It's actually false. The edges of the four quadrilaterals do not really meet. Try it with a piece of cardboard. The picture has been "adjusted" to make it look like they do. Notice that 5,8 and 13 are part of the Fibonacci sequence. Cassini's identity allows for any nxn square to be rearranged into an (n-1)x(n+1) rectangle, with the loss of a small, unit-sized area between the diagonals, approximately the size of a parallelogram.
Thanks Rage........That Hurt Even More...... Did I mention that all the algebra floating around in my head evaporated at the end of the semester?:rolleyes::dead:
From here: http://brainden.com/forum/index.php?showtopic=139 64 = 65 Geometry Paradox - solution It looks like a triangle, because a thick line was used. Hypotenuse of the composite triangle is actually not a straight line – it is made of two lines. Forth cusps are where the arrows point (c9, l6). The 64 = 65 paradox arises from the fact that the edges of the four pieces, which lie along the diagonal of the formed rectangle, do not coincide exactly in direction. This diagonal is not a straight segment line but a small lozenge (diamond-shaped figure), whose acute angle is arctan 2/3 - arctan 3/8 = arctan 1/46 which is less than 1 degree 15' . Only a very precise drawing can enable us to distinguish such a small angle. Using analytic geometry or trigonometry, we can easily prove that the area of the "hidden" lozenge is equal to that of a small square of the chessboard. Perhaps another pic http://i39.tinypic.com/258bgh3.gif
ummm yeah Moiz, im with Captain, i got as far as trying to figure how many inches in a gallon and gave up. and apparently Kimsland is all over it too.
I Feel Feint....And My Vision is Starting to Blur...... Has anyone ever died from acute mathematics overdose.....? :dead: @Red1776.....Think and take baby steps, figure out how many inches are in a pint and multiply by 8.....! Oh, BTW, good one.... :haha: :wave:
ohhhh yes Captain, your right, im replete with embarrassment having forgotten the basics, now lets see...thats the F.O.I.L. method correct? :wave:
Thanks for clarifying Rage and kimsland. I find this sort of stuff kind of interesting. Maybe I'm mathematically inclined or something
No problems. Actually I think I linked the right site but wrong pic above (not sure) Anyway I have a logical mind myself, which is good for computer servicing, but I can get caught up trying to prove something is correct (or not )
Good solutions, if tougher than the problem. Well, it is well known that cures can be worse than the malady. Or, as somebody, unfortunately not bobcat, aptly put it, The main cause of problems is solutions. Anyway, kudos guys, or should I say kudi or something, because “kudos” is actually Greek and singular, of whose plural I’m not sure, maybe the captain can throw some light when he recovers from his maths headache. By the way, trying to work out how many inches to a pint is the wrong approach, because the problem has to do with areas, whose unit is…er…er…degrees I think…er…No! That’s the alcohol in my brandy, which I blame for my slight confusion. Gosh! I sure am glad I’ve found an excuse!
Actually adding an i to make a word plural is from Latin, and it applies to words ending in us like cactus, which becomes cacti.
OK, could you make this teensy little clarification...? Would that be pronounced, "fun-guy" or "funge-eye"? @kimsland: Your point as I understand it is this: If you have a fat enough marker and a sufficiently dull pair of scissors, then the math doesn't matter anyway. @Bobcat: Perhaps we could split and augment the pluralization efforts by adding the Spanish suffix "oso", which would indicate a rather large, sincere, but conceivably flatulent "kudo". After which a simple "s" would suffice for the onerous chore of pluralization. BAD PUN WARNING:.... "Bear in mind that "oso", by itself, also means "bear" in Spanish. @ red1776: "DILLIGAF" begins with "FOIL", but you round off to the nearest 500, (or so).
It appears that the grammar, that is the plural of kudos, is presenting more problems than the maths of the original problem. I see further headaches on the way, but now we know that a brandy is an excellent cure for headache. In fact, I always keep a supply of brandy handy (it even rhymes) in case I get a headache from a maths problem - which I also keep handy. And now, I give my solution to the original problem – prepare for more headaches! I remind that we want to know where the extra (65th) square came from. Well, consider the following related problem. You will notice that this time there is a square missing or disappearing. Well, this square that disappears from here should be the extra square appearing in the 1st problem. This is an example of the “Principle of preservation of squares”. And this way we’ve solved 2 problems in one go. The principle in operation this time is: If you can’t solve a problem, create another one. (bobcat) Now, where is that brandy again?
I'm not sure. I've always said fun-guy, but my biology teacher says funge-eye, so that is probably the correct way.
You guys crack me up! It's always fun to find a bunch of fellow geeks (geeki/geekum/geekonians?) pondering over preposterous propositions of the polygonal persuasion. Greek grammar getya's gotta get yer goat too, eh? I'm lookin' forward to some interesting brain-ulation on these forums and finding others that are mathematically reclined like myself popping up with puzzles that Martin Gardner never published. he he he...
Greetings; glad that geeks, gurus, graphical guessing games, the great Gardner & Greek grammar get you going. gobcat…er…bobcat
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