Sum of Dots
The Grand Master paints 3 dots on one logician's forehead, 2 on another, and 2 on the last. Each logician cannot see his own forehead but can see the others'. The Master announces, "At least one of you has an odd number of dots. I painted 6, 7 or 8 dots. The first to correctly announce the number of dots on his forehead passes." After a brief silence, one shouts out the correct answer. How many dots are there on his forehead? How did he know?
What did he shout?
The logician with three dots on his forehead shouts.
How did he know?
The logician with three dots sees two even-numbered sets of dots. Knowing that at least one logician has an even number of dots, he concludes that he has an even number of dots. Two evens must sum to an even, and an even with an odd must sum to an odd. The only possible odd number is 7. Thus, he sees 4 dots and concludes he must have 3.