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Question 1045151: 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??
Found 2 solutions by KMST, ikleyn:
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Each day the total number of fruits goes down by one.
One day, there will just be two fruits to pick, and the gardener will pich the last pair of fruits as a single fruit will appear.
The next day, there will be just one fruit, and the gardener will pick just one fruit: the last fruit.
At the start there is an odd number of bananas:  bananas.
Any day that there is an odd number of bananas,
if the gardener picks only one banana,
he is picking two of different fruits,
and a new a banana appears,
so the next day the number of bananas is the same odd number.
Any day that there is an odd number of bananas,
if the gardener picks two bananas,
he is picking two of the same fruits,
and an orange appears,
so the next day, the number of bananas is  less, but still an odd number.
Every day there is an odd number of bananas.
The last day, when there is only one fruit,
there is also an odd number of bananas: just  .
The last fruit picked up ias a  .
Answer by ikleyn(52794)  (Show Source):
You can put this solution on YOUR website! .
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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Answer. BANANA.
Solution
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.
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.
Answer. The last fruit will be BANANA.
Solved.
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Notice. It is clear that the last day is special. It does not obey the general procedure rule.
So, accurately speaking, I don't know how the procedure works in the last day.
I only know (and it is proved) that the BANANA will be on the miracle tree in the last day.
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