Find the solution for the following: There are 3 hats with 11, 6, and 4 marbles, respectively. On a single move we can remove one marble from one of the hats, one marble from some other hat, and place both removed marbles to the third hat. Can we reach a situation where there are 7 marbles in each hat after some number of moves? If so, show the way. If not, provide argument why not?
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hat 1 = 11 marbles
hat 2 = 6 marbles
hat 3 = 4 marbles
step 1 - take one marble each from hats 1 and 2 and place in hat 3 to get
hat 1 = 10 marbles
hat 2 = 5 marbles
hat 3 = 6 marbles
repeat step 1 to get
hat 1 = 9 marbles
hat 2 = 4 marbles
hat 3 = 8 marbles
step 2 - take one marble each from hat 1 and 3 and place in hat 2 to get
hat 1 = 8 marbles
hat 2 = 6 marbles
hat 3 = 7 marbles
in order to get 7 marbles in each hat all we would need to do is move just ONE marble from hat 1 to hat 2. but because we are moving 2 marbles at a time the numbers in the 3 hats will always be different and not the same
hat 2 = 6 marbles
hat 3 = 4 marbles
step 1 - take one marble each from hats 1 and 2 and place in hat 3 to get
hat 1 = 10 marbles
hat 2 = 5 marbles
hat 3 = 6 marbles
repeat step 1 to get
hat 1 = 9 marbles
hat 2 = 4 marbles
hat 3 = 8 marbles
step 2 - take one marble each from hat 1 and 3 and place in hat 2 to get
hat 1 = 8 marbles
hat 2 = 6 marbles
hat 3 = 7 marbles
in order to get 7 marbles in each hat all we would need to do is move just ONE marble from hat 1 to hat 2. but because we are moving 2 marbles at a time the numbers in the 3 hats will always be different and not the same