mersenneforum.org  

Go Back   mersenneforum.org > Fun Stuff > Puzzles

Reply
 
Thread Tools
Old 2005-12-15, 15:24   #1
fatphil
 
fatphil's Avatar
 
May 2003

3·7·11 Posts
Default Mortar-less brick constructions

How many bricks are required to build a structure such that one of the bricks overhangs at least one brick-length beyond the base of the structure, assuming nothing apart from normal gravity holds the structure together. (By an overhang of at least one brick-length, I mean that the nearest part of that brick to the base is horizontal distance >epsilon from the base, and the furthest part is horizontal distance >1+epsilon, where 1 is the brick's length, it's longest dimension.)

What about an overhang of 2?
fatphil is offline   Reply With Quote
Old 2005-12-15, 15:47   #2
xilman
Bamboozled!
 
xilman's Avatar
 
"๐’‰บ๐’ŒŒ๐’‡ท๐’†ท๐’€ญ"
May 2003
Down not across

101011011110012 Posts
Default

Quote:
Originally Posted by fatphil
How many bricks are required to build a structure such that one of the bricks overhangs at least one brick-length beyond the base of the structure, assuming nothing apart from normal gravity holds the structure together. (By an overhang of at least one brick-length, I mean that the nearest part of that brick to the base is horizontal distance >epsilon from the base, and the furthest part is horizontal distance >1+epsilon, where 1 is the brick's length, it's longest dimension.)

What about an overhang of 2?
Ah, another old favourite.

It's the minimum i such that the harmonic series sum
exceeds 2 (resp. 3).

The sum reaches 2.083333333333333333333333333 at i=4 and
3.019877344877344877344877344 at i=11.

To get an overhang of ten, you need 33617 bricks, I believe.



Paul
xilman is offline   Reply With Quote
Old 2005-12-15, 19:07   #3
axn
 
axn's Avatar
 
Jun 2003

34·5·13 Posts
Default

Five, counting the base brick
axn is offline   Reply With Quote
Old 2005-12-16, 08:16   #4
fatphil
 
fatphil's Avatar
 
May 2003

E716 Posts
Default

The base brick indeed is counted, and with that you have the right answer.
It's pretty difficult to achieve with imperfect bricks.
Here's me missing by 1, but able to get plenty of extra overhang from the extra "brick":
http://fatphil.org/maths/bricks.jpg
fatphil is offline   Reply With Quote
Old 2005-12-16, 13:36   #5
rogue
 
rogue's Avatar
 
"Mark"
Apr 2003
Between here and the

198016 Posts
Default

Quote:
Originally Posted by fatphil
The base brick indeed is counted, and with that you have the right answer.
It's pretty difficult to achieve with imperfect bricks.
Here's me missing by 1, but able to get plenty of extra overhang from the extra "brick":
http://fatphil.org/maths/bricks.jpg
What are those ?
rogue is offline   Reply With Quote
Old 2005-12-16, 13:41   #6
Greenbank
 
Greenbank's Avatar
 
Jul 2005

2×193 Posts
Default

Things without uniform weight distribution.

HTH.
Greenbank is offline   Reply With Quote
Old 2005-12-16, 14:46   #7
fatphil
 
fatphil's Avatar
 
May 2003

3·7·11 Posts
Default

The thicker plastic casing is heavier than the thin aluminium, and so I was at a _disadvantage_ that way round.

I just tried with some fairly uniform 'plastic':
http://fatphil.org/maths/cards.jpg
and it was much easier; I managed to hit the goal of 5.
fatphil is offline   Reply With Quote
Old 2005-12-18, 11:14   #8
mfgoode
Bronze Medalist
 
mfgoode's Avatar
 
Jan 2004
Mumbai,India

22×33×19 Posts
Thumbs up Mortar-less brick constructions


This problem is better explained by using a pack of standard cards which is more practical. The problem is beautifully treated in the very first chapter by John Derbyshire in his excellent book โ€˜Prime obsessionโ€™
The total overhang is with just 4 bricks (not counting the base card).
The total over hang is 1/2 + 1/4 +1/6 + 1/8 = 25/24.
Yes your are right Paul its half the harmonic series.
Some interesting facts he gives are: For 51 cards the total overhang is a shade less than 2.2594065907334.
For 100 cards itโ€™s a tad less than 2.58868875887.
For a trillion cards we have a bit more than 14.10411839041479 overhang.
The total over hang increases very slowly as the number of cards goes up but never comes to an end as the harmonic series is divergent See proof by Nicole dโ€™Oresme.
mathworld.wolfram.com/HarmonicSeries.html - 32k - Cached - Similar pages
Mally.
P.S. 'Prime Obsession' is a must read for anyone interested in the RH.
mfgoode is offline   Reply With Quote
Reply

Thread Tools


All times are UTC. The time now is 20:58.


Wed Jan 19 20:58:44 UTC 2022 up 180 days, 15:27, 1 user, load averages: 1.13, 1.29, 1.28

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2022, 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.

โ‰  ยฑ โˆ“ รท ร— ยท โˆ’ โˆš โ€ฐ โŠ— โŠ• โŠ– โŠ˜ โŠ™ โ‰ค โ‰ฅ โ‰ฆ โ‰ง โ‰จ โ‰ฉ โ‰บ โ‰ป โ‰ผ โ‰ฝ โŠ โŠ โŠ‘ โŠ’ ยฒ ยณ ยฐ
โˆ  โˆŸ ยฐ โ‰… ~ โ€– โŸ‚ โซ›
โ‰ก โ‰œ โ‰ˆ โˆ โˆž โ‰ช โ‰ซ โŒŠโŒ‹ โŒˆโŒ‰ โˆ˜ โˆ โˆ โˆ‘ โˆง โˆจ โˆฉ โˆช โจ€ โŠ• โŠ— ๐–• ๐–– ๐–— โŠฒ โŠณ
โˆ… โˆ– โˆ โ†ฆ โ†ฃ โˆฉ โˆช โŠ† โŠ‚ โŠ„ โŠŠ โŠ‡ โŠƒ โŠ… โŠ‹ โŠ– โˆˆ โˆ‰ โˆ‹ โˆŒ โ„• โ„ค โ„š โ„ โ„‚ โ„ต โ„ถ โ„ท โ„ธ ๐“Ÿ
ยฌ โˆจ โˆง โŠ• โ†’ โ† โ‡’ โ‡ โ‡” โˆ€ โˆƒ โˆ„ โˆด โˆต โŠค โŠฅ โŠข โŠจ โซค โŠฃ โ€ฆ โ‹ฏ โ‹ฎ โ‹ฐ โ‹ฑ
โˆซ โˆฌ โˆญ โˆฎ โˆฏ โˆฐ โˆ‡ โˆ† ฮด โˆ‚ โ„ฑ โ„’ โ„“
๐›ข๐›ผ ๐›ฃ๐›ฝ ๐›ค๐›พ ๐›ฅ๐›ฟ ๐›ฆ๐œ€๐œ– ๐›ง๐œ ๐›จ๐œ‚ ๐›ฉ๐œƒ๐œ— ๐›ช๐œ„ ๐›ซ๐œ… ๐›ฌ๐œ† ๐›ญ๐œ‡ ๐›ฎ๐œˆ ๐›ฏ๐œ‰ ๐›ฐ๐œŠ ๐›ฑ๐œ‹ ๐›ฒ๐œŒ ๐›ด๐œŽ ๐›ต๐œ ๐›ถ๐œ ๐›ท๐œ™๐œ‘ ๐›ธ๐œ’ ๐›น๐œ“ ๐›บ๐œ”