Hi ,
I have this logic puzzle I've been trying to solve for quite some time and I'd appreciate any help: It goes like this:
You have a round table with 20 seats all around it . You sit 10 men and 10 women in a totally random order. In the middle there is a round cake divided into 20 slices where each slice is in front of one person sitting around the table . Some of the slices have a cherry on them and some don't. The number and the arrangment of cherries is also totaly random (the number can't be more then 20). Men are happy if they see a cherry on the slice in front of them and women are happy if they see a slice without a cherry.
it's a prove or dismiss question (I think the direction goes towards proof). The taurem is "for any arrangment of people around the table and of cherries on the cake we should be able to make at least half the people around the table happy only by rotating the cake (as many degrees as we want)" . I'm looking for a clear proof of a dismissing example
2006-08-29 05:06:32 · 6 answers · asked by Ran M 1 in Science & Mathematics ➔ Mathematics
For any one arrangement, there will be a number pleased, n(p), and a number displeased, n(d), but always
n(p) + n(d) = 20.
If we look at all 20 positions of the plate,
total (n(p) + n(d)) = 20 x 20
= 400
If there are c cherries, then in the 20 positions each woman is pleased c times and displeased 20 - c times, making a total of 10c times that women are pleased. In the same way each man is pleased 20 - c times, making a total of 200 - 10c times that men are pleased.
Thus the total number of pleased responses is
10c + 200 - 10c
= 200.
If more than half the people were displeased in every arrangement, the total pleased would be less than 200. Since this is not the case, there must be at least one arrangement where at least half the people are pleased.
I now realise that this is basically the same as begoner's solution, but I find it clearer to think in terms of totals than averages.
2006-08-31 00:24:42 · answer #1 · answered by Hy 7 · 0⤊ 0⤋
Putting it simply, whether the people are happy of not depends on if they see a cherry in front of them.
For 0 cherries - you can make 50% of the people happy (ten women).
For 1 cherry - you can make 55% of the people happy (one man and ten women - fact).
For 2 cherries - you can make 60% of the people happy (maybe).
For 3 cherries - you can make 65% of the people happy (maybe).
And so on.
For 20 cherries - you will only have 50% of the people happy(Fact).
To obtain 100% of the people to by happy, you need a maximum of 10 cherries. But they must be in a way as to match up with the position of the men.
If you can show a possibility of an order of people and cherries that make less then 50% of the people happy, then you have yourself a possible dismissal.
On the spur of the moment there's only one thing I would suggest.
Make a list of all the possible combination of the men and women - where they sit about the table. Also make a similar list of all the possible arrangements of the cherries on the cake. Then compare all of the sequences and if you can show a majority of cases where the people are not happy 50% or more of the time, you have your dismissal.
2006-08-29 13:15:41 · answer #2 · answered by Brenmore 5 · 1⤊ 0⤋
It's actually extremely easy. Say there are x cherries. There are also 20 possible ways to turn the cake for any given combination of men and women. That means that the men will be happy x/20 of the time and the women will be happy (20-x)/20 of the time. On average, they will both be happy [x + (20-x)]/40, or 1/2 the time. That means that if you try every possible arrangement of the cake, in the end, each person will be satisfied (on average) 1/2 the time. Thus, it is impossible to try every signle possible arrangement of the cake and find that each time, less than half the people are happy because you know that overall, 1/2 of them will be happy. I didn't explain it too well, but it's really quite simple.
2006-08-29 13:24:23 · answer #3 · answered by Anonymous · 2⤊ 0⤋
Not true, Debbie. I can think of **far** more fun ways to spend a weekend than breaking in a cherry âº
As for the question...... That's a damn good one. I'm gonna have to think about that for a while. Probability isn't exactly my strong suite.
Doug
2006-08-29 12:14:40 · answer #4 · answered by doug_donaghue 7 · 0⤊ 2⤋
just put twenty cherries or dont put any at all either way half of the people will be happy however u rotate it .simple!!
2006-08-29 12:18:20 · answer #5 · answered by bookwormanu 2 · 0⤊ 1⤋
all men are happy when they get a cherry.. lol
2006-08-29 12:11:14 · answer #6 · answered by debbie 3 · 0⤊ 1⤋