A Candy Game (Change Discharged)
A number of students sit in a circle while their teacher gives them candy. When the teacher blows a whistle, each student divides his/her candies into two equal piles, discharging one candy if such is left over. Simultaneously students give one of their piles to the neighbor on the right. Show that no matter how many pieces of candy each student has at the beginning, after a finite number of iterations of this transformation all students have the same number of pieces of candy.
|What if applet does not run?|
(In the applet, the number of candies can be set either randomly or by modifying each of the circled entries. To modify a number, click a little to the right or left of its centerline. In the symmetric variant, each fellow gives the remaining pile to the left neighbor.)
There is another version of the game wherein each student starts with an even number of candies. In this variant, if - after an iteration - a student ends up with an odd number of candies, the teacher adds a candy to his/her hoard thus insuring that at all times all students have an even number of candies. The reasoning that applied in that case works as well for the present variant.
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