The rules are... total of 10 marbles, red or blue no 3 consecutive marbles taken out are the same color the last x marbles cannot have $ x/2$ red and $ x/2$ blue marbles my spot notation..._ _ _ _ _ _ _ _ _ _ 1 is the first underscore, 10 is the last marble chosen Case 1We are assuming that we will have enough marbles because the Cases will cover all cases (in other words, I was going to write a completely rigorous proof but I decided not to) Case 1: The middle two are blue. It now looks like _ _ _ R B B R _ _ _ Case 1a: B is the last one taken out. The second to last taken out has to be blue or the game would have ended sooner. Spot 8 has to be red to avoid the 3 consecutive rule. However, because the last four marbles have an equal number of marbles, we have a contradiction. Case 1b: R is the last one taken out. We need an a red in spot 9. We need a blue in spot 8. However, the last 6 marbles (spots 5-10) have an equal number of blue and red marbles, a contradiction. Case 1b done. Case 1 done. Case 2Case 2: The middle two are red. _ _ _ B R R B _ _ _ Case 2a: B is the last one taken out. Therefore, a blue goes in spot 9, a red goes in spot 8. However, the last six marbles have an equal number of blue and red marbles again, a contradiction. Case 2a done. Case 2b: R is the last one taken out. Therefore, a red goes in spot 9, a blue goes in spot 8. However, the last four marbles have an equal number of blue and red marbles, a contradiction. Case 2b done. Case 2 done. We have covered all the cases, regardless of how many of each marble we have. Therefore, because we have proved that the middle two cannot be the same color, they must be different colors. QED

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