We have a bag of red, green and yellow apples?
Initaiily the bag contains red, green and yellow apples. Moving involves removing any two apples of different color of the bag, and substituting an apple for the additional third color. After the series blows the last bag contains a red apple. Prove that if another round of shots was applied to the same bag, it could Neeve terminate in a single apple green or yellow at the end. For example, if the bag initially contained 4 apples rrgy, we have series of moves rrgy RYY Gy r
I like this problem. I think the question posed to us is that one of a whole class of questions that need answers cool ... if I have time Later I might put a little thought to other matters such. We will start by attributing a number to each color. Virtually every assignment will, but I will use the first one I thought of: -1: yellow 0 red 1 green We'll consider the sum of the numbers in the bag. For example, in your example, our figures are 0 -2 0 0 Consider possible relocation, and how it affects our money. If we remove and add RG Y, we removed a 1 and add a -1 for a change net -2. Remove RY G add, we have the net change of +2. Remove YG add R, and we have a net change of 0. So any movement can change the parity of this sum. Then, if the bag can be finished with a red apple, we have a final sum of 0 and then every step along the way we had a sum still In particular, we started with an amount yet. Then, since other sequences of movements, we also have a further sum for each not, so we do not end with a single yellow or apple green, for fear that our money must be -1 or 1. Note that does a parallel argument with another color, So if the bag can be reduced to a single bone of any color, it can not be reduced to an apple or another color other. However, it can be blocked after a few shots, leaving us with multiple heads all the same color. In fact, it can be also stuck as to never be able to achieve a single apple. Consider, for example, starting with an apple in each color. Then one our 3 possible moves leaves us with two apples of the same color, and we're stuck. (Note that using the same number assignment, we guaranteed to receive a sum yet to such an arrangement.)
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