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 2005-03-14, 15:31 #1 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 22×33×19 Posts Problems of Antiquity I have been out of touch with Mersenneforum for about two months as I was on a business cum holiday trip to Ratlam (India). where there are few cyber cafes. Amongst others I took with me two ‘companions’ viz: 1) The ‘ Mathematical Experience’ by Davis and Hersh 2) A ‘Concise history of Mathematics’ by Dirk J. Struik. I read them both for the second time around.. I would recommend these books to anyone who is interested in Maths. .In them I found four challenging problems known to the ancient Egyptians, Greeks and Orientals before Christ which can even confound math scholars today. 1) Given the sum (x + y) and product x y. Find x and y 2) The square number 16 and the rectangular no. 18 are alone of plane numbers, which have their perimeters equal to the areas enclosed by them. Prove it. 3) Find the volume of the frustrum of a square pyramid where ‘a’ and ‘b’ are the lengths of the sides of the squares and ‘h’ is its height., Assume vol. of square pyramid is known to be one third the Area of the base multiplied by its height.. 4) Prove Herons formula for the Area (A) of a triangle is A = Sqr. Root of [ s (s-a) (s-b) (s-c) ] where a, b, c, are its sides and ‘s’ = semi- perimeter. I spent a few pleasant evenings deriving/solving them without referring to text books or the Net. Try your hand at them too and prove yourself equal to the ancients! Mally.
 2005-03-14, 16:09 #2 akruppa     "Nancy" Aug 2002 Alexandria 2,467 Posts 1. is easy. x+y=a, xy=b, x=(a +- sqrt(a^2 - 4*b))/2 Alex
2005-03-14, 16:38   #3
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

40048 Posts
\problems of Antiquity

Quote:
 Originally Posted by akruppa 1. is easy. x+y=a, xy=b, x=(a +- sqrt(a^2 - 4*b))/2 Alex
Excellent Alex ! x=y = a/2 +- sqr.rt.[ (a/2)/2)^2 - b ] when simplified further

Mally.

 2005-03-14, 16:58 #4 Ken_g6     Jan 2005 Caught in a sieve 2×197 Posts Don't forget If x==0 or y==0 <=> b==0, the system has two solutions. Either x=0 and y=a or y=0 and x=a. 3 looks nice and linear.
 2005-03-14, 17:41 #5 jinydu     Dec 2003 Hopefully Near M48 2×3×293 Posts I remember proving Heron's Formula in high school. It took nearly a page. Basically, it can be done using the cosine rule and lots of brute force algebra.
 2005-03-14, 22:55 #6 cheesehead     "Richard B. Woods" Aug 2002 Wisconsin USA 22×3×641 Posts 2a) square number 16. For each square number n^2, the area is n^2 and the perimeter is 4n. n^2 = 4n => n=4. But it could be that n^2 = 16 falls out as a special case of rectangular number, which proof I haven't completed. Last fiddled with by cheesehead on 2005-03-14 at 23:06
2005-03-15, 03:33   #7
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

1000000001002 Posts
Problems of Antiquity

Quote:
 Originally Posted by mfgoode Excellent Alex ! x=y = a/2 +- sqr.rt.[ (a/2)/2)^2 - b ] when simplified further Mally.
Correction: x=a/2 + [..........]. and y = -[........ ]if x>y

Mally

 2005-03-15, 07:31 #8 cheesehead     "Richard B. Woods" Aug 2002 Wisconsin USA 22·3·641 Posts Does this thread belong in the Puzzles section of the "Fun stuff" subforum?
2005-03-19, 14:28   #9
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

205210 Posts
Problems of Antiquity

Quote:
 Originally Posted by cheesehead 2a) square number 16. For each square number n^2, the area is n^2 and the perimeter is 4n. n^2 = 4n => n=4. But it could be that n^2 = 16 falls out as a special case of rectangular number, which proof I haven't completed.
[Quote=cheesehead] Does this thread belong in the Puzzles section of the "Fun stuff" subforum]

This theorem was quoted by Plutarch [40A.D. - 120A.D.] who blended sacred history with mathematical theorems. It was well known by the ancients Pythagoras included.

It was recommended by Davis and Hersh to try to prove it.
"This is a nice theorem, Prove it."

I am still waiting for you to complete it Then decide if it should be shoved into. the Puzzles section of the "Fun Stuff" subforum!!!

Mally

2005-03-21, 16:47   #10

"Richard B. Woods"
Aug 2002
Wisconsin USA

1E0C16 Posts

Quote:
 Originally Posted by mfgoode I am still waiting for you to complete it Then decide if it should be shoved into. the Puzzles section of the "Fun Stuff" subforum!!!
Advisory: Completion is way down on my priority list, like ... end-of-summer-ish.

2005-03-21, 16:56   #11
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

205210 Posts
Problems of Antiquity

Quote:
 Originally Posted by cheesehead Advisory: Completion is way down on my priority list, like ... end-of-summer-ish.
Suit yourself buddy!. Im waiting for Christmas

Mally

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