|
< Earlier Kibitzing · PAGE 3 OF 3 ·
Later Kibitzing> |
May-31-14
 | | Domdaniel: <FSR> - < Incidentally, 325 is the smallest number that is expressible as the sum of two perfect <squares> in <three> different ways.>
Well, yes and no. Every prime of the form 4n+1 is expressible as the sum of two squares: the sequence begins 5, 13, 17, 29, 37, 41, 53...
A product of two such (different) numbers is expressible as a sum of squares in two ways. If the primes are p = a^2 + b^2 and q = c^2 + d^2, then the product pq = (ac + bd)^2 + (ad-bc)^2, and (ac-bd)^2 + (ad+bc)^2. The lowest product of two distinct 4n+1 primes is 5x13 = 65. The smallest product of 3 such primes is 5x13x17 = 1105. 65 = 8^2 + 1^2 = 7^2 + 4^2
1105 = 33^2 + 4^2 = 32^2 + 9^2 = 31^2 + 12^2 = 24^2 + 23^2 When all prime factors are distinct and of form 4n+1, the number of ways in which the product equals the sum of two squares is 2^n, where n is the number of distinct factors. 325 = 5x5x13 is a special case where two of the prime factors are the same.
In these cases, the number of ways in which the product equals the sum of two squares is less than 2^n. A simple formula also exists in such cases. Ramanujan is a very well-known name. His insights were extraordinary. I'm not sure whether the player here, with a similar name, deserves the so-called pun. |
|
| May-31-14 | | john barleycorn: <Domdaniel: <FSR> - < Incidentally, 325 is the smallest number that is expressible as the sum of two perfect <squares> in <three> different ways.> Well, yes and no.> Which number smaller than 325 can be expressed as the sum of 2 perfect squares? I can't conclude it from the math you have given. |
|
May-31-14
| | Domdaniel: <jb> 65 = 5x13 = 8^2 + 1^2 = 7^2 + 4^2 is the smallest number that can be expressed as the sum of two squares, though 50 = 2x5x5 = 7^2 + 1^2 = 5^2 + 5^2 and 25 = 5x5 = 3^2 + 4^2 = 5^2 + 0^2 may qualify if you don't stipulate that the square roots must be distinct and >0.
The minimum for 3 sets of squares is 325. For 4, it's 1105. |
|
| May-31-14 | | auh2o: An excellent biography of the mathematical genius Ramanujan is "The Man Who Knew Infinity," available here: http://www.amazon.com/Man-Who-Knew-... Ramanujan sometimes ended his name with an "m" but mostly with an "n," as his biographer explains. The taxicab story arises from G. H. Hardy, another great Cambridge mathematician, who brought Ramanujan to Cambridge from India and who was his lifelong collaborator. See G. H. Hardy's "A Mathematician's Apology" (which contains the original account of the taxicab number). |
|
| May-31-14 | | morfishine: FWIW: I give "The Taxi Cab Game" a 'thumbs up' |
|
May-31-14
 | | Richard Taylor: <FSR: <Richard Taylor> This is the only game in the database with 3...e6 4.e3 a6. Much more common is 3...Nf6 4.Nf3 a6. Opening Explorer In the latter case, Black can develop his QB, responding to 5.c5 with 5...Bf5. Nonetheless, in the present game, 5...e5! looks OK to me. > Yes. Thanks. I saw the other options but wondered if what Black played was valid. e5 looks thematic for sure. As it was he just got too cramped. |
|
May-31-14
| | Richard Taylor: < goldenbear: That's not the only good story about 1729... http://www.ee.ryerson.ca/~elf/abacu...
>
Well I read the story but it was mostly beyond me I read (part of) a book by Feynman and what I liked most was one funny incident when he was in hospital abusing the doctors under anaesthetic. But mostly mathematics is beyond me but he was clearly a genius or a freak of some kind. |
|
| May-31-14 | | anjyplayer: non sense |
|
Jun-01-14
 | | AylerKupp: <goldenbear> A good story, but if the salesman was Japanese he was probably not using an abacus (or suanpan) but a soroban. The soroban is a streamlined version of an abacus, with only one bead instead of two in the upper part and four beads instead of five in the lower part. It is more efficient; a long, long time ago I learned to use both and I could solve problems with a soroban significantly faster than with an abacus, possibly because I'm not too coordinated and with the soroban I had less beads to move. Feynman probably didn't know the difference and I think that's good news for us ordinary folks. Just because you're a brilliant Nobel Prize winner it doesn't mean that you know everything. Or maybe he did know the difference, and he was joking with us. Oh dear, I never thought of that! And the story is probably a good reminder for those of us who use chess engines extensively to remember that we also need to use our heads and some old fashioned thinking. |
|
| Jun-01-14 | | Conrad93: Perfidious, believe it or not I never claimed to know everything. And believe it or not, that was exactly my point.
Some morons were mentioning the mathematician, when this sort of name is really not that uncommon for Indians. |
|
| Jun-01-14 | | Conrad93: If it wasn't for the name, this game would not even be GOTD. |
|
Jun-01-14
 | | perfidious: Of course not. |
|
Jun-01-14
 | | northernfox: Apropos of this topic, some lite mathematical entertainment: Tom Lehrer -- Lobachevsky
http://www.youtube.com/watch?v=UQHa... |
|
| Jun-02-14 | | epicchess: Conrad93, you seem to not understand what chess is really about. The name is in fact the only thing about this game that is not good, and the game is better than most GOTD's. Just because it was played by a 2000 doesn't matter, since anyone can learn from this game. |
|
Jun-02-14
| | perfidious: <epicchess: Conrad93, you seem to not understand what chess is really about.> This sentence gets my nomination for Understatement of the Year Award. |
|
| Jun-06-14 | | Conrad93: I understand that chess is a children's game taken too seriously by adults. |
|
| Jun-06-14 | | SimonWebbsTiger: chess is a fun game that can be loved by children and great grandparents. It is loved, and has been, by people of all nations, colours, religions, gender, able bodied and disabled. Nope Conrad, you don't know a thing about chess, which your famous notes on e.p. prove. |
|
| Jun-06-14 | | Conrad93: A "fun" game?
What planet do you live on? |
|
| Jun-06-14 | | Conrad93: Chess is about stroking your ego. It's bragging about how high your rating is, or seeing how many opponents you can wreck. |
|
| Jun-07-14 | | SimonWebbsTiger: <Conrad>
Maybe some, be they at the top (e.g. Fischer) or lower skilled (e.g. AJ Goldsby) are so damaged they need chess to destroy others to support an unhealthy ego. Ppl don't like to lose, it's hard to be angry when you win after all, but you'll be surprised how many love chess for the fun of the fight, the beauty/aesthetics. |
|
Jun-07-14
| | perfidious: <Simon> For those cursed with such damage to the psyche, life is a thoroughly unpleasant business indeed; for all of us run into someone bigger, tougher or more skilled in some fashion. |
|
| Jun-07-14 | | N0B0DY: yeah, cry baby drools constantly, is dissatisfied, does not sleep, being misunderstood, rejected, or laughed at ... |
|
| Jun-07-14 | | Conrad93: <Maybe some, be they at the top (e.g. Fischer) or lower skilled (e.g. AJ Goldsby) are so damaged they need chess to destroy others to support an unhealthy ego. Ppl don't like to lose, it's hard to be angry when you win after all, but you'll be surprised how many love chess for the fun of the fight, the beauty/aesthetics.> 99% of OTB players would disagree with you. |
|
| Sep-28-14 | | BOSTER: I'd prefer to play crazy 31.Bg8+ to make the taxy story more interesting. |
|
| May-10-15 | | Imran Iskandar: If White played 31. Bg8, then he would have the pleasure of setting up a windmill, killing Black's pieces one by one, then ending the game with checkmate. |
|
 |
|
|
< Earlier Kibitzing · PAGE 3 OF 3 ·
Later Kibitzing> |
Kibitzing
sac: 28.Rxe4