How to prove theorems ?

Many years ago, I received an email listing these wonderful ways to prove theorems. I am sure the mail would still be circulating in some form or the other. This is my small attempt to preserve the best mathematical lesson I had for posterity. Thanks to the anonymous author.

1. Proof by example

The author gives only the case n = 2 and suggests that it contains most of the ideas of the general proof.

2. Proof by intimidation

"Trivial."

3. Proof by vigorous handwaving

Works well in a classroom or seminar setting.

4. Proof by cumbersome notation

Best done with access to at least four alphabets and special symbols.

5. Proof by exhaustion

An issue or two of a journal devoted to your proof is useful.

6. Proof by omission

"The reader may easily supply the details."
"The other 253 cases are analogous."
"..."

7. Proof by obfuscation

A long plotless sequence of true and/or meaningless syntactically related statements.

8. Proof by wishful citation

The author cites the negation, converse, or generalization of a theorem from literature to support his claims.

9. Proof by funding

How could three different government agencies be wrong?

10. Proof by eminent authority

"I saw Karp in the elevator and he said it was probably NP-complete."

11. Proof by personal communication

"Eight-dimensional colored cycle stripping is NP-complete [Karp, personal communication]."

12. Proof by reduction to the wrong problem

"To see that infinite-dimensional colored cycle stripping is decidable, we reduce it to the halting problem."

13. Proof by reference to inaccessible literature

The author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Slovenian Philological Society, 1883.

14. Proof by importance

A large body of useful consequences all follow from the proposition in question.

15. Proof by accumulated evidence

Long and diligent search has not revealed a counterexample.

16. Proof by cosmology

The negation of the proposition is unimaginable or meaningless. Popular for proofs of the existence of God.

17. Proof by mutual reference

In reference A, Theorem 5 is said to follow from Theorem 3 in reference B, which is shown from Corollary 6.2 in reference C, which is an easy consequence of Theorem 5 in reference A.

18. Proof by metaproof

A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.

19. Proof by picture

A more convincing form of proof by example. Combines well with proof by omission.

20. Proof by vehement assertion

It is useful to have some kind of authority in relation to the audience.

21. Proof by ghost reference

Nothing even remotely resembling the cited theorem appears in the reference given.

22. Proof by forward reference

Reference is usually to a forthcoming paper of the author, which is often not as forthcoming as at first.

23. Proof by semantic shift

Some standard but inconvenient definitions are changed for the statement of the result.

24. Proof by appeal to intuition

Cloud-shaped drawings frequently help here.

Normality test

Well, you just apply the Kolmogorov Smirnov test for normality, which consists of drinking vodka and looking at your data from time to time until it looks normal enough to you.

- Ronan Conroy

Biostatistician, Dublin

Anand's blindfold win over Leko

Vishy won his first round game in Amber Blindford tournament against Hungarian Peter Leko. The game ended with a beautiful combination from Anand. Although the start was impressive, he ultimately finished joint 4-5 at 6.5/11 (+4-2=5), behind Aronian, Kramnik and Carlsen, who shared 1-3 in this section. All the leaders scored 7/11.

Here is the game. Enjoy.

PS: If you are thinking why did I put this old game on the blog today, then the answer is: I was experimenting with inserting a chess game on blog. This is the first time I could successfully do it. :)

Gulaal - Anurah Kashyap's coming of age

Anurag kashyap's new movie Gulaal is a class apart from the usual hindi movies dealing with more serious stuff like romancing the heroine around the trees. The story builds on the pain and agony of some former Rajput kings who feel cheated by the nation which promised them indefinite privy purse in 1947-49 but withdrew it in 1976. They now intend to return to the old "glory" of their kingdom. The story also portrays naked college politics and how it is influenced by parties pursuing their own agenda.

Kay Kay Menon plays the role of the leader of the movement attempting to go back in history. I last saw him in "Honeymoon Travels Pvt Ltd". He played a orthodox bengali gentleman in that movie. His screen play and transformation as a confident rajput royal "Duki Banna" in Gulaal is amazing. Apart from him, Raja Choudhary's acting is also worth many accolades.

There are subtle political/ philosophical issues scattered all over the movie. Prthvi, the brother of Duki banna wears a locket of John Lennon. The hanging of Ran Sa wears a stark resembles to the public hanging of erstwhile Afghan president Nazibullah. Kiran, the lady in the hostel sits in a dark room with 'nihilism' written on the wall. The Ranaji song is hilariously anti-US.

The movie itself is definitely worth watching but I am specially impressed by the lyrics of the songs. Piyush mishra is truly original in the songs. I do not remember any movie in recent times which has used such beautiful word play. Gulaal also includes a stranza from Ramdhaari Singh Dinkar's epic "Rashmirathi" (दो न्याय अगर तो आधा दो, पर इसमें भी यदि बाधा हो, तो दे दो केवल पांच ग्राम, रखो अपनी धरती तमाम, ...).

The following are only two examples of the songs of this movie. I hope you enjoy it.

(1)
आरंभ है प्रचंड, बोलें मस्तकों के झुंड, आज जंग की घडी की तुम गुहार दो |

आरंभ है प्रचंड, बोलें मस्तकों के झुंड, आज जंग की घडी की तुम गुहार दो |
आन बान शान, या की जान का हो दान, आज एक धनुष के बाण पे उतार दो || (२)
आरंभ है प्रचंड...

मन करे सो प्राण दे, जो मन करे सो प्राण ले, वही तो एक सर्व शक्तिमान है |
विश्व की पुकार है, ये भागवत का सार है, की युद्ध ही तो वीर का प्रमाण है ||
कौरवों की भीड़ हो, या पांडवों का नीड़ हो, जो लड़ सका है वो ही तो महान है |
जीत की हवस नहीं, किसी पे कोई वश नहीं, क्या जिन्दगी है ठोकरों पे मार दो ?
मौत अंत है नहीं तो मौत से भी क्यों डरें, ये जाके आसमान में दहाड़ दो ||
आरंभ है प्रचंड ...

हो दया का भाव या की शौर्य का चुनाव, या की हार का वो घाव तुम ये सोच लो | (२)
या की पूरे भाल पर जला रहे विजय का लाल, लाल ये गुलाल तुम ये सोच लो |
रंग केसरी हो या मृदंग केसरी हो या की केसरी हो ताल तुम ये सोच लो ||
जिस कवि की कल्पना में जिन्दगी हो प्रेम गीत, उस कवि को आज तुम नकार दो |
भीगती नसों में आज, फूलती रगों में आज, आग की लपट का तुम बगार दो ||

आरंभ है प्रचंड, बोलें मस्तकों के झुंड, आज जंग की घडी की तुम गुहार दो |
आन बान शान, या की जान का हो दान, आज एक धनुष के बाण पे उतार दो |
आरंभ है प्रचंड ...

(2)
राणाजी म्हारे, गुस्से में आये, एसो बल खाए, अगिया बरसाए, घबराए म्हारो चैन | (२)
जैसे दूर देस के,
जैसे दूर देस के, टावर में घूस जाए रे एरोप्लेन ||
राणाजी म्हारे ...

राणाजी म्हारे, एसो गुर्राए, एसो थर्राए, भर आये म्हारे नैन |
जैसे सरे आम ही,
जैसे सरे आम इराक में जाके जम गए अंकल सेम ||
राणाजी म्हारे...

राणाजी म्हारी सास ननद के ताने, राणाजी म्हारी जेठ ससुर की बानी |
राणाजी थापे भूत परेत की छाया, राणाजी थापे इल्बिल जिन का साया ||
सजनी को डियर बोले, ठर्रे को बियर बोले, मांगे है इंग्लिस बोली, मांगे है इंग्लिस चोली |
मांगे है इंग्लिस जेपुर, इंग्लिस बीकानेर ||
जैसे बिसलेरी की ,
जैसे बिसलेरी की बोतल पिके बन गए इंग्लिस मेन |
राणाजी म्हारे...

राणाजी म्हारी सौतन को घर ले आये |
पूछे तो बोले फ्रेंड हमारी है हाये ||
राणाजी ने ठंडा चक्कू यूँ खोला |
बोले की हाये ठंडा मने कोका कोला ||
राणाजी बोले मोरों की बस्ती में है शोर रानी |
क्यों की ये दिल मांगे मोर, मोर रानी, मोर रानी, मोर रानी ..
म्हारी तो बीच बजरिया, हाये बदनामी हो गयी |
म्हारी तो लाल चुनरिया, सरम से धानी हो गयी ||
म्हारो तो धक् धक् होवे, जो जो बीते रे
जैसे हर एक बात पे ...
जैसे हर एक बात पे डिमोक्रिसी में लगने लग गयो बेन ...

जैसे दूर देस के, टावर में घूस जाए रे एरोप्लेन
जैसे सरे आम इराक में जाके जम गए अंकल सेम
जैसे बिना बात अफगानिस्ता का
जैसे बिना बात अफगानिस्ता का बज गया भैया बेंड |

जैसे दूर देस के, टावर में घूस जाए रे एरोप्लेन ||
राणाजी म्हारे ...

Management Lessons

FIVE MANGEMENT LESSONS
(COURTESY: PANCHATHANTRA)

CASE 1:

A crow was sitting on a tree, doing nothing all day.
A small rabbit saw the crow, and asked him, "Can I also sit like you and do nothing all day long?
The crow answered: "Sure, why not." so, the rabbit sat on the ground below the crow, and rested.

All of a sudden, a fox appeared, jumped on the rabbit and ate it.

Management Learning:

For sitting and doing nothing, you must be sitting very, very high up.

******************************

*******************************

CASE 2:

A turkey was chatting with a bull.
"I would love to! be able to get to the top of that tree," sighed the turkey, "but I haven't got the energy.

"Well, why don't you nibble on some of my droppings?" replied the bull."They're packed with nutrients."

The turkey pecked at a lump of dung and found that it actually gave him enough strength to reach the first branch of the tree.

The next day, after eating some more dung, he reached the second branch.
Finally after a fourth night, there he was proudly perched at the top of the tree. Soon he was promptly spotted by a farmer, who shot the turkey out of the tree.

Management Learning:

Bullshit might get you to the top, but it won't keep you there.
*****************************************************

CASE 3:

A little bird was flying south for the winter. It was so cold, the bird froze and fell to the ground in a large field.
While it was lying there, a cow came by and dropped some dung on it.

As the frozen bird lay there in the pile of cow dung, it began to realize how warm it was.
The dung was actually thawing him out! He lay there all warm and happy, and soon began to sing for joy.
A passing cat heard the bird singing and came to investigate.
Following the sound, the cat discovered the bird under the pile of cow dung, and promptly dug him out and ate him!

Management Learning:

1) Not everyone who drops shit on you is your enemy.
2) Not everyone who gets you out of shit is your friend.
3) And when you're in deep shit, keep your mouth shut!

***********************************************

CASE 4:

The boy rode on the donkey and the old man walked.
As they went along, they passed some people who remarked "it was a shame

the old man was walking and the boy was riding".
The man and boy thought maybe the critics were right, so they changed positions.
Later, they passed some people that remarked, "What a shame, he makes that little boy walk."
They decide d they both would walk!
Soon they passed some more people who thought they were stupid to walk when they had a decent donkey to ride.
So the both rode the donkey! Now they passed some people that shamed them by saying "how awful to put such a load on a poor donkey".
The boy and man said they were probably right so they decided to carry the donkey.
As they crossed a bridge, they lost their grip on the animal and he fell into the river and drowned.

Management Learning:

If you try to please everyone, you will eventually lose your ass

*******************************************************

CASE 5:

Once PVNR (PV Narasimha Rao), L.K.Advani and Laloo Prasad Yadav were travelling in an autorickshaw.
They met with an accident and all three of them died. Yama was waiting for this moment at the doorstep of death.
He asks PVNR and Advani to go to HEAVEN. But, for Laloo, Yama ha! d already decided that he should be sent to HELL.
Laloo is not at all happy with this decision. He asks Yama as to why this discrimination is being made.
All the three of them had served the public. Similarly, all took bribes, all misused public positions, etc. Then why the differential treatment?
He felt that there should be a formal test or an objective evaluation before a decision is made; and should not be just x-udd on opinion or pre conceived notions.

Yama agrees to this and asks all the three of them to appear for an English test.
PVNR is asked to spell "INDIA" and he does it correctly.
Advani is asked to spell "ENGLAND" and he too passes.
It is Laloo's turn and he is asked to spell "CZECHOSLOVAKIA".
Laloo protests that he doesn't know English. He says this is not fair and hat he was given a tough question and thus forced to fail with false intent.

Yama then agrees to conduct a written test in Hindi (to give another chance assuming that Laloo should at least feel that Hindi would provide an equal platform for all three).

PVNR is asked to write "KUTTA BOLA BHOW BHOW". He writes it easily and passes.
Advani is asked to write "BILLY BOLI MYAUN MYAUN". He too passes.
Laloo is asked to write "BANDAR BOLA GURRRRRR....."
Tough one. He fails again. Laloo is extremely unhappy.

Having been a student of history (which the other two weren't), he now requested for all the 3 to be subjected to a test in history.
Yama says 'OK', but this would be the last chance and that he would not take any more tests.
PVNR is asked: "When did India get Independence?".
He replied "1947" and passed.
Advani is asked "How many people died during the independence struggle?".
He gets nervous. Yama asked him to choose from 3 options: 100,000 or 200,000 or 300,000.
Advani catches it and says 200,000 and passes.
It's Laloo's turn now. Yama asks him to give the! Name and Address of each
of the 200,000 who died in the struggle.
Laloo accepts defeat and agrees to go to HELL.

Management Lesson:

"IF YOUR MANAGEMENT HAS DECIDED TO SCREW YOU, THERE IS JUST NO ESCAPE"