mersenneforum.org  

Go Back   mersenneforum.org > Fun Stuff > Puzzles

Reply
 
Thread Tools
Old 2006-02-14, 21:36   #12
xilman
Bamboozled!
 
xilman's Avatar
 
"π’‰Ίπ’ŒŒπ’‡·π’†·π’€­"
May 2003
Down not across

2·72·109 Posts
Default

Quote:
Originally Posted by Patrick123
It is fascinating, this is how I originally worked it out.
This is my solution.

If the question has a unique answer, and it phrased as if it must, the solution must be independent of the diameter of the sphere or the radius of the hole. In particular, the special case of a hole of zero radius and hence zero volume must yield the unique answer. This hole must clearly be drillled through a 6" diameter sphere.

Paul
xilman is offline   Reply With Quote
Old 2006-02-15, 01:03   #13
mfgoode
Bronze Medalist
 
mfgoode's Avatar
 
Jan 2004
Mumbai,India

22·33·19 Posts
Thumbs up Volume od a sphere

Quote:
Originally Posted by xilman
This is my solution.

If the question has a unique answer, and it phrased as if it must, the solution must be independent of the diameter of the sphere or the radius of the hole. In particular, the special case of a hole of zero radius and hence zero volume must yield the unique answer. This hole must clearly be drillled through a 6" diameter sphere.

Paul
Kindly re-read my post no.10.
Any sphere of diameter 6 inches and Above will yield a residue of 36 pi for a 6 inch long hole, including our earth! Fascinating isn't it?
Mally
mfgoode is offline   Reply With Quote
Old 2006-02-15, 22:18   #14
Fusion_power
 
Fusion_power's Avatar
 
Aug 2003
Snicker, AL

95910 Posts
Default

Draw a square. It can be any size. Now draw a circle so that it fills the square. What is the relationship of the area of the circle to the area of the square?

Now draw the same square and put 4 identical circles inside it so they fill the square. What is the relationship of the area of the circle to the area of the square?

Now draw the same square with 9 circles and figure the areas.

What would it be with 16 circles?

Hint, an easy approach to this is to use a square with sides 6 long. Use inches, cm, etc, whatever makes you happy.

Fusion
Fusion_power is offline   Reply With Quote
Old 2006-02-16, 04:41   #15
drew
 
drew's Avatar
 
Jun 2005

2×191 Posts
Default

Quote:
Originally Posted by Fusion_power
Draw a square. It can be any size. Now draw a circle so that it fills the square. What is the relationship of the area of the circle to the area of the square?

Now draw the same square and put 4 identical circles inside it so they fill the square. What is the relationship of the area of the circle to the area of the square?

Now draw the same square with 9 circles and figure the areas.

What would it be with 16 circles?

Hint, an easy approach to this is to use a square with sides 6 long. Use inches, cm, etc, whatever makes you happy.

Fusion
Is this another puzzle?

The way you described, the picture can still be reduced to smaller squares, so the ratios of areas are the same due to similarity. The only way to improve this is to change the packing. Hexagonal packing will be an improvement over the square packing you described.

You can do even better if you allow circles of various sizes.

Drew

Last fiddled with by drew on 2006-02-16 at 04:42
drew is offline   Reply With Quote
Old 2006-02-16, 06:50   #16
Fusion_power
 
Fusion_power's Avatar
 
Aug 2003
Snicker, AL

7·137 Posts
Default

Its not a packing puzzle, its a relationship demonstration. You will see the relationship if you solve the elementary math involved. Leave the result in the form X(pi). You should also see the relationship to the above about a hole drilled into a sphere.

Fusion
Fusion_power is offline   Reply With Quote
Old 2006-02-16, 14:17   #17
drew
 
drew's Avatar
 
Jun 2005

38210 Posts
Default

Quote:
Originally Posted by Fusion_power
Its not a packing puzzle, its a relationship demonstration. You will see the relationship if you solve the elementary math involved. Leave the result in the form X(pi). You should also see the relationship to the above about a hole drilled into a sphere.

Fusion
The ratio of circle area to square area is pi/4 for all cases. (pi*R2/(2R)2)

Like I said. The ratio will always be the same due to similarity. It can always be divided into a number of circles inscribed in squares.

Drew
drew is offline   Reply With Quote
Old 2006-02-18, 20:46   #18
Numbers
 
Numbers's Avatar
 
Jun 2005
Near Beetlegeuse

18416 Posts
Default

Fusion,
When you say the circle "fills" the square, do you mean the sides of the square are tangent to the circle, or do you mean the circle touches the square at its corners?

Thanks,
Numbers is offline   Reply With Quote
Old 2006-02-19, 08:15   #19
mfgoode
Bronze Medalist
 
mfgoode's Avatar
 
Jan 2004
Mumbai,India

1000000001002 Posts
Exclamation

Quote:
Originally Posted by Numbers
Fusion,
When you say the circle "fills" the square, do you mean the sides of the square are tangent to the circle, or do you mean the circle touches the square at its corners?

Thanks,
Ah numbers, you mean the circle is inscribed in the square or it circumscribes the square. With the right math terminology you cant go wrong!
Mally
mfgoode is offline   Reply With Quote
Old 2006-02-20, 15:00   #20
Fusion_power
 
Fusion_power's Avatar
 
Aug 2003
Snicker, AL

11101111112 Posts
Default

Inscribed in the square meets the conditions stated. However, a modified set of conditions would give a similar result for a circumscribed circle(s).

Fusion
Fusion_power is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Pseudo-mathematical gibberings (volume 9a) storflyt32 storflyt32 81 2015-04-22 16:06
Hole and sphere. mfgoode Puzzles 32 2007-09-15 12:55
4D-Volume of a 4D-Sphere davar55 Puzzles 16 2007-07-05 22:16
Triangles on a sphere davieddy Puzzles 15 2007-04-06 20:16
Cylinder in sphere Greenbank Puzzles 17 2006-01-26 17:18

All times are UTC. The time now is 01:59.

Wed May 12 01:59:24 UTC 2021 up 33 days, 20:40, 0 users, load averages: 2.09, 2.15, 2.32

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.