Discover the Key Idea to Solve the 10 Coins Puzzle Elegantly
You have 10 coins all heads up. On your first turn, flip every coin. Next, flip every second coin, and so on. After 10 turns, which coins will be heads up?
The 10 Coins Puzzle
You have 10 coins in a row, all heads up. On your first turn, flip every coin. On your second turn, flip every second coin, on the third turn flip every third coin, and so on. After 10 turns, which coins will be heads up?
Solution to the 10 coins puzzle
You have to find which coins will be heads up after a series of flips, each flip reversing the coin’s face up.
Start with the basic idea: Each coin is flipped once for every factor its position number has. For example, coin #3 with its position factors of 1 and 3 is flipped on 1st and 3rd turns.
Reasoning chain:
The first question you ask,
How will a coin turn out to be heads up?
Answer: Only the coins flipped an even number of times will end up as heads up.
Second question you ask,
Which coins will get an even number of flips?
While answering, you think for a moment to discover the key to the solution:
The solution’s key idea is:
The number of factors of its position number determines how many times each coin gets flipped. For example, coin #8 is flipped on turns 1, 2, 4, and 8 (its factors). It only remains to find the position numbers having the number of factors as EVEN.
Answer: The heads up coins will be in positions 2 (factors 1, 2), 3 (factors 1, 3), 5 (factors 1, 5), 6 (factors 1, 2, 3, 6), 8 (factors 1, 2, 4, 8), and 10 (1, 2, 5, 10).
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