This site makes extensive use of JavaScript.
Please enable JavaScript in your browser.
Live
PTR
10.2.7
PTR
10.2.6
Beta
A slightly mathematical riddle.
Post Reply
Return to board index
Post by
Hyperspacerebel
P.S. to hyper the OP ask what is the least amount he would need to buy. your answer is correct but it is not the least.
But I already demonstrated
why
my number is the least.
Burning a candle is defined in the problem as burning (vaporizing) 3 units of candle. So if we are burning 1600 candles, we are consequently burning at least 4800 units.
The same can be applied to the metal. Making a square is defined (by extension) as vaporizing 3 bars. As such, to make 1600 squares, we need to vaporizes 4800 units.
Post by
351418
This post was from a user who has deleted their account.
Post by
Hyperspacerebel
And oh duh! I figured out what you guys are doing wrong.
Every 5th candle is free, not every 4th.
That would change your solution to 320 +64 + 12.8 + 2.56 + 0.512 + 0.1024 + .... =~ 400
1600 - 400 = 1200 = 4800 units.
Then +1 for the final candle stump that is left over for a grand total of 2801.
Post by
Hyperspacerebel
I thought he asked how many units were needed to make the candles not how many units were consumed.
Think about it.
What's the minimum amount needed? Well, theoretically it would be the same as the amount consumed. But in this case there is always a candle stump left over. So the minimum needed is the amount used + 1.
Post by
351418
This post was from a user who has deleted their account.
Post by
Hyperspacerebel
following your math I get 1599 free units.
Nope, my math gives you exactly 1600 free units. You realizes that you have to continue that series into infinity, right? It's an infinite series that ultimately equals 1/4 (400/1600 candles in this case).
When all is said and done, 1/4 of the candles are free. But that's
after
taking into account the free candles you get from the free candles.
Post by
351418
This post was from a user who has deleted their account.
Post by
Hyperspacerebel
Nope, my math gives you exactly 1600 free units. You realizes that you have to continue that series into infinity, right? It's an infinite series that ultimately equals 1/4.
no it never reaches 0 the infinite series always grows closer but doesnt reaches 0.
so the last unit is not a complete unit.
The third poster put is simply it takes 4 units to make the first candle after that it takes one stub and 3 new units
(1599*3)=4797+4 (first candle) =4801
i think u agreed with him
Yes the series = 400 ... and this example is practical proof that it does.
Methinks you should read about
infinite series
.
Post by
351418
This post was from a user who has deleted their account.
Post by
Hyperspacerebel
From your link
It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, ½, ¼, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼.
This argument does not prove that the sum is equal to 2
(although it is), but it does prove that it is at most 2. In other words, the series has an upper bound.
the upper bound or "limit" of the problem is 400 but the infinite series never reaches the limit. if it did it wouldnt be infinite.
on a graph it never reaches the upper bound
Look at the very thing you quoted...."
This argument does not prove that the sum is equal to 2
(
although it is
)"
The sum = 2 (or in our case 400/1600).
All the argument above proves is that the sum is equal to at most 2...further proofs are made to narrow it down to exactly 2.
Post by
484763
This post was from a user who has deleted their account.
Post by
Hyperspacerebel
No,the note is referring to the fact that it is 2 is given.
It's refering to the fact that it
is
2. There is a reason we use equal signs in infinite series.
And it states afterwards that this only proves that
it is at most
2
Thank you for saying the exact same thing I said.
And basic math says
1600*4=6400 (converting to units from candles)
6400-1600=4800 ( here we assume there are 1600 free units as you say)
Practical proof shows that the answer 4801
6400-4801=1599 , this is proof that the free units are 1599.
I'm not sure what the heck you are talking about.
Let me illustrate my 2 proofs again since you don't seem to understand them.
First Proof:
This is I guess what you're referring to as a 'practical proof.'
Burning a candle is defined as completely consuming 3 units. Since we are burning 1600 candles, we will be consuming 4800 (ie 1600 * 3) units. However there will be a single stump left due to the fact that you can only burn 3/4 of the candle. Thus our final answer is
4801
.
Second Proof:
Every 5th candle is free, so given 1600 candles, 1600/5 or 320 will be free. But then 1/5 of those or 64 (which is 1/25th) are free, and so on (ad infinitum).
So we can set this up like this (i'm not totally sure how to write this correctly given this forum's formatting, but if you understand calculus, you'll get it):
∑(n=1 to infinity) of 1 / (5^n)
We solve by plugging into a / (1 - r)
a = the first element 1/5
r = the ratio, also 1/5
(1/5) / (1 - ) = 1/4
So that means that exactly 1/4 of the candles are free, taking them all into account.
1600 / 4 = 400
1600 - 400 = 1200
And again, since we defined burning a candle as consuming 3 units, we multiply by 3
1200 * 3 = 4800
And again, the final candle must leave a stump, so we add 1.
4800 + 1 = 4801
Now putting calculus aside, we can also do the above calculations by intuition.
320 = 320
320 + 64 = 384
320 + 64 + 12.8 = 396.8
320 + 64 + 12.8 + 2.56 = 399.36
320 + 64 + 12.8 + 2.56 + 0.512 = 399.872
320 + 64 + 12.8 + 2.56 + 0.512 + 0.1024 = 399.9744
320 + 64 + 12.8 + 2.56 + 0.512 + 0.1024 + 0.02048 = 399.99488
320 + 64 + 12.8 + 2.56 + 0.512 + 0.1024 + 0.02048 + 0.004096 = 399.998976
320 + 64 + 12.8 + 2.56 + 0.512 + 0.1024 + 0.02048 + 0.004096 + 0.0008192 = 399.999795
Now coupling the fact that we know this will continue to grow infinitely with the fact that we are well within the range of experimental error by this point, we can safely assume 400 as our answer.
From that point we would continue as above and get the same answer:
4801
________________________________
TL;DR I don't know how you guys are getting 2 different answers. There is 1 answer to the problem and it is 4801.
Post by
124027
This post was from a user who has deleted their account.
Post by
239153
This post was from a user who has deleted their account.
Post by
Hyperspacerebel
Umm isnt this actually really easy? I think you guys are complicating it a lot.
Neither of the ways I did it are complicated.
And what you did is pretty much exactly what I did in my first method.
Post by
Haxzor
I like this thread
Post by
484763
This post was from a user who has deleted their account.
Post by
Hyperspacerebel
Given : the answer is 4801 minimum units required ( you agree on this and have provided many ways to prove it).
There are 1600 candles.
Each candles divides into 4 units ( hereafter called candle-units)
Prove: there are 1599 free units. ( Bonus , disprove there are 1600 free units)
1600 candles * 4 (units) = 6400 candle-units
6400 candle-units-4801 (minimum required candle-units)=1599 is the maximum free candle -units we can get
Bonus :
First way: 1599<1600 , since the maximum is 1599 and 1600 is larger 1600 is incorrect
Second way:
1600 candles * 4 (units) = 6400 candle-units
6400 candle-units-1600 (supposed free units) = 4800 minimum units required
Since given : 4801 minimum units required and 4800=/=4801 ,1600 is incorrect.
Extremely simplified : 1600*4=6400-1599=
4801
which is the answer so 1599 is correct.
1. Answers are not 'given.' Again, not sure what you're talking about.
3 + 5. Wrong. There are two possible definitions of 'candle' in this problem and you're confusing them. One is defining the candle as it is when you buy it like you are doing (4 units). The other is do define a candle by the amount needed to 'burn a candle' (which is 3 units). We don't need 1600 candles by the first definition (which is the one your using). We need 1600 candles by the second definition.
6. Subtracting 2 unrelated numbers means nothing.
8. What do you mean by '1600 is incorrect'? That's in the original problem!
10. Same problem as 3 + 5.
11. You just did something more or less right for once. As I demonstrated in my example exaclty 1/4 of the candles are free, so by subtracting 1/4, you got 4800 just like me. But then you stopped there and didn't continue. There has to be a single stump left over because you can only burn 3/4 of the last candle. Therefore the answer is 4801. That doesn't change the fact that 1600 units were free.
12. You forgot to add the last stump.
13. You fail at math.
I've now used elementary maths to prove it in very simple , mediocre , and slightly advanced form to prove it, I also used the american school system of algebraic questions to make it more understandable.
I've never seen math like that anywhere in America.
Post by
484763
This post was from a user who has deleted their account.
Post by
484763
This post was from a user who has deleted their account.
Post Reply
You are not logged in. Please
log in
to post a reply or
register
if you don't already have an account.