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This is how I would do it:
N is the number you are trying to find.
You are told it is a 6 digit number and the left most number is 1.
To remove the 1 is to say N-100000.
To add the 1 on the right is to multiply the thing by 10 and then add 1. This would be then: 10*(N-100000)+1
We are told that this new number 10*(N-100000)+1 must equal 3* the original number (which was N). So we have an equation:
10*(N-100000)+1=3*N
solving: 10*N - 10*100000 + 1 = 3*N
then: 10*N-3*N - 1000000 + 1 = 0
then: 7*N = 999999
finally: N=999999/7 = 142857
Does it help if you just look at squares of 3? This way the fraction can be combined.
Define a subset of x:
Christ on a bike, I wouldn't trust them as quantity surveyors.
Excuse me JurassicPlank3? My answer is the same as yours, just approached in a different way. I'm not sure what your problem is....
There is a 1/3 chance of the goat being behind the door selected. When a door is revealed to have one goat, that leaves the 2 remaining doors (one has a goat and one a prize - a car? I cant remember). Anyway, now you have a 1/2 chance of winning.
NanaDee - what you say here is wrong. When you picked the door you did, the odds were 1:3 that you chose the car. EVEN WHEN another door is revealed, your chances of already having the car are STILL 1/3. This is a fundamental truth, the showing of the goat does not change that, because Monty Hall KNOWS what is behind the doors, and he makes sure that he reveals a goat and not the car. If a random person in the audience picked another door, and revealed what was behind it, that would be different. In this new case, the chance of a car being revealed is 1/3, and if a car was NOT revealed then NOW your chances are 1/2 that the car is behind the door you chose.
But, back to the original question, here are the scenarios (there are 6 permutations):
One goat is black (B), one goat is white (W), and of course there is the car (C).
The possible distribution of objects (the first column is the one you picked):
C B W you already have the car , B or W goat is shown, if you switch you lose 1/6
C W B you already have the car , B or W goat is shown, if you switch you lose 1/6
B C W you chose the black goat, white goat is shown, if you switch you win 1/6
W C B you chose the white goat, black goat is shown, if you switch you win 1/6
B W C you chose the black goat, white goat is shown, if you switch you win 1/6
W B C you chose the white goat, black goat is shown, if you switch you win 1/6
NOTICE - that Monty Hall has to be very careful which door he opens. In 4 out of the 6 scenarios, if he opens the wrong door, the car will be revealed!
If you stick with your original choice, your odds of winning are 1/6+1/6 = 1/3
If you switch, then they are 1/6+1/6+1/6+1/6 = 2/3
mathsmypassion - I think you have made several mistakes.
First, if the roll diameter is 18inches, with a 2inch core, then the radius is 9inches, with 1inch not used, leaving 8 inches to be wrapped. With 1/8inch per wrap, that gives 64 wraps (for your N) - NOT 128.
Second, the formula should be worked out as:
each wrap is 2 pi R
the first wrap is approximately the radius 1and1/8inches.
the next wrap is then the radius 1and2/8inches
and so on, giving the summation
1+1/8 + 1+2/8 + 1+3/8+ .... + 1+N/8
which equals
N*1 +1/8+2/8+3/8+....+N/8
=N+1/8*(N*(N+1))/2 -----------you used N*(N-1)
=N+N*(N+1)/16
Now put the summation with the circumference formula and we have the total length as:
2 pi *{ N + N*(N+1)/16}
=pi * {2N + N*(N+1)/8}
As you can see, my answer differs from yours mainly because I have 1/8 and you had 1/16.
So I would say that the roll is theoretically (using N=64) 2036inches long, or 170feet
With 2 and a half foot width of the roll, then the square footage is 425.
As you say, it will be smaller than this theoretical limit because the cardboard does not wrap infinitely close together.
It was a bit of hit-and-miss, but it seemed the easiest way to explain.
We could use the 30 high column also:
On the 30 high column, we would need to move 4 units on the 3hr candle (4x6min) and 3 units on the 4hr candle (3x8min).
TAKE A PIECE OF GRAPH PAPER. Draw each candle on the paper 60 units high.
The 4hr candle will burn 15 units per hour. - mark these on the graph as hours 0,1,2,3,4. 0 at top, 4 at bottom. You can think of each unit as 4 minutes (4x15=60min)
The 3hr candle will burn 20 units per hour -mark them also from 0 to 3. You can think of each unit here as 3 minutes (3x20=60min)
You can see that at hour 2 the 4hr candle is at 30 units up, and the 3hr candle is at 20 units up. It is just after this hour that the 3hr candle will be half the height of the 4hr candle.
NOW - burn each candle for 12minutes, which is 3 units on the 4hr candle, getting to 27 units, AND 4 units on the 3hr candle, getting to 16 units. This is not quite half (16/27).
Burn each candle AGAIN for 12 minutes. The 4hr candle goes to 24 units, and the 3hr candle goes to 12. Now we are at half (12/24).
The time elapsed is 2hrs plus 12minutes plus 12 minutes.
I hope this is clear enough without showing you my drawing.
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