zeus, so Zeus gathers an infinite number of demons and issues them with the following commands. Demon 1: if Prometheus is not dead in one hour kill him, Demon 2: If Prometheus is not dead in the half an hour kill him, Demon 3: if Prometheus is not dead in quarter of an hour kill him, and so on. As it turns out Prometheus was dead within the hour (as he didn’t really have much chance). The council of gods was not happy about Zeus's killing of Prometheus and pressed Zeus on the point. But on the question of whodunit, none of his demons could be found guilty, as, for each positive integer n, it was not possible for the nth demon to have killed Prometheus because the (n + 1)th demon should already have done so.

A variant, in a civilian setting:

A hotel has infinitely many rooms 1,2,3,..... etc. Every room is full when a new guest arrives. The clerk has the following brilliant solution to ensure that the newcomer as well as all the current occupants are provided accommodation. He moves the occupant of room n to n+1 to make room for the new guest in room 1. So far so good. But now comes an onslaught(perhaps engineered by the devil). An hour later another guest arrives and the clerk repeats the process. 30 minutes later a third guest arrives and the process is repeated. Then 15 minutes, 7.5 minutes, etc., until two hours after the first new guest infinitely many guests have arrived and been accommodated. The clerk is very pleased with himself for dealing with these infinitely many guests, when he notices to his horror that all the rooms are empty! All the guests have mysteriously disappeared!

[Eric Demopheles]

Hmm... turns out the solution is not much different from the Josephine puzzle.

What is the Josephine puzzle?

[Eric Demopheles]

Next one.

Simplify (x-a)(x-b)(x-c).....(x-z).

[Jayram]

Hah x-x !!

[Nandu]

Hmm... turns out the solution is not much different from the Josephine puzzle.

What is the Josephine puzzle?

It is listed in the induction puzzles wiki page I linked above.
Also known as the cheating husbands puzzle, or the cheating wives puzzle, it is sometimes discussed in distributed computing courses as an example of acquiring "common knowledge". Goes like this:


In a kingdom, all men and women are married. One day the queen (Josephine) announces "At least one man among you is cheating. If a woman discovers that her husband is cheating, she should shoot him at midnight following her discovery. You may not discuss your husbands amongst yourself". The kingdom is an island small enough that a gunshot anywhere will be heard by everybody. Assuming a) there are no gunshots unrelated to this puzzle, b) all women are perfect logicians, c) all women know about the faithfulness of every man except her own husband, and d) there are exactly 100 cheating husbands, what happens on the midnight following the announcement, the midnight after that, and so on....


Interestingly, the wiki page on common knowledge contains exactly the blue eyed islander puzzle.

http://en.wikipedia.org/wiki/Common_knowledge_(logic)

Here is cheating husbands discussion from a CS course: http://www.cnds.jhu.edu/courses/cs437/Week12/sld005.htm

[Nandu]

A hotel has infinitely many rooms 1,2,3,..... etc. Every room is full when a new guest arrives. The clerk has the following brilliant solution to ensure that the newcomer as well as all the current occupants are provided accommodation. He moves the occupant of room n to n+1 to make room for the new guest in room 1. So far so good. But now comes an onslaught(perhaps engineered by the devil). An hour later another guest arrives and the clerk repeats the process. 30 minutes later a third guest arrives and the process is repeated. Then 15 minutes, 7.5 minutes, etc., until two hours after the first new guest infinitely many guests have arrived and been accommodated. The clerk is very pleased with himself for dealing with these infinitely many guests, when he notices to his horror that all the rooms are empty! All the guests have mysteriously disappeared!

Yeah, he should have moved each guest from room n to room 2n, and given the m-th new guest room 2m-1.

[Eric Demopheles]

Yeah, he should have moved each guest from room n to room 2n, and given the m-th new guest room 2m-1.

Well, here you first create an infinite number of vacant rooms. Why would you do that, unless you were knowing in advance about the infinite number of incoming guests?

Anyway, here's yet another deal with the devil, from the paper arXiv:math.LO/9808093.

In a more entertaining example, let’s suppose that you have infinitely many one dollar bills (numbered 1, 3, 5, . . .) and upon entering a nefarious underground bar, you come upon the Devil sitting at a table piled high with money. You sit down, and the Devil explains to you that he has an attachment to your particular bills and is willing to pay you a premium to buy them from you. Specifically, he is willing to pay two dollars for each of your one-dollar bills. To carry out the exchange, he proposes an infinite series of transactions, in each of which he will hand over to you two dollars and take from you one dollar. The first transaction will take 1/2 hour, the second 1/4 hour, the third 1/8 hour, and so on, so that after one hour the entire exchange will be complete. The Devil takes a sip of whiskey while you mull it over; should you accept his proposal? Perhaps you think you will become richer, or perhaps you think with infinitely many bills it will make no difference? At the very least, you think, it will do no harm, and so the contract is signed and the procedure begins. How could the deal harm you?

It appears initially that you have made a good bargain, because at every step of the transaction, he pays you two dollars and you give up only one. The Devil is very particular, however, about the order in which the bills are exchanged. The contract stipulates that in each sub-transaction he buys from you your lowest-numbered bill and pays you with higher-numbered bills. Thus, on the first transaction he accepts from you bill number 1, and pays you with bills numbered 2 and 4. On the next transaction he buys from you bill number 2 (which he had just paid you) and gives you bills numbered 6 and 8. Next, he buys bill number 3 from you with bills 10 and 12, and so on. When all the exchanges are completed, what do you discover? You have no money left at all! The reason is that at the nth exchange, the Devil took from you bill number n, and never subsequently returned it to you. So while it seemed as though you were becoming no poorer with each exchange, in fact the final destination of every dollar bill in the transaction is under the Devil’s ownership. The Devil is a shrewd banker, and you should have paid more attention to the details of the supertask transaction to which you agreed. And similarly, the point is that when we design supertask algorithms to solve mathematical questions, we must take care not to make inadvertent assumptions about what may be true only for finite algorithms.

[Eric Demopheles]

Yes, and the proof is really very simple, suppose if (p1,p2,p3,.... p_n) are the only prime numbers.. take $(p1)(p2)(p3)..(p_n)+1$ is either prime, or not divisible by any of the primes above (it will leave a remainder or 1) .. in any case you found a prime larger than p_n ...so you found 1 more... QED.

How about this slight variation:

Suppose if p1, p2, p3 ,.... ,p_n are the only prime numbers. Consider $(p1)(p2)(p3)..(p_n)+1$ and consider its prime factorization. Since the previous list consists of all possible primes, some prime p_i with i between 1 and n, would divide this number. But then p_i would divide their difference, ie p_i would divide 1, which is absurd.

[Eric Demopheles]

Here's a nice classroom experience story, based on the principle that a "generic" matrix has zero oops, sorry, nonzero determinant:

http://www.thehcmr.org/issue1_1/gaitsgory.pdf

[Nandu]

So, was it because it was "generic" or because it had a particular structure that made the columns linearly dependent?

[Eric Demopheles]

So, was it because it was "generic" or because it had a particular structure that made the columns linearly dependent?

Sorry, I made a mistake above. I had meant to write that a generic matrix had nonzero determinant.

[svenkat]

Hey Amber G.

Can you please suggest some magazines which publish articles (or book reviews) about recent changes in Mathematics for non-experts like myself (I am a science graduate but not a Mathematician).

I have noted that the Science and Nature magazines do not publish much about Mathematics.

Thanks.

Many many years ago,I asked this question to a mathematics guru.He said 'mathematical intelligencer'.I never went anywhere near it.Maybe it will interest you.

[abhishek_sharma]

Many many years ago,I asked this question to a mathematics guru.He said 'mathematics intelligencer'.I never went anywhere near it.Maybe it will interest you.

Thanks.

[Eric Demopheles]

some magazines which publish articles (or book reviews) about recent changes in Mathematics for non-experts like myself (I am a science graduate but not a Mathematician).

mathematical intelligencer

Try "Notices of the american mathematical society" also, available online at:

www.ams.org/notices/

[abhishek_sharma]

^ Okay. Thanks.

[wig]

fascinating reading:
Large cardinals: maths shaken by the 'unprovable' A shocking discovery has unsettled the world of numbers, says Richard Elwes.
source:
http://www.telegraph.co.uk/science/8118 ... vable.html

the esoteric world of mathematical logic, a dramatic discovery has been made. Previously unnoticed gaps have been found at the very heart of maths. What is more, the only way to repair these holes is with monstrous, mysterious infinities.

To understand them, we must understand what makes mathematics different from other sciences. The difference is proof.

Other scientists spend their time gathering evidence from the physical world and testing hypotheses against it. Pure maths is built using pure deduction.

[Eric Demopheles]

maths shaken by the 'unprovable'

This matter of "unprovability" and its ramifications has affected philosophy more than mathematics itself, I would say. Kurt Goedel proved that it is not possible to prove consistency of systems like arithmetic, and thus, most of mathematics. A concrete example of such an "unprovable" statement is Goodstein's theorem: http://video.ias.edu/voevodsky-80th . This, though appearing unintuitive at first, becomes more credible after a few low numbers are tried out.

Here is an excellent video by Vladimir Voevodsky at Princeton Institute for Advanced Studies, on the "unprovability" issues of mathematics:

http://video.ias.edu/voevodsky-80th

Hope it piques your curiosity.

[wig]

Infosys prize for Chandrashekhar Khare, the maths wizard
The Infosys Award carries the highest cash prize for scientific research in India — Rs 50 lakh for each of the winners. The award is an annual feature that recognises outstanding contributions by scientists, researchers, engineers and social scientists in India. By recognising and rewarding extraordinary accomplishments, the award aims to elevate the prestige of pure and applied sciences research in India.

http://www.tribuneindia.com/2010/20101114/edit.htm#4

[Amber G.]

Infosys prize for Chandrashekhar Khare, the maths wizard

http://www.tribuneindia.com/2010/20101114/edit.htm#4

^^^Congrats. Nice to see that Indian Newspapers covered this story ..About 2 weeks ago, ...Khare's award was mentioned in this BRF thread too. ( with a comment " No one seem to have mentioned it anywhere in BRF . MMS will personally award this to prof Khare in coming January.

Others to be honored with this prize are:
Sandip Trivedi, (TIFR) (prize in the Physical Sciences category - superstring theory)

Ashutosh Sharma (IITK) (Engineering and Computer Science for his fundamental contributions in materials science).

Chetan E. Chitnis ( Malaria Group, (ICGEB)) (Life Sciences - viable malarial vaccine.)

Amita Baviskar ( Institute of Economic Growth) and Nandini Sundar (Delhi School of Economics) (Social Sciences)

[Amber G.]

Wrt Khare - He is well known for proving Serre conjecture which BTW also proves the famous Fermat's last theorem.

In honor of number theory and his work, here is a small problem:
(Actually this comes from Brahmagupta's Brahma-sphuta-siddhanta - Translating and putting it in modern terms)

Given, both m and n are natural numbers (positive integers) and:

m = 2 + 2 * sqrt ( 28*n^2 + 1)

prove m is a perfect square.

(May or may not be a hint: Try to find some values of n where m is a natural number too.)

[Amber G.]

Folks if you don't think math is beautiful...
Take a=15.9372539
Take the integer closest to a, a^2, a^3 etc ...
(we get 16, 254, 4048 etc...)
squares of these numbers satisfy m in the problem above. (n will come out, integer too)
(for example we have 16^2=256=2+2sqrt(28*24^2+1))

Why does this happens? (what is special about a?)

[Nandu]

Shouldn't that wiki entry say Khare's Theorem now?

[Amber G.]

^^^ IMO Khare's work proves the conjecture proposed (and thus as "Serre's Conjecture" ) by Serre..It also deals with (proves), say "Taniyama-Shimura conjecture", deals with "Galois (French Math genius) representations of ..." etc.... all those terms are well known so it is okay... (It is similar to Grigory Perelman proving Poincaré conjecture... remember all the excitement in the world..)

Impressive part is Chandrashekhar Khare published a proof of the level 1 case of Serre conjecture,(2005) and later (2006) proof of the full conjecture (linear algebraic groups) in collaboration with Jean-Pierre Wintenberger. Pretty impressive. May be the theorem be called after Khare (and(?) Wintenberger).. ( For example, there is a theorem many refer as "Quillen–Suslin theorem" which proves Serre's conjecture - (and hence, still the proof is some times referred as "Serre's conjucture) ... BTW objects use of "his" name with this as according to him he did not proposed the specific conjucture etc.
.
Serre BTW Serre also is quite gifted (honored with Fields Medal etc) and worked expertly in Galois representations, linear algebraic groups and many other fields..

[ramana]

AmberG, I would like you to write occassional general interests posts or short papers introducing high fund maths for lay people. For Eg what do linear lagebraic groups do?

If you want I'll post some ocassional queries?

[chetak]

I am not of any great religious bent of mind but I thought that this was remarkable. In another version the word BULLSHIT had a numerical value of >100

The Beauty of Mathematics and the Love of God! This is cool!

I received this e-mail and thought it was pretty cool! Keep scrolling it gets better. The Beauty of Mathematics! !!!!!!

1 x 8 + 1 = 9

12 x 8 + 2 = 98

123 x 8 + 3 = 987

1234 x 8 + 4 = 9876

12345 x 8 + 5 = 98765

123456 x 8 + 6 = 987654

1234567 x 8 + 7 = 9876543

12345678 x 8 + 8 = 98765432

123456789 x 8 + 9 = 987654321

1 x 9 + 2 = 11

12 x 9 + 3 = 111

123 x 9 + 4 = 1111

1234 x 9 + 5 = 11111

12345 x 9 + 6 = 111111

123456 x 9 + 7 = 1111111

1234567 x 9 + 8 = 11111111

12345678 x 9 + 9 = 111111111

123456789 x 9 +10= 1111111111

9 x 9 + 7 = 88

98 x 9 + 6 = 888

987 x 9 + 5 = 8888

9876 x 9 + 4 = 88888

98765 x 9 + 3 = 888888

987654 x 9 + 2 = 8888888

9876543 x 9 + 1 = 88888888

98765432 x 9 + 0 = 888888888

Brilliant, isn't it?

And look at this symmetry:

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321

1111 x 1111 = 1234321

11111 x 11111 = 123454321

111111 x 111111 = 12345654321

1111111 x 1111111 = 1234567654321

11111111 x 11111111 = 123456787654321

111111111 x 111111111 = 12345678987654321

Mind Boggling....



Now, take a look at this...

101%

From a strictly mathematical viewpoint:

What Equals 100% ?

What does it mean to give MORE than 100% ?

Ever wonder about those people who say they

Are giving more than 100% ?

We have all been in situations where someone wants you to

GIVE OVER 100%...

How about ACHIEVING 101%?

What equals 100% in life?

Here's a little mathematical formula that might help

Answer these questions:

If:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Is represented as:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26.

Then:

H-A-R-D-W-O- R- K

8+1+18+4+23+ 15+18+11 = 98%

And:

K-N-O-W-L-E- D-G-E

11+14+15+23+ 12+5+4+7+ 5 = 96%

But:

A-T-T-I-T-U- D-E

1+20+20+9+20+ 21+4+5 = 100%

THEN, look how far the love of God will take you:

L-O-V-E-O-F- G-O-D

12+15+22+5+15+ 6+7+15+4 = 101%

[chilarai]

Made in Pakistan fake money floods India

sometime back someone posted the german tank problem on this thread.
Now can the same be applied to estimate the amount of fake currency in circulation. I know the serial numbers in this case is prolly not serialised. but would it still be possible to get good estimates ?

[Amber G.]

Happy New year.
There is quite a bit of excitement about Ken Ono (There is a public lecture at 8 p.m. Friday on the Emory campus) work regarding finite formula and other new theories regarding partition numbers.

Partition function is something which almost every elementary math students learns.
(Basically it is number of ways a number can be divided, for example partition number for 4 is 5 because (4 = 3+1= 2+2 = 2+1+1=1+1+1+1 .. 5 ways) -- one can see wiki article to get more info). Yet there are no easy formulae.

Euler was first to give recursive technique, which most students use, for computing the partition values but major work, on this was done by Srinivasa Ramanujan (and G. H. Hardy) which yields good approximation for larger numbers.

One of major unfinished work by Ramanujan (He noted the pattern but, up till now, no one was able to prove it. Prof Ono seems to able to explain it now) was a pattern involving just three prime numbers (5,7 and 11)..(Again for details and clarity please check out "Ramanujan's congruences" (in wiki or math book).. Ramanujan's quote was little understood because he did not explain exactly what he meant for ordinary folks ...
and many mathematicians worked on this since 1920's ...

The "eureka moment" as Prof Ona told, came while hiking in northern Georgia and they started noticing odd patterns in clumps of trees (they began thinking about what it would be like to "walk" through partition numbers.. they say ..and is related to another branch of math (fractals)... believe it or not, just a few posts ago, this thread talked about Mandelbrot!

Anyway there will be many mainstream news stories - you can check, for example here:
http://esciencecommons.blogspot.com/?
or http://blogs.plos.org/badphysics/2011/01/20/ono/
or http://www.eurekalert.org/pub_releases/ ... 011911.php

[SwamyG]

...Sorry for trespassing....but I thought this is interesting. A search for India’s mathematical roots
teaser...

As Indians become more confident and self-assured of their place in the world, there is a renewed interest in tracing India’s intellectual history. Many individuals as well as private and government-aided organizations are putting together teams to delve into the work of ancient scientists. For instance, the Indian government is in the process of setting up a 22-member committee led by Anil Kakodkar, former chairman of the Bhabha Atomic Research Centre, to “identify literature on the history of Indian science and technology, interpret India’s cultural expressions and their development in scientific and technological perspective”.

At the IIT department, members are excited about this revival of interest in their work: exploring India’s scientific history, especially of the Kerala School of astronomy and mathematics that was founded by Madhava, a famous mathematician, and included geniuses such as Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati and Achyuta Panikkar.

Explaining what the Kerala School means, Ramasubramanian said: “Mathematics in India has a long, hallowed history. In old times, every ritual, journey and life event was planned in tune with the stars and this necessity forced them to learn precise mathematics and astronomy. But in the past, travel was difficult. And since astronomy was such an integral part of everyday life, mathematicians in different regions had to do independent investigations.” Over time, mathematicians of different regions coalesced into structured schools of thought that were named after the region.

[darshhan]

Guys I am planning on starting vedic mathematics.Can you name some good authors for this subject?Also online resources would be extremely welcome.

[svenkat]

Guys I am planning on starting vedic mathematics.Can you name some good authors for this subject?Also online resources would be extremely welcome.

http://forums.bharat-rakshak.com/viewto ... 88#p853588

[ArmenT]

Interesting way that a statistician cracked the lottery:
http://www.wired.com/magazine/2011/01/ff_lottery/all/1

[abhishek_sharma]

Memories of Martin Gardner

http://www.ams.org/notices/201103/rtx110300418p.pdf

[disha]

Guys I need help. I am presenting evolution of zero to a very wide audience and I have collected some material. I would like to have access to more materials as possible, can you please help me.

If I can get photos of Bhakshali manuscript or any references to zero in an image, that will help. Other thing I am looking for is examples of mathematical problems that cannot be solved by zero and the philosophical discourse behind zero. Third, the etymology of zero (it is from shunya, but more references the better)

This forum is my only recourse!