just to spread a little of my math i learn, though this is really just for fun, its an application from our last lecture:

1. A teacher paints a spot on each of three students’
foreheads.
2. The students know each spot is either black or white.
3. Each student can see each other's forehead (but not his
own).
4. Anyone who sees a black spot on another person's
forehead must raise a single hand.
5. The first person who knows the colour of his own spot
must announce immediately to the rests.
6. If the three spots are actually all black (unknown to the students), will the students be able to tell whether the spot on their own forehead is black or white?

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you can highlight the below portion to read the solution.:
Every student will see 2 black spots. So everyone will
raise their hands. There are two possible cases
a. Exactly 2 black spots
b. Exactly 3 black spots
If there are exactly 2 black spots, then those with black spots will know it and announce immediately. (suppose student C's spot is white.then student A will think: If my spot is white, then B will not raise his hand since he will not see any black spot, as C's spot is white.) Hence, if no one announce immediately, then there cannot be exactly 2 black spots. So there must be 3 black spots!


get it? i think its super cool. this is exactly why i love math.