Logic/Math problem

Here's a logic problem for you to dwell on please don't post solutions for a few hours after this post to give everyone a chance to work it.

What you have:
12 cannon balls. A scale that measures 1 side vs the other (no numbers just the left side is heavyer or the right side is heavyer)

The Problem: 1 of the cannon balls is different (heavyer or lighter), you need to determine which cannon ball is different you only get 3 weighings to figure this out.
You don't know if it's heavyer or lighter.

also: if you like this problem I can post more at later dates so send me feedback.

For 1 heavier:

6 on each side > discard the lighter side
devide the heavier side 3 on each side > discard the lighter side
place 1 on each side the heavier one will shoe if in the scale, if it dosn't it's in your hand

reverse the discard process for lighter

you don't know if it's lighter or heavyer

"you don't know if it's lighter or heavyer" oh crud, missed that line. that brings it up to 4 tries back to the drawing board.

Quote:

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Originally posted here by Tedob1
For 1 heavier:

6 on each side > discard the lighter side
devide the heavier side 3 on each side > discard the lighter side
place 1 on each side the heavier one will shoe if in the scale, if it dosn't it's in your hand

reverse the discard process for lighter

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If you need to determine "heavier or lighter," think you need a fourth weighing... if the second weighing balances, you need to move to the other six.

Then again, I've once again been up like over 24 hours.

Take the 12 balls and divide them into 3 groups of four and number them from 1-12....

1)Compare 1,2,3,4 to 5,6,7,8 if equal then the ball is in the third group of 9,10,11,12.

2)Compare 1,2,3 with 9,10,11 if equal, the answer is 12 and compare it to 1 to find out if it is heavier or lighter.

3)If step 2 is not equal then compare 9 and 10, if equal answer is 11. If step 3 is not equal choose the one that matches the fulcrum tilt in step 2.

Now for the tricky part

4)If step 1 is not equal compare balls 1,2,3,5 to 4,10,11,12. You know from step one which way the tilt was and we will assume it tilted down on the left..it doesn't matter.

5)If step 4 is equal then compare 6 to 7, if equal the answer is 8 and it is the light one. If not equal take the lighter one of 6&7.

6)If step 4 was not equal and tilted down on left side, compare 1 to 2, if equal the answer is 3 and it is heavy. If not equal choose the heavier of 1 & 2.

7) If step 4 tilted down on right side compare 4 to 9, if equal then 5 is the the light ball, if not then 4 is the heavy ball.

I would prop the tilt up with my knee and roll all the cannonballs down at the same time. There should be one that rolls at a slower or faster rate than the others. Thats the culprit.

Since this obviously won't work with comparing 6 to 6, I started a little different (credit where credit is due, my younger brother came up with most of this - I DO understand how it works though, and I take credit for that :D ).

Here goes...

Let's give the balls a number from 1 to 12 to make this easy (easier...).

First weighing: 1,2,3,4 against 5,6,7,8

A. 1,2,3,4 balances with 5,6,7,8 --> the answer is in 9,10,11,12

--> Second weighing: 1,2,3 against 9,10,11

A1. 1,2,3 balances with 9,10,11 --> The answer must be 12 (since 1 - 11 all balance).

--> Third weighing: compare 12 to one of the other 11 to find out if 12 is heavier or lighter.. Problem solved.

A2. 1,2,3 does not balance with 9,10,11 --> The answer must be in 9,10,11.

--> Third weighing: compare 9 to 10.
If 9 balances with 10, the answer is 11, and assuming the balance went down on the left with our 1,2,3 vs. 9,10,11 weighing, 11 is lighter. If the balance went down on the right with our 1,2,3 vs. 9,10,11 weighing, 11 is heavier. Problem solved.

If 9 does not balance with 10, the lighter one of the two is the answer if the balance went down on the left with our 1,2,3 vs. 9,10,11 weighing. If the balance went down on the right with our 1,2,3 vs. 9,10,11 weighing, the heavier one of the two is the answer. Problem solved.

That's all for the case where 1,2,3,4 = 5,6,7,8.

B. 1,2,3,4 does not balance with 5,6,7,8 --> the answer is in 1,2,3,4,5,6,7,8

Second weighing: 1,2,3,5 against 4,10,11,12.

B1. 1,2,3,5 = 4,10,11,12 --> The answer must be 6,7 or 8.
Third weighing: 6 compared to 7
If 6 = 7, the answer is 8. If the balance went down on the left with our 1,2,3,4 vs. 5,6,7,8 weighing, 8 is lighter. If not, 8 is heavier. Problem solved.
If 6 and 7 don't balance, the lighter one of the two is the answer. Problem solved.

B2. 1,2,3,5 does not balance with 4,10,11,12.
Since the answer must be in 1,2,3,4,5,6,7,8, and 6,7,8 are already covered in the previous case, the answer must be in 1,2,3,4 or 5.

Third weighing: 1 compared to 2
If the balance goes down on the left, compare 1 to 2. If they balance, 3 is the answer (and is heavier). If they don't balance, the heavier of the two is the answer (and is, of course, heavier). Problem solved.

Third weighing: 4 compared to 9
If the balance goes down on the right, compare 4 to 9. If they balance, 5 is the answer (and is lighter). If they don't balance, the heavier of the is the answer. Problem solved.


Understanding it wasn't the hardest part... writing it down was... This is not something I wanna do every day... ;)

you put 6 balls on an counter ... witch part is easier or heavyer and then you put
the remaining 6 balls on the counter ... same .. 3 balls easier or heavyer ....
then you have 3 balls . ... one of them is easy or heavy that the rest ..
soo you put 2 balls on the counter .... if the balls are equal then the ball that remains is
the diferent one ... or one of the balls ai easi or heavy and you get the ball you want .
(sorry for my language .... but i'm kindda lazy :) )

Quote:

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Originally posted here by xmaddness
I would prop the tilt up with my knee and roll all the cannonballs down at the same time. There should be one that rolls at a slower or faster rate than the others. Thats the culprit.

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Actually the only thing that would cause a cannonball to roll faster or slower would be friction between the cannonball and the board. So since they do not discuss the texture of the balls, or the scale, this would not work. They would all roll at the same rate regardless of weight. Everything falls at the same rate 22ft/sec/sec unless acted upon by another force. That's the beauty of gravity.

Mohaughn: I haven't taken physics in quite some time, but isn't it 32 ft/s/s? (I may be wrong, but 32 sounds more familiar than 22)

AJ

Quote:

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Originally posted here by avdven
Mohaughn: I haven't taken physics in quite some time, but isn't it 32 ft/s/s? (I may be wrong, but 32 sounds more familiar than 22)

AJ

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Thanks for the correction.. It is 32.. Not sure if that was a typo or a brainfart..

mohaughn, i had to study your logic to understand it, when i did, it all came so clear, that was brilliant!

Thanks zepherin for an intriguing post. do it again! yeah! do it again!

mohaughn, correct me if i'm wrong, but thats in freefall, not on an in-cline. Its the friction your talking about that will make it work. The different cannonball will have different inirtia. This difference in inirtia will cause it to roll against the friction faster or slower. 32/ft/s^2, absolutly not on an incline. I could make the incline 1 degree below horizontal and it would take about ten minutes for it to get up to that speed. 32ft/s^2 only in freefall

Neg's post is dead on.

The quickest way to come to that conclusion (but lacking the incredibly thorough detail Neg included in his post) is by the simple idea that for each unknown, you are dividing the pile once.

In this scenario, there are two unknowns: whether the ball is lighter/heavier, and which ball it is.

So we need to perform 2 equal divisions, and end up with three identical piles. Negative's method is dead on. :)

I was just curious. Didn't I make a post about copying and pasting earlier today, and how quoting sources is illegal. Gee, maybe Negative is right and I do have to write a tutorial on it. It is really bad when you don't even cite your source.....

We (my logic class group) solved it in a similar way to negative, http://www.antionline.com/showthread...=541833#541833 my math teacher (different class) used a wicked line of semitry rule and stacked them like bowling pins (10 of them) and kept restacking them, and then wrote a formula on how many minimum times it would take for any n amount of cannon balls.

Well, since he never did post a source......
http://www.greenspun.com/bboard/q-an...?msg_id=003Tgo
does that look kinda familiar to anyone....

Quote:

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It appears I am going to have to defend the army here and show where
the country gets its smarts from...no offence navy...
Take the 12 balls and devide them into 3 groups of four and number
them from 1-12....now try and keep up navy boy...
1)Compare 1,2,3,4 to 5,6,7,8 if equal then the ball is in the third
group of 9,10,11,12. 2)Compare 1,2,3 with 9,10,11 if equal, the
answer is 12 and compare it to 1 to find out if it is heavier or
lighter. 3)If step 2 is not equal then compare 9 and 10, if equal
answer is 11. If step 3 is not equal choose the one that matches the
fulcrum tilt in step 2.

Now it gets tricky...4)If step 1 is not equal compare balls 1,2,3,5
to 4,10,11,12. You know from step one which way the tilt was and we
will assume it tilted down on the left..it doesn't matter. 5)If step
4 is equal then compare 6 to 7, if equal answer is 8 and it is the
light one. If not equal take the lighter one of 6&7. 6)If step 4
was not equal and tilted down on left side, compare 1 to 2, if equal the answer is 3 and it is heavy. If not equal choose the heavier of
1 & 2. 7) If step 4 tilted down on right side compare 4 to 9, if
equal then 5 is the the light ball, if not then 4 is the heavy ball.

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about 3/4 of the way down the page.....

Quote:

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Originally posted here by xmaddness
mohaughn, correct me if i'm wrong, but thats in freefall, not on an in-cline. Its the friction your talking about that will make it work. The different cannonball will have different inirtia. This difference in inirtia will cause it to roll against the friction faster or slower. 32/ft/s^2, absolutly not on an incline. I could make the incline 1 degree below horizontal and it would take about ten minutes for it to get up to that speed. 32ft/s^2 only in freefall

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The mass of an object is in no way a factor of it's inertia. The two factors that do come into play are the incline of the ramp and the inertia on the object caused by gravity. Galileo used ramps to prove that all objects fall at the same rate regardless of weight as long as the amount of friction is negligible.

http://www.lightandmatter.com/htmlbook/bk1ch03.html

or search for Galileo's incline

Wow... the memories of taking physics. lol I never thought I could actually learn (or at least get refreshed) in Physics here. :-)

AJ