A recent problem was about someone who figured out a puzzle she was working on, and your job (if you’re a subscriber, that is) was to figure out how she figured it out. In the solution, we encountered two problem-solving strategies: putting yourself in someone else’s shoes, and making a table. A strange combination of ideas, to be sure, but there are, strangely, other problems that make use of these very same strategies! Here’s another, which again focuses on how someone actually managed to solve a puzzle; it’s definitely one of our favorites. Read it, but before you read on to see our solution, think about the problem, and see if you can solve it yourself!
A mad scientist created an army of smart robots (called, appropriately, “smart-bots”). They’re not quite as smart as he’d like, however; after much hard work, he has finally created his first “genius-bot”. He’s hoping his genius-bot is much more intelligent than his smart-bots, so he devised a way to test this.
He arranged his genius-bot and two smart-bots, all currently switched off, so each robot can see the other two. He then glued a red ball to the top of each robot’s head, so that each robot can see the other two balls, but not its own.
Now the scientist turns on the robots, and instructs them: “Don’t move. All I want you to do right now is see, and think. Look at each other; each of you has either a red ball or a green ball glued to the top of your head. If you see a red ball, raise your hand.”
All the robots raised a hand. “Now then,” said the scientist, “who knows the color of the ball on top of their own head?”
The robots were silent for some time. Then the genius-bot suddenly said, to the surprise and joy of the scientist, “I have a red ball!” How did it know?
Our task it to figure out how the genius-bot determined the color of its ball. Now remember, they all had red balls on their heads, so of course they all raised their hands. But the smart-bots were confused; they didn’t know what color ball was on their heads. But the genius-bot saw exactly what they saw…. So, somehow, the genius-bot must have taken advantage of the fact that the smart-bots were confused… but how did it do this?
The genius-bot wondered just what we’re wondering: “why can’t the smart-bots figure out their own color?” What the genius-bot did next was put himself in the shoes (or, wheels, whatever) of one of the smart-bots. Here’s what the genius-bot was thinking: “It might be wise to make a list of all the possibilities of who has what color ball, from the perspective of what smart-bot #1 sees. It saw a red ball on smart-bot #2, and it saw my ball, whatever color that might be, red or green. It doesn’t know its own color, so from its point of view, its column in the table should be blank.”
|smart-bot #1||smart-bot #2||genius-bot|
“Ah, so there are only two rows to my table! Simple enough, but now let’s think. If I were smart-bot #1, I would see the other two robots raise their hands, which means they each saw at least one red ball. But consider the situation in the second possibility… again, smart-bot #2 raised his hand, so it saw a red ball. But where? He must have seen it on smart-bot #1! Therefore, in this situation, smart-bot #1 would know that he had a red ball on its head. But it didn’t know! That means that the second row of the table does NOT describe our situation. The only other possibility is the first row, where my ball is red, like the others!”
Each smart-bot, evidently, only thought about what it saw. They each saw two red balls, and so expected everyone to raise their hands, and learned nothing. But the genius-bot thought about what the other bots were seeing and thinking; he thought about the situation as a whole. From the bots in this problem, we are reminded once again of the value of being able to see through the eyes of another. Beyond that, we are reminded that being able to see the big picture as well as the individual parts is also a useful skill to have in our mathematical toolbox. And, one might argue, these skills are not only useful mathematical tools, but useful tools in all aspects of daily life.
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