…
Ali… That wasn’t very nice.
hmm… Chinese food… drools
fish
Ali, the difference between a set and a sequence is that a set is unordered and a sequence is ordered. There is no difference between a sequence and a series.
Also, the form of induction you posted is “mathematical induction”, which is not at all the same thing that is called “induction” in logic. In fact, “mathematical induction” is a form of what’s called “deduction” in logic. Logic “induction” involves extrapolating, and never “prooves” anything, whereas deduction proves the result provided the argument is properly constructed and the premises are true (“soundness” and “validity”, but I forget which is which!)
An example of induction: The sun has risen every day my whole life. Therefore, it will rise tomorrow.
This concludes today’s lecture!
| Quote (learjeff @ Feb. 04 2005,08:40) |
| Ali, the difference between a set and a sequence is that a set is unordered and a sequence is ordered. There is no difference between a sequence and a series. Also, the form of induction you posted is "mathematical induction", which is not at all the same thing that is called "induction" in logic. In fact, "mathematical induction" is a form of what's called "deduction" in logic. Logic "induction" involves extrapolating, and never "prooves" anything, whereas deduction proves the result provided the argument is properly constructed and the premises are true ("soundness" and "validity", but I forget which is which!) An example of induction: The sun has risen every day my whole life. Therefore, it will rise tomorrow. This concludes today's lecture!
|
You tell 'em, Learjeff! I love the fact that mathematical induction ain't really induction! (well that perhaps reveals my parochialism.)
Nergle, I thought mathematical induction invovled any use of a base case with a property plus a rule that preserves the property, so that I had my base case (the interval between the numbers) and the rule (add 1 each iteration to those intervals) preserving a property of uniqueness. Like I said, I'm not a mathematician...but we do use mathematical induction in soundness/completeness proofs in logic.
| Quote (nergle @ Feb. 01 2005,20:25) |
| OK, here's one for you Fish, and all the rest of you brainboxes out there. In the following series of numbers, which is the odd one out? 1,3,5,8,9,12,17,21,22,25,27,30. Ali |
Is that perhaps some intervals for a chord from some Beatles song?
How about the number of days between smokes while weening off cigarettes?
No? I didn’t thinks so.
LOTS of 12 stuff can go in the list.
Looks like some sort of musical intervals to me…
…
…
Good point: when we use the term “series” in math, we’re generally using it as shorthand for “sum of sequence of terms”, or equivalently, “the sum of a series of terms”. However, you might also be talking about “the product of a sequence/series of factors”. Sum and product are just two operators that can be used and that have handy Greek symbols. (For additition, it’s a capital sigma. For multiplication, I don’t know the greek letter, but it looks like a big C.)
In English (this side of the pond or that), the terms “sequence” and “series” mean pretty much the same thing, an ordered set. But the term “series” implies more of a relationship between the successive members. It’s used this way in physics, chemistry, and geology. So, you’re right, they’re not precisely the same thing, and “sequence” would be the better term.
OK, I couldn’t resist–I needed to exercise my brain a little (well, a lot for me, a little to some of you guys 
). I looked around the net to see if I could find mention of this problem just to make sure Ali was for real this time 
, and found it mentioned once, but with no answer provided, so I promise I didn’t cheat . Anyway, this solution looks like it works, but involves a decision tree, so I kinda put it in an outline form in my own lame “code” to shorten the explanation. I may put it in plain prose later if possible, but right now I don’t want to look at this problem anymore
Oh, and like Ali, I promise it’s not a joke answer–if it’s wrong, yes, I’m really that dumb; sorry for wasting your time (but please point out my error(s)!). And if there’s a simpler correct answer, ditto.
(BTW if you like this kind of stuff, check out this site.)
Tony
Notation:
<w#> : number of the weighing where # is 1 for first, 2 for 2nd, and 3 for 3rd
abc/xyz–pqr: balls a,b,&c are being weighed on the left side against x,y,&z on the right, with p,q,&r left out.
(abcd): balls in parentheses (a,b,c,&d here) are known to NOT be the odd one out
(a): ball a is known as NOT the odd one out
[n]: ball n IS the odd one out
-b- : case in which weighed balls balance the scale
-u- : case in which weighed balls are unbalanced on scale
status: gives current status of “safe” balls vs. questionable ones
NOTE1: Direction of tilt must be observed/recorded in all unbalanced weighings for reference
NOTE2: Indentations give the hierarchy of the weighing results and subsequent weighing decisions
The Weighings:
: 123/456–789 (<–i.e., 123 on left weighed against 456 on right, 789 left out)
-b- (i.e., balanced-scale case)
- status: 789 (123456)
7/8–9
-b-
- [9] is odd one out; determine heavy/light
(7)/[9]–(1234568)
-b- (balanced case not possible)
-u- if [9] goes up, it’s lighter, down it’s heavier (duh)
-u-
- status: 78 (1234569)
7/(9)
-b- [8], tilt of determines heavy/light
-u- [7], tilt determines heavy/light
-u- (i.e., unbalanced-scale case)
- status: 123456 (789)
: 12/34–56 (789)
-b-
- status: 56 (1234789)
: 5/6–(1234789)
-b- (balanced case not possible)
-u- compare tilt direction to
- [6] if tilts as in
- [5] if tilts opposite from
-u-
- status: 1234 (56789)
- NOTE: ball 3 changed sides from , so tilt direction gives more info:
- same tilt as :
status: 124 (356789)
1/2–4 (356789)
-b- [4], tilt of tells light/heavy
-u-
- [2] if tilt changed from
- [1] if tilt same as
- [3] if tilt opposite from
- refer to previous weighing to determine heavy/light
…
So I'll give you the benefit of the doubt. (Is this the strategy you used to get through school? LOL).
Gah! Why didn't I think of that?
Divide the balls into 3 groups of three, called A, B and C.
Ok, I've seen enough--looks like we're in agreement on this Actually, though, everything does seem to fall into place after you've made that initial decision. I also looked at 4-4-1 and 2-2-5.
OK, your prize is a night spent with Halle Berry, (or Brad Pit, (or both!)) just tell her/him/both that Ali sent you, and it'll be fine, honest! lol
Ali
Sorry, Ali, but my wife said I can't go unless I take the kids...and leave Brad Pitt with her. I knew I shouldn't have asked for permission!
Tony
Gah! Why didn't I think of that?
Divide the balls into 3 groups of three, called A, B and C.
Ok, I've seen enough--looks like we're in agreement on this
Actually, though, everything does seem to fall into place after you've made that initial decision. I also looked at 4-4-1 and 2-2-5.OK, your prize is a night spent with Halle Berry, (or Brad Pit, (or both!)) just tell her/him/both that Ali sent you, and it'll be fine, honest! lol
Ali
Sorry, Ali, but my wife said I can't go unless I take the kids...and leave Brad Pitt with her. I knew I shouldn't have asked for permission!
Tony
…
I don’t do mixed drinks Ali. Sorry. The best riddle I can come up with for you guys is this…
You are walking down the river in a rowboat and the wheels fall off. How many pancakes does it take to roof a doghouse?
Take your time…
TG
| Quote (nergle @ Feb. 08 2005,18:43) |
| After a long period, (and some sacrificial undercover work by one of the girls), it transpired that he was an organ player who worked in a crematorium! I just had this vision of the recently deceased tootling along the conveyor belt to their ultimate toasting, (or pen-ultimate, depending on your beliefs), being accompanied by some guy trying to play Maxwell's equations propped up on top of his Hammond! LOL |
Wow, sounds like a tough crowd to me. ba-dum-dum.. That's one gig where I would be hoping and praying NOT to hear any applause. ba-dum-dum...
About the problem, this is pretty much just a straight math problem, right? Unless there is something really obvious I missed, which is always a possibility. Actually I think I've heard this one before, but I probably spent too much time looking at the girl sitting next to me as well, and never bothered to solve it. Oh well, I'm always prepared to make a fool of myself, so:
Common sense answer: Drink alcohol and make pass at girl sitting next to me in class. (har har har)
Short math-based answer: They are contaminated equally.
Long math-based answer:
Container W = container originally containing 100% water
Container A = container originally containing 100% alcohol
INITIAL STATE:
- Container W = 1 litre of pure water
- Container A = 1 litre of pure alcohol
1) Take .005L of water out of W
results:
- purity of W = 100%, but volume is drops to .995 litre
- purity of A = 100%, volume is 1 litre
2) Put .005 litres of water into container A (alcohol)
results:
W = purity/volume unchanged from step 1
A = 1 litre alcohol, .005 litre water
- % of alcohol is 1/1.005
- % of water is 1 minus alcohol purity, or (1-(1/1.005))
3) take .005 litre of alcohol/water mix out of container A
- this doesn't change the purity/contamination of the water, just the volume
- removed portion contains (.005 *(1/1.005)) litres of alcohol
- removed portion also contains (.005 * (1-(1/1.005)) litres of water
- W still unchanged
- volume of A drops to 1 litre now, purity is unchanged
4) Pour removed liquid in step 3 into container W
results:
Volume of W is now 1 litre again
W = (.995 litres) + (.005 litres *(1-(1/1.005))) litres of water
A = (1/1.005) litres of alcohol (as per step 2)
Being lazy algebra-wise, I plug these into my trusty calculator and find:
Water purity is 0.9950248756218905472636815920398
Alcohol purity is 0.9950248756218905472636815920398
These are close enough that I feel I don't need no stinking algebra to tell me they're the same! In other words, I fail for lack of clarity.
Now please, Ali, no more of these things! Besides their being wildly OT, I need to get some work done! Errr..., unless you know Charlize Theron as well...
Tony
…
Ok, no more questions Tony, you're much too sharp.
Uh oh, here it comes...
However, an easier answer.
Ouch!
...
(humblingly simple/clear solution removed)
...
See! there wasn't even any need to specify the size of the tea-spoon.
*silence*
Ahem...very elegant! However, mine is more obfuscated by unnecessary calculations, making it more impressive and intimidating to the unaware, so that should count for something. I knew there was a general case in there somewhere, but after the first problem, I didn't have enough brain power left to bother! Anyway, thanks for the brain teasers (including the "prize")
(I'm not sure if I want to touch gtr4him's riddle...smell's like a trap)
Tony
Uh oh, here it comes...
However, an easier answer.
Ouch!
...
(humblingly simple/clear solution removed)
...
See! there wasn't even any need to specify the size of the tea-spoon.
*silence*
Ahem...very elegant! However, mine is more obfuscated by unnecessary calculations, making it more impressive and intimidating to the unaware, so that should count for something. I knew there was a general case in there somewhere, but after the first problem, I didn't have enough brain power left to bother! Anyway, thanks for the brain teasers (including the "prize")
(I'm not sure if I want to touch gtr4him's riddle...smell's like a trap)
Tony