Save the red flowers
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.
Hi. My name is Mister Kvakk. I live in a very big house in the country. Outside my house I have a lot of big gardens. In one of them I have ten beautiful, red flowers. These red flowers grow in a 5x2 rectangular shape:
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The flowers had always lived in peace in this garden, until the day I allowed my stupid dog to go into the garden. This dog has only one goal in its life: it wants to eat as many red flowers as possible.
But the flowers are not normal flowers. They can think, and are able to talk to each other. And they now need a good plan to be able to save as many flower-lives as possible.
Each of the ten red flowers has the option to develop poison. If the dog is so unlucky to eat a poisonous flower it will surely die, and thus cannot eat any more flowers. So, when the dog arrives in the garden, it will start to eat flowers in random order as long as it is alive (the dog cannot see which flowers contain poison). If none of the flowers make poison, the dog will certainly eat all of them.
But how many flowers should make poison? Unfortunately, it's not possible to save every flower. The problem is that the creation of poison is a hard task to do. When one flower develops poison, it will use all the food and water that is in the ground around it. This will kill the flower to the left (if there is a flower to the left).
The flowers are not sure about how many flowers should make poison. They think that maybe only the two flowers to the very left should do it, but they are not sure. Therefore, they need your help.
Question 1: What is the maximum number of flowers that without risk can be guaranteed to survive?
Question 2: The flowers are risk neutral, and want to minimize the expected number of dead flowers. How many flowers should make poison to afford this?
HintQuestion 1: This is an easy question. As many flowers as possible should make poison to do this.
Question 2: Remember that expected value is the sum of each outcome times its probability. The expected value of a roll of a die is for instance:
1/6 * 1 + 1/6 * 2 + 1/6 * 3 + 1/6 * 4 + 1/6 * 5 + 1/6 *6 = 3.5
For each possible strategy, you have to calculate the expected number of flowers eaten, as well as the number of flowers killed by a flower to the right. You must find the probability that 1 flower will be eaten, 2 flowers will be eaten, and so on. If only ONE flower makes poison, this is fairly easy, because the probabilities will be the same for each possible outcome (as in the dice example). But if more than one flower make poison, this is much harder. To give you some numbers: if zero flowers make poison, the expected number of flowers eaten is 10 (all of them). If one makes poison, the expected number of flowers eaten is 5.5.
I solved this using Microsoft Excel, but you should be able to solve it without a computer.
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Answer
Question 1: To eliminate all risk, as many flowers as possible should make poison. This means that 6 flowers will make poison. This will kill 4 flowers (symbolized by an x):
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and 1 flower will be eaten by the dog. Consequently, 5 flowers is the maximum number of flowers that can be guaranteed to survive.
Question 2: If zero flowers make poison, the expected number of flowers eaten is 10 (all of them, no risk). If one makes poison, the expected number of flowers eaten is 5.5. If two make poison, the expected number of eaten flowers is 3.67.
If more than two flowers should make poison, the flowers have to suicide-kill one of their own friends. If three flowers make poison, 1 is killed by its neighbour, and 2.5 are expected to be eaten, thus the number of flowers lost will be 3.5.
If four flowers make poison, 2 will be killed, and 1.8 expected to be eaten. Expected number of lost flowers will be 3.8.
If five make poison, 3 are killed and 1.33 expected to be eaten, which gives 4.33 lost.
And if six make poison, 4 are killed and 1 is eaten, giving 5 lost flowers.
As we see, three flowers should make poison. The two flowers to the very left, and one of the six right-most others.
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Comments
lauramarie
Jun 03, 2003
| very difficult and very interesting |
mrsbloom
Jun 09, 2003
| I thought this was really clever! It quite hard and i had to think for ages to get the answers! |
rose_rox  
Jun 09, 2003
| I love your teasers! You fun rating should be a lot higher than what it is. |
batti_s
Jun 19, 2003
| Yes, the expectation model is applicable, and has been put to good use. The case is not infinitely projectable, and so thanks for giving the numeric solutions!! |
noturfriend421
Feb 26, 2004
| your answer is VERY wrong. the gog only eats red flowers, right? well dogs are color blind, so, depending on the size of the garden, then the chance of the dog eating a red flower lower dramatically. just thought you you'd like to know, good teaser nonetheless  |
(user deleted)
May 12, 2004
| Well, sure, but it was still an awesome teaser.  |
Poker
Aug 01, 2004
| The question is, which of the six potential suicide candidates will be killed by the flowers? Hopefully a volunteer.  |
mr_brainiac
Jan 10, 2006
| I suddenly feel like a complete peabrain. I haven got a clue how to even approach this one.  |
javaguru
Dec 15, 2008
| I did it by calculating the expected number of survivors instead of expected casualties, but got the some probabilities.
I thought it was interesting, and somewhat unexpected, that the probabilities for 2 (no sacrifices) to 4 (2 sacrifices) were so similar. With the small population, my intuition was that sacrificing a flower would have had a larger effect on the probability. |
Zag24
Feb 20, 2009
| My answer was to have all the flowers turn which way they are facing, so that "to their left" was always a spot that is currently empty!
Just kidding, I understood what you meant. This was a good puzzle and a fun presentation.
By the way, in reference to an earlier comment about dogs being color-blind. Not all dogs are -- there are breeds that can see some colors. |
precious1026
Nov 20, 2009
|  I dont know enough Math to complete the problem. Furthermore, es much as I love all types of Teaser, hard questions should be reserved in another section of
Braingle. And, reserve this section for the Fun Stuff. I know Math can be and is Fun for many people, but this Teaser just put damper in my funnybone.
A Teaser...  |
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