### Brain Teasers

# Save the red flowers

Probability
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?

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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?

### Hint

Question 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.

### 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):FxFxF

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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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