This is a fun little puzzle to sort out. It reminds me of Kriegsspiel.
Both sides play Kriegsspiel blind via an umpire. It is a lot like poker in some ways. What does the enemy know? What do they not know? What do we know about the enemy? What does their known action reveal about their motive and intent?
These are all the types of questions you constantly puzzle over while playing Kriegsspiel. It is exactly the type of thinking you need to solve this problem:
There is a secret prize hidden under one of these five objects. The teacher privately tells the girl what the correct shape is. The teacher privately tells the boy what the correct color is.
Next the teacher brings them both together. The first one to pick the correct object gets the prize.
Teacher: Do either of you know where the prize is?
(silence)
Teacher: Do you know now?
(silence)
Teacher: Do you know now?
Boy and Girl: Yes!
Where is the prize?
It is indeterminate, because you do not provide information about the “picking” process. You only say that there is silence. There are any number of probable answers.
If you’ve reprinted this puzzle, check the original source because you missed a key piece of information. If it’s original, check your unstated assumptions – your solution requires an assumption which you are making, but which isn’t being stated and which the reader cannot make without further support.
There may be a number of assumptions… They do not cheat. They do not talk to each other. There is really a prize. They both do want to win.
They only get 1 chance to answer. If they are wrong, they cannot guess again.
I’ll also say that they do not guess. I will also say that when they say ‘yes’, they do both know and they are both correct.
Who gets to guess first? That’s a rule that they know, and we don’t. For an example.
I’ll go ahead and spell it out. The solution requires that each child know what the other child knows. The girl knows that the boy knows the color, and the boy knows that the girl knows the shape. Thus, they can draw inferences from the other’s lack of action during the previous turn.
There’s a large problem with this and an enormous one. The large problem is there are no defined turns, so there is no actual way for them to know that their rival has “passed” on the chance to pick – the rival could just be slow-witted.
The enormous problem is that the puzzle specifies that the teacher PRIVATELY tells them this information. Without knowing what the other person knows, neither of them has a clue no matter what their rival’s behavior is.
Ah yes! They DO know beforehand that the girl knows the correct shape and the boy knows the correct color.
There are no turns. Whoever speaks first wins. Neither of them speak for the first 2 questions. They both simultaneously answer for the third.
Blue triangle.
1st round.
Girl knows shape. Boy knows color (assume not color blind)
If shape square or color yellow, G or B would immediately know. As neither know…and each now knows the _other doesn’t know_, we are left with blue and red triangles and blue circle.
2nd Round
If shape is circle, girl would know.
If color is red, boy would know.
Neither knows for sure when asked second time. Thus, shape cannot be circle, and color cannot be red. This leaves….
3rd round
Blue triangle.
Yes! perfect.