A prize is hidden under one of these five objects. Shilpa is privately told the shape and Colin is privately told the color . Shilpa and Colin are mathematicians who use perfect logical reasoning, and the initial setup is common knowledge to both of them.
The host asks, “Do either of you know where the prize is?”
Again both of them are silent.
The host asks again, “Do you know now?”
Both simultaneously say, "Yes!"
The host asks, “Do either of you know where the prize is?”
Under which object is the prize?
Shilpa is told the shape, so she can consider where the prize might be based on which shape she is told.
Shilpa
triangle: green triangle, blue triangle
circle: yellow circle, green circle
square: blue square
Similarly, Colin is told the color, so he can consider where the prize might be based on which color is told.
Colin
green: green triangle, green circle
blue: blue square, blue triangle
yellow: yellow circle
Each time the host asks a question, each of them can inductively eliminate options until they can solve for which object has the prize.
Round 1
The host first asks, “Do either of you know where the prize is?” And both of them are silent.
If Shilpa had been told “square,” she would know the object was under the blue square. So she must not have been told square initially.
Similarly, if Colin had been told “yellow,” then he would have concluded the object is under the yellow circle. So he might not have been told yellow initially.
Shilpa
triangle: green triangle, blue triangle
circle: yellow circle, green circle
Colin
green: green triangle, green circle
blue: blue square, blue triangle
Because Shilpa reasons perfectly, she can also conclude that Colin was not told yellow. Thus she can eliminate the yellow circle from her set of options.
Shilpa
triangle: green triangle, blue triangle
circle: green circle
Similarly, since Colin reasons perfectly, he can reason that Shilpa was not told square initially, and he can eliminate the blue square from his set of options.
Colin
green: green triangle, green circle
blue: blue triangle
Round 2
The host then asks, “Do you know now?” Again both of them are silent.
If Shilpa had been told circle initially, then by this stage she would have concluded the prize was under the green circle. Since she was still silent, she must not have been told cirlce initially and was instead told triangle.
Shilpa
triangle: green triangle, blue triangle
If Colin had been told blue initially, then by this stage he could have concluded the prize was under the blue triangle. But again he was silent, so he must have been told green initially.
Colin
green: green triangle, green circle
But now Shilpa knows Colin must have been told green initially, and Colin knows Shilpa must have been told triangle initially. So they are ready for the answer once the host asks them again!
Round 3
The host asks again, “Do you know now?” At this point both Shilpa and Colin simultaneously say, “Yes!”
Shilpa must have been told triangle initially, and Colin must have been told green initially. Thus the prize is under the green triangle.