The easiest way to think about this problem is not to think of number of marbles in each color but the differences in number between colors. Let’s say “a”, “b” and “c” are the pairwise differences in colors. Their values are 10, 10 and 20 to begin with.
Let’s see what happens whenever there is a “hit”.
If the marbles are of same colors, of course no change in count happens – so a, b and c remain the same.
If the marbles are of different colors – then those two colors go down by 1 each and the third color goes up by 2. Which means the pairwise differences either remain the same (the two colors that hit) or they change by 3 (one color goes down by 1 and the resulting color goes up by 2).
So, the first concept to realize is that a, b and c always change in steps of 0 or steps of 3. There are no other step changes they can have.
Now, let’s say what if the marbles actually did become all of one color. What as the step right before it? We would have had to have all marbles of the same final color excepting two – one each of the other two colors. Thus if finally everything is green, in the penultimate step, there has to be one red, one blue and rest green. The red hits the blue to become green.
Note that in that penultimate step, the pairwise difference between two colors (that have one marble left each) is 0. And that is the second concept to realize.
Now putting these two concepts together…
Well, we started with a, b and c being 10, 10 and 20. Going in steps of 0 and 3, can any of these numbers ever get to 0? No. Therefore it will never reach the penultimate step.
Which means the marbles will never be of the same color !!
This story is rather embarrassing for me. I was too young to remember this. But my dad often told this story to razz me. And it usually worked.
So, this goes back – way, way too many years… 51 to be precise. Now to fully get the story, I have to give a little background here.
Back in India, all my life I grew up having my meals while squatting on the floor. In fact, even in Kalyani, when I would be with my parents – including my last meal with them – we would all sit down on the floor and eat.
The next thing you might want to know is that when we were growing up, we were sufficiently modest in our income that a square meal meant rice, lentils – called “daal” – and some vegetables. On a couple of days in the week, we had fish and then we had some special Sundays when we would have meat (always goat meat those days).
So, now, to hear my dad’s version… Apparently, we had finished lunch one day. And by “we”, I mean my parents, my sister and myself. My brother was not yet around. Mom had cleaned up the floor with a wet cloth (“nyata”, as we referred to it). Apparently, she missed a particular piece of lentil that had flung afar. That piece of lentil, over a couple of hours, had turned from a juicy bit to a hard and dried morsel well stuck to the floor.
My sister, ostensibly crawling around without any rhyme or reason, had spotted that hapless bit of lentil. And as every kid of that age does, she promptly decided that the lentil needed to be transferred from the floor to her mouth.
Like those best laid plans of mice and men, she was completely stopped in her tracks by that rather stubborn (“thhnyata”, as we would say) piece of lentil. It simply refused to budge. After giving it a few heave ho-s with her tender fingers, she decided to call in for reinforcements. In this case, her reinforcement refers to yours truly.
Like a knight in a shining armor, I appeared on the scene. After carefully evaluating the scenario, I apparently also proceeded to dig with my fingers. Except that my fingers were not that tender. At least not as hers. A few determined attempts and I succeeded in uprooting it!! The “daal” (lentil piece) was in my hand. I have no doubt that I was as proud as proud could be.
And this is where the story gets a little murky and somewhat anti-climactic. In an alternate universe, I handed over the dirty lentil to my sister who put it in her mouth and savored it as she dotingly looked up to her brother shimmying out of the scene.
In this real universe however, that knight in shining armor decided that since he took all the pains, he should get all the gains. And promptly put the dirty lentil in his mouth!!
In this universe, my sister bawled a lot, my father told me!!
There is a large tray with red, blue and green marbles in it. In fact, there are 20 red marbles, 30 blue marbles and 40 green marbles. In that tray, the marbles are continuously moving around in Brownian (random) motion – often hitting each other.
If two marbles of same color hit each other, they just bounce off and continue. However, if two marbles of different color hit each other, then, both change to the third color and then bounce off and continue. All such hits are always between two marbles only. (no simultaneous three or more marbles hitting each other).
So a red and blue marble hitting each other will make both of them turn green and so on….
Is it ever possible for all the marbles in the tray to become of one color? If so, which color will it be?