When two red amoebas join, the result in one blue amoeba; when a red amoeba and a blue amoeba join, they turn into three red amoeba; and when two blue amoeba join, they become four red amoeba. Fernando observes a test tube with initially blue amoebas and red amoebas.
Determine, in terms of and , all possible quantities of amoebas in the test tube, specifying the quantities of amoebas of each color.
Solution
If the number of blue amoebas is and the number of red amoebas is then is invariant: indeed, whenever one blue amoeba appears/disappears, two red amoebas disappear/appear.
If there is only one amoeba, then it will never change. Otherwise, change all blue amoebas into red amoebas: this is possible because one can fuse a blue amoeba with a red amoeba to get three red amoebas repeatedly. So at this point we have amoebas. Then change back red amoebas into blue amoebas. If we change red amoebas, , we will get blue amoebas and red amoebas, in a total of amoebas.
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