Lesson The gardener's problem on a miracle tree in an enchanted garden

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<H2>The gardener's problem on a miracle tree in an enchanted garden</H2>

This lesson is advanced. &nbsp;It is intended only for those to whom it is interesting.

<H3>Problem 1</H3>In an enchanted garden there was a miracle tree. &nbsp;It had &nbsp;25&nbsp; bananas and &nbsp;30&nbsp; oranges. 
Everyday the gardener picked two fruits and a new one appeared at the same time. 
If he picked two of the same fruits an orange appeared. &nbsp;If he picked two of different fruits a banana appeared. 
What was the last that he picked??

<B>Answer</B>.  BANANA.

<B>Solution</B>

First, &nbsp;it is clear that the total number of fruit decreases in 1 each day.
So, &nbsp;it will take &nbsp;25 + 30 = 55 &nbsp;days to complete the process.

<pre>
Let us ask the gardener to write "-1" on each banana, 
                       and write "+1" on each orange.

Let us ask the gardener, further, to write "-1" on each new banana and/or "+1" on each new orange, whichever appears.

Let us calculate the product of all "+1" and "-1" for the original/initial status. The product will be -1: 25 factors of "-1" and 30 factors of "+1".

If the gardener picks two oranges, two "+1" come out and one new "+1" appears.
    The product of all factors remains without change. It is still "-1".

If the gardener picks two banana, two "-1" come out and one new "+1" appears.
    The product of all factors remains without change. It is still "-1".

If the gardener picks banana AND orange, one "-1" comes out, one "+" comes out and one new "-1" appears.
    The product of all factors remains without change. It is still "-1".

So, we can conclude that the product of all factors remains without change ALL THE TIME, day after day till the last day.

It means that in the last day it will be one banana on the miracle tree.

<U>Answer</U>.  The last fruit will be BANANA.
</pre>

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<B>Notice</B>. &nbsp;It is clear that the last day is special. &nbsp;It does not obey the general procedure rule.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So, &nbsp;accurately speaking, &nbsp;I don't know how the procedure works in the last day. &nbsp;Probably, &nbsp;it doesn't work at all.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I only know (and I proved it) that a BANANA will be on the miracle tree in the last day.

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