Thursday, August 14, 2008

Round 274

Date Released: August 14th, 2008.
Date Rewarded: August 21st, 2008.

A Chomby was walking down a path one day when Jhudora appeared in front of it. "Solve this puzzle, and you may pass," she said.

Suppose you have a bunch of wooden beds, each weighing 200 pounds each, arranged in a pyramid. The base layer is 5 beds by 5 beds, the layer above is 4 beds by 4 beds, and so on until the top layer of one bed. They are arranged in such a way that each bed is resting on four beds below it, one leg on each bed.

If the weight on each bed is always equally distributed to all four legs, and each bed leg can only support 200 pounds, how many pounds can you place on top of the topmost bed? Please round to the nearest pound, and plese submit only a single number as your answer.


Prize


Wooden Bed

Click to show/hide solution

If the weight on each bed is always equally distributed to all four legs, and each bed leg can only support 200 pounds => Each bed can support maximum 200 x 4 = 800 pounds.

Each bed weighs 200 pounds. Let x be the extra weight you can put on the top of the topmost bed. We will go from top to bottom level.

Top level: total weight = x + 200. Each bed leg can withstands: (x + 200) / 4.

2nd top level: see the below figure for visual help, the four red dots are the four legs of the top level bed. The total weight of each bed (including itself) = (x + 200) / 4 + 200 = (x + 1000) /4. Each bed leg withstands: (x + 1000) / 16.


3rd level: see the below figure for visual help, the red dots are the 2nd top level beds' legs.

* middle bed: the bed withstands 4 legs of the above 4 beds (one of each). Therefore, the total weight should be: (x + 1000) / 16 * 4 + 200 = (x + 1800) / 4. Each bed leg withstands: (x + 1800) / 16.

* side beds: each bed withstands 2 legs of the above 2 beds (one of each). Therefore, the total weight should be: (x + 1000) / 16 * 2 + 200 = (x + 2600) / 8. Each bed leg withstands: (x + 2600) / 32.

* corner beds: each bed withstands 1 leg of the above 1 bed. Therefore, the total weight should be: (x + 1000) / 16 + 200 = (x + 4200) / 16. Each bed leg withstands: (x + 4200) / 64.


4th level: see the below figure for visual help, the red dots are the 3rd level middle bed's legs, the yellow dots are the 3rd level side beds' legs and the blue red dots are the 3rd level corner beds' legs.

Since we only concern about the bed that withstand the most weights, we can ignore the side and corner beds and only consider the 4 middle beds which contributes most weights into the middle bed on the bottom level (see the bottom level figure). Each of the 4 middle beds withstand 1 leg from a 3rd level corner bed, 2 legs from 3rd level side beds, 1 leg from the 3rd level middle bed. Therefore, the total weight of each 4th level middle bed should be: (x + 1800) / 16 + (x + 2600) / 32 * 2 + (x + 4200) / 64 + 200 = (9x + 34600) / 64. Each bed leg withstands: (9x + 34600) / 256.


Bottom level: see the below figure for visual help, the red dots are the 4th level middle beds' legs, the yellow dots are the 4th level side beds' legs and the blue red dots are the 4th level corner beds' legs.

Similar to the 4th level, we only concern with the bed that has the most weight on. That bed is the middle bed. Therefore, the weight of on that bed is: (9x + 34600) / 256 * 4 + 200 = (9x + 47400) / 64. Each bed leg withstands: (9x + 47400) / 256.


We know that each beg leg can only withstand maximum 200 pounds. From all the levels, the bed that has most weights on is the middle bed of the bottom level. Therefore:
(9x + 47400) / 256 <= 200 <=> x <= 3800/9 = 422.22

Therefore, the answer is 422.

Results: 1432 people guessed the correct answer earning themselves 1397 NP each.

22 comments:

Rodrunner said...

i am confused with the word extra.

camila said...

I´m confused with the word extra too. I though the challenge was to say not the extra weight on bed, but how much wight you can place without breaking the pyramid.

I answered 400 pounds, but I really don´t know if this is the correct answer, but one thing I know: If you put +800 pounds on the topmost bed, the pyramid will crush.

victor said...

hey, look this, on the lc says that u "equally" distributs the weight of the bed on the four legs, if the 2X2 on the top uses 4 legs to support the top bed and the bed has 200 pounds you should divide the pounds by 4, thats all i can say the rest you do guys!!

LC Solver said...

the answer is between 1500-2000 pounds
---
this is not true. If it is, then the topmost bed already broken.

chrischois said...

ha. very true. then you guys got the answer already.

cassby916 said...

My answer came out to over 5000 pounds... o_0

jake said...

im serious!! i dont get this at all!!! please, can someone please tell me the answer?? i want the trophy so bad!

camila said...

....
My answer gave 1,648. Can someone say what I did Wrong?

9x+47400/256=200
9x=200-47400/256
9x=200-185,15
9x=14,84
x=14,84/9
x=1,648

Rodrunner said...

i submitted a wrong answer but i will help you with this:
let me ask you first if u are aware of cross-mulitiplication.
if so, cross multiply the denominator of the fraction part to the right side of the equation.

(9x+47400)/256=200
9x +47400 =200*256
9x=(200*256)-47400
x= [(200*256)-47400]/ 9

camila said...

xD I didn´t remember this part! Thanks!

chrischois said...

actually, i think now that my answer between 1500-2000 is correct. this weight won't break the topmost bed because the topmost bed is not the only bed supporting this weight. whatever weight you place on top will be distributed to the bottom, and thus it is not one bed on the top that is holding the extra weight, but the entire pyramid supporting it. in terms of Newton's Laws, the normal force acting on the topmost bed is not >200 per leg, though the weight is far greater.

DarkAngel said...

I got an answer between 300 and 500 is that right and i thought an interger was a negative? does that mean you round to whole number??

wouldnt you be able to have more than 800 pounds since the bed is just forcing its weight on the other beds and since they can hold weight would you need to add the weight they can hold to what the top bed can hold. diffrent theory though.

Sworhdo Flare said...

Lol at first I tried to figure out a formula of my own, then i got the same as LC Solver's, so I slowly worked it out. At first I got like 1.6 pounds when rounded off (oops!) ah then I realized where I went wrong.

Camilla I also made that mistake. You cannot transfer a x/y over to the right side like that. That was my answer too haha.

Just follow rodrunner. He has... how do I put it... simplified it for us.

Don't worry, between 300 to 500 poudns is right. Don't reveal more than that if you want a good share from the pool.

XD P.S Camila, your first answer was closer by a lot. 1.648pounds is impossible.

Tip: After you get your answer, sub it back into the equation or work it back out to see if it works.

ponylover9 said...

i'm really stuck. I'm only in year three so i don't know cross multiplication. I barely know what cross multiplication means. Please help!! PLEASE!!

Deuce Loosely said...

I took the question to ask what the maximum amount of weight can be placed on the top bed without breaking the bottom level. To that end, I figured this...

Bed = 200lbs each
Each leg = 200lbs support
Max support of each bed = 800lbs
Total support of 55 beds =

* Since the four corner beds of the bottom level each only carry one leg of the bed above them, that would equal 50lbs it's carrying, leaving 750lbs free.
* All beds along the edge between the corners each carry two legs of the beds above them, totaling 100lbs each and leaving 700lbs of support unused.
* The nine remaining beds in the center each support four legs, equaling 200lbs and leaving 600lbs of support unused.
* There is a total of 25 beds on the lowest level, each of which can carry up to 800lbs. The maximum amount of weight this level could support would be 20000lbs.
* With four beds leaving behind a total of 3000lbs support, 12 leaving 8400lbs, and the remaining nine leaving 5400lbs, I figured the total of unused support to be 16800lbs.

I took that number and set it aside before figuring for the next level up...
* Max support =
16 beds x 800lbs support =
12800 lbs
* Four corners = 200lbs supported
* Eight edge beds = 800lbs
* Four center beds = 2400lbs
* Support used = 3400lbs
* Leftover support =
12800 - 3400 =
9400lbs

Level 3 by this method...

200 corner + 400 edge + 200 center = 800 supported; subtracted from possible 7200 support = 6400 left.

Level 4 only supports the top bed = 200lbs, leaving 3000lbs support unused.

My guess for maximum amount of weight to be placed on top bed without crushing the legs of the lower level...

16800 + 9400 + 6400 + 3000 + 800 =
36400lbs.

I entered a wrong number for my guess and don't know why...

RITBedBugger said...

ponylover9, say you have an equation like this:

a c
- = -
b d

cross multiplication is when you multiply the top parts by the opposite bottom parts, like this:

a * d = c * b

Also, remember that if there is no bottom number (divisor) then use 1 as the divisor.

RITBedBugger said...

Sorry, that looks dumb because they deleted some spaces, I meant

a/b = c/d

ponylover9 said...

I still don't get it!! Waah!!

jake said...

ya, i dont get it either, well, there doesnt seem much of a point of trying to figure out the answer now, cause more than 250 people have probably already got it right.

Sworhdo Flare said...

Cross Multiplication = if you have an equation involving 2 fractions, a/b = c/d, sry i can't put fraction format here

Imagine a is at the top left, c is top right, b is bottom left, and d is bottom right, you draw a cross, and multiply a by d and c by b which leads to ad = cb

hard to explain here. anyway, chrischoris or wadeva your name is, talking about newton's laws as a general term doesn't make you look really smart. If I wanted to I could say something like "The laws of trigonometry states that sinX = opposite/hypothenuse..." etc. and your answer of 1500 pounds to 2000 pounds is definitely wrong.

P.S ponylover if you still don't understand, you can check it up on wikipedia ^^ or if you're still young you'll learn in it school later.

camila said...

I´m really curious to know the answer this week!

Marcella said...

I've just got to say...WOW. I love math, but when it comes to problems like this, I tend to tackle them visually. When I realized this problem was three-dimensional, I went crazy! I realized I'd need to figure out some equations, but my brain didn't feel like doing it (still in summer mode, school hasn't started yet lol). But anyway, your breakdown makes perfect sense, and thank you for explaining it!