Martin is a gardener in Mathland.

There are 100 poisonous flowers in his garden

that he wants to destroy.

On their own,

the flowers do not change in number.

Martin has a spray to kill the poisonous flowers,

but the spray works only in a specific way.

The spray has settings to destroy exactly 3,

5, 14, or 17 flowers instantly.

The spray only works if

there are at least as many flowers as the setting.

If there are two flowers, for example,

the setting for 3 does nothing.

Furthermore, if at least one flower survives

after he sprays them,

the flowers instantly grow back

based on how many died.

If 3 die, then 12 grow back.

If 5 die, then 17 grow back.

If 14 die, then 8 grow back.

If 17 die, 2 grow back.

If the number of flowers is ever exactly zero,

then the flowers never grow back.

Can Martin ever get rid of the

poisonous flowers in his garden?

You can assume he has an unlimited amount of spray.

Can you figure it out?

Give this problem a try, and when you're ready,

keep watching the video for the solution.

So, from the initial 100 flowers,

you can try experimenting the different combinations

of the flowers you can spray,

and keep track of how many flowers remain.

But no matter how you try,

you're not gonna be able to get rid of all the flowers.

We can understand why

by thinking forward and reasoning backwards.

So in order for Martin to succeed,

there needs to be 3, 5, 14, or 17 flowers

just before the last time he sprays the flowers.

This means from the initial 100 flowers,

Martin has to decrease the number of flowers

by exactly 97, 95, 86, or 83 flowers.

Let's consider the net change after each setting of the spray.

If he uses a setting of 3,

he kills 3, but 12 grow back,

which leads to plus 9 flowers.

For the setting of 5,

there is a net change of plus 12 flowers.

For 14 setting,

he decreases the number of flowers by 6,

and for the 17 setting,

he decreases the number of flowers by 15.

On each setting of the spray,

Martin can only change the number of flowers

by +9, +12, -6, or -15,

and there's a pattern to these numbers.

They are all multiples of 3.

Since Martin only changes the number of flowers

by a multiple of 3,

the number of flowers has to change by 3x

for certain integer values of x.

Now, since the numbers 97, 95, 86, and 83 are not multiples of 3,

that means Martin is not capable

of decreasing the initial number

by exactly any of those numbers.

Therefore, we can conclude

Martin can never get rid of

all of the poisonous flowers.

Did you figure it out?

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