document.write( "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?? \n" ); document.write( "
Algebra.Com's Answer #660566 by ikleyn(52797)\"\" \"About 
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\n" ); document.write( "In an enchanted garden there was a miracle tree. It had 25 bananas and 30 oranges.
\n" ); document.write( "Everyday the gardener picked two fruits and a new one appeared at the same time.
\n" ); document.write( "If he picked two of the same fruits an orange appear.
\n" ); document.write( "If he picked two of different fruits a banana appeared.
\n" ); document.write( "What was the last that he picked??
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\n" ); document.write( "\n" ); document.write( "Answer. BANANA.\r
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\n" ); document.write( "\n" ); document.write( "First, it is clear that the total number of fruit decreases 1 in each day.
\n" ); document.write( "So, it will take 25 + 30 = 55 days to complete the process.\r
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document.write( "Let us ask the gardener to write \"-1\" on each banana, \r\n" );
document.write( "                       and write \"+1\" on each orange.\r\n" );
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document.write( "And let us ask the gardener to write \"-1\" on each new banana and/or \"+1\" on each new orange, whichever occurs.\r\n" );
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document.write( "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\".\r\n" );
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document.write( "If the gardener picks two oranges, two \"+1\" come out and one new \"+1\" occurs.\r\n" );
document.write( "    The product of all factors remains without change. It is still \"-1\".\r\n" );
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document.write( "If the gardener picks two banana, two \"-1\" come out and one new \"+1\" occurs.\r\n" );
document.write( "    The product of all factors remains without change. It is still \"-1\".\r\n" );
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document.write( "If the gardener picks banana AND orange, one \"-1\" comes out, one \"+\" comes out and one new \"-1\" occurs.\r\n" );
document.write( "    The product of all factors remains without change. It is still \"-1\".\r\n" );
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document.write( "So, we can conclude that the product of all factors remains without change ALL THE TIME, day after day till the last day.\r\n" );
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document.write( "It means that in the last day it will be one banana on the miracle tree.\r\n" );
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document.write( "Answer.  The last fruit will be BANANA.\r\n" );
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\n" ); document.write( "Notice.  It is clear that the last day is special.  It does not obey the general procedure rule. \r
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\n" ); document.write( "\n" ); document.write( "             So,  accurately speaking,  I don't know how the procedure works in the last day.\r
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\n" ); document.write( "\n" ); document.write( "             I only know (and it is proved) that the BANANA will be on the miracle tree in the last day. \r
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