Never Walks
What can run but never walks,
Has a mouth but never talks,
Has a head but never weeps,
Has a bed but never sleeps?
SOLUTION
The answer is RIVER.
Difficulty: Advanced
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Missing Pawns
White to play and mate in 4 moves.
Remark: The position on the diagram is one which occurs in actual play.
SOLUTION
Notice that the black queen and the black king have switched positions. However, this can happen only if some pawns have been moved. Therefore, we can conclude that the bottom row on the diagram is actually the 8th row of the chessboard. All black and all white pieces have reached their respective opposite sides of the board.
Now, White’s first move is Kb8-d7. The only moves black can play are with the knights. If Black plays Kb1-a3, Kb1-c3 or Kg1-h3, white mates in 2 more moves – Kd7-c5 and Kc5-d3. If Black moves Kg1-f3, then after Kd7-c5 Black can delay the mate by playing Kf3-e5. However, after the white queen takes it with Qxe5, Kc5-d3 is unavoidable.
Burn the Ropes
You have two ropes and a lighter. Each of the ropes burns out in exactly 60 minutes, but not at a uniform rate – it is possible for example that half of a rope burns out in 40 minutes and the other half in just 20. How can you measure exactly 45 minutes using the ropes and the lighter?
SOLUTION
First, you light up both ends of the first rope and one of the ends of the second rope. Exactly 30 minutes later the first rope will burn out completely and then you have to light up the other end of the second rope. It will take 15 more minutes for the second rope also to burn out completely, for a total of 30 + 15 = 45 minutes.
Missionaries, Cannibals
Three missionaries and three cannibals must cross a river with a boat which can carry at most two people at a time. However, if on one of the two banks of the river the missionaries get outnumbered by the cannibals, they will get eaten. How can all 6 men cross the river without anybody gets eaten?
Remark: The boat cannot cross the river with no people on board.
SOLUTION
Label the missionaries M1, M2, M3 and the cannibals C1, C2, C3. Then:
1. M1 and C1 cross the river, M1 comes back.
2. C2 and C3 cross the river, C2 comes back.
3. M1 and M2 cross the river, M1 and C1 come back.
4. M1 and M3 cross the river, C3 comes back.
5. C1 and C2 cross the river, C1 comes back.
6. C1 and C3 cross the river.
Now, everyone is on the other side.
Friends and Enemies
Show that in each group of 6 people, there are either 3 who know each other, or 3 who do not know each other.
SOLUTION
Let’s call the people A, B, C, D, E, F. Person A either knows at least 3 among B, C, D, E, F, or does not know at least 3 among B, C, D, E, F.
Assume the first possibility – A knows B, C, D. If B and C know each other, C and D know each other, or B and D know each other, then we find a group of 3 people who know each other. Otherwise, B, C, and D form a group in which no-one knows the others.
If A doesn’t know at least 3 among B, C, D, E, F, the arguments are the same.