Question 1045151
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In an enchanted garden there was a miracle tree. It had 25 bananas and 30 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 appear. 
If he picked two of different fruits a banana appeared. 
What was the last that he picked??
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<U>Answer</U>.  BANANA.


<B>Solution</B>


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


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

And let us ask the gardener to write "-1" on each new banana and/or "+1" on each new orange, whichever occurs.

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" occurs.
    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" occurs.
    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" occurs.
    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>

Solved.



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<U>Notice</U>. &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;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I only know (and it is proved) that the BANANA will be on the miracle tree in the last day.