# How is this puzzle solved?

Three university maths students are taken from their lecture and sat in a room facing each other. The organiser comes in and tells them...

"I'm going to paint a red or a white dot on your forehead. If you see any red dots at all you must put your hand up. If you see only white dots, you must leave it down."

The winner is the first person to work out what dot they have on their forehead.

The organiser paints only red dots on their foreheads. They all see two red dots on the two foreheads opposite and consequently all put their hands up.

After a while for thought, one of them shouts out to the organiser and correctly tells her that he has a red dot on his forehead. How did he work it out?

There's no trick, no info I haven't told you, there is no mirror, nobody tells anybody, nobody saw their own dot, the red paint feels the same, nobody saw that the organiser only had one pot of paint or one brush, the individual just worked it out. All the info is there in the question.

• Surely this one has been done before? – Dr Xorile Apr 8 '17 at 17:54
• I believe its called a 'common knowledge problem' – Ewan Apr 9 '17 at 9:00

This is solved by noticing it's the same as the island of blue and brown eyes, or the many "logicians and hats" puzzles we have here. At first, you see two red dots, so you put your hand up and think you have no more information.

But after a while you think, what if I had a white dot? These other students would see one white and one red dot. What would they think about that?

You pick one and think, that other guy, he must be wondering if he has a white dot too. If he saw one white one plus possibly himself, he would be expecting the third student to put her hand up if his is red (because in this line of thought, mine is white) and leave it down if we're both white. But she put her hand up. So if my dot is white, the other guy knows his dot is red.

But the other guy hasn't announced that his dot is red. So mine can't be white. So mine is red.

On the matter of timing: if, after I go through all this, I think that perhaps the reason the other guy hasn't answered yet is that he is a slower thinker than me, I can consider the third student, who may not be a slower thinker than me. But after allowing enough time for however dense I think they are, and still nothing, I can safely call out. Therefore the one who answers is probably the one who believes "I am the slowest thinker in this group. If I have worked this out and they haven't, I must have a red dot." Those who consider themselves fast will allow time for the "slower" ones, who will probably shout out in that gap.

• Nice job. In a nutshell, supposing your spot was white, it would be obvious to one of the others, seeing the other other's hand up, and your spot being white, that they had a red spot and therefore they would shout out. Their failure to do so means you cannot have a white spot. – samerivertwice Apr 8 '17 at 17:40

Call the 3 students Student A, Student B and Student C.

The one who answered is student A. Let's begin by saying that B and C are average students but Student A is far superior

Student A sees 2 red dots. Student A considers the situation where he has a white dot... then both B and C will see one red and one white. Each of these students knows that if they're also white then the other one should have their hand down, but they don't, so they know they're red, in which case they would call out their color.

But they don't.

Therefore student A has a red dot and is the smartest one.

If student A saw one white dot he'd also use this reasoning if all 3 hands were up.

• Welcome to Puzzling! Thanks for your answer! Why don't you check out Kate Gregory's answer, they've reached the same conclusion. While you're at it, if you take the site tour, you'll also get a badge. – boboquack Apr 8 '17 at 23:21
• @Noel +1 what you have done, which most fail to realise, and which inferior versions of this problem fail to, is that the intellects of the participants are integral to the solution. And you're right, the brightest student shouts out. – samerivertwice Apr 9 '17 at 7:02
• @RobertFrost I don't see that. If I am smarter than the others, their silence may only mean they haven't thought it through yet. I have to wait until they would have figured it out in order to benefit from their silence. – Kate Gregory Apr 9 '17 at 18:37
• @KateGregory If you didn't know the people you were with, or more specifically what their deductive capabilities were, you would not be able to deduce that their failure to shout out, was because they could see two red spots. Only knowing that they are at least reasonably intelligent can that deduction be made. – samerivertwice Apr 9 '17 at 20:27
• @RobertFrost no argument. However, knowing they are logical and able to work it out does not make me the smartest or brightest in the room. In fact if I know myself to be the slowest in the room, I will shout out for sure. – Kate Gregory Apr 9 '17 at 20:37

Because no-one is left with their hand still down, then no-one can see any white dots. Therefore there are no white dots and the only alternative is for you to have a red dot on your forehead.

If there were two red dots or two white dots.

Everyone would still have their hand up if there are two reds. However, you can see a white dot and a red so you have your hand up. The person with the red dot either has their hand up or has their hand down. If up, they can see a red which must be you, if down they can't see any reds so you are white.

Or the only other alternative of you being red the others white

They have their hands up but you keep yours down. You can realise from this that they can both see a red dot but you can't see any so you must be red