Discover the Key Idea to Solve the 10 Coins Puzzle Elegantly
Master the discovery process behind solving the 10 coin puzzle. Learn how to think critically and uncover the key idea leading to the elegant solution.
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?
Modified puzzle: after you solve for 10 coins, you must solve for 20 coins to show that you really could get hold of an elegant, efficient method.
Recommended time to solve 20 minutes.
Hint: Persistently think to discover the fundamental rule behind the number of times a coin will be flipped after all flips are over.
Solution to the 10 coins puzzle: Discovering the Key Idea Behind the Number of Times a Coin will be Flipped at the End
What we know:
- Starting position: 10 coins in a row, all heads up.
- Flipping actions: 1st turn: flip every coin, 2nd turn: flip every second coin, 3rd turn: flip every third coin....last 10th turn: flip only the 10th coin.
- Result of a coin flip: its face up reverses.
- To find: which coins will be heads up after 10 turns.
Discovering the key idea behind a coin finishing heads up after 10 flips:
Question you ask: What is the rule behind a coin finishing heads up after a few flips?
Answer: And this is crucial realization: A coin starting from heads up will also finish heads up after a few flips, only when the number of flips is an EVEN number.
This is the first important clue you discover. And it raises the next question.
Question: Under what conditions, the number of flips of a coin will finish as an even number?
The answer to this question is not immediately obvious, and so you make a few mental analytical trials (this is knowing by trial, or more commonly known prototyping method).
- Analytical trial 1: The 1st coin will be flipped only once, and never again.
- Analytical trial 2: The 2nd coin and the 3rd coin will be flipped exactly twice, on 1st turn and on 2nd and 3rd turn respectively. These will end as heads up.
- Analytical trial 3: The 4th coin will be flipped three times: on first turn, second turn and fourth turn. It will end face down.
- Analytical trial 4: The 5th coin flip result will behave exactly as the flip results of the 2nd and 3rd coin.
Reasoning chain:
- From trial 2 and trial 4, it is easy to discover the reason behind the number of flips being 2: the coin position numbers for all three 1, 3 and 5 are prime numbers that cannot be broken up as a product of smaller numbers. Each of these have factors 1 and the number. That's why in the 1st turn and in the 2nd, 3rd and 5th turns, coins in these positions 2, 3 and 5, are flipped again. These coins will finish heads up being flipped twice.
- What about trial 3? In this case, the coin in position 4 will be flipped in 1st turn, 2nd turn and the 4th turn and will end up face down because:
- The position number 4 has factors 1, 2 and 4—the number of factors and the number of flips both odd.
Discovery of the key mechanism: Efficient solution to the 10 coin and 20 coin puzzle:
Fact: The coin flipped an even number of times will end as heads up.
Rule governing these coins ending as heads up: The coins in positions with numbers that have an even number of factors will be flipped even number of times and will end as heads up.
Answer for 10 coins puzzle: 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), 7 (factors 1, 7), 8 (factors 1, 2, 4, 8) and 10 (factors 1, 2, 5, 10).
7 coins will end as heads up. Exceptions are the coins in positions 1, 4 and 9 with an odd number of factors.
Answer for 20 coin flips:
Besides the coins in positions 2, 3, 5, 6, 7, 8 and 10, the coins that will end as heads up in a 20 coin flip puzzle are in positions:
11 (factors 1, 11), 12 (factors 1, 2, 3, 4, 6, 12), 13 (factors 1, 13), 14 (factors 1, 2, 7, 14), 15 (factors 1, 3, 5, 15), 17 (factors 1, 17), 18 (1, 2, 3, 6, 9, 18), 19 (factors 1, 19) and 20 (factors 1, 2, 4, 5, 10, 20).
The only exception among the coins 11 to 20 will be the 16th coin with factors (1, 2, 4, 8, 16)—an odd number of factors that makes this coin end up face down.
The total number of coins ending as heads up will be 16 among the total of 20 coins.
Curious pattern:
- In 10 coin flip, 7 out of 10, that is 70% of the coins end as heads up—a significantly large proportion indeed!
- In 20 coin flip, the percentage of coins ending heads up in this range increases to 90%—a more curious pattern.
Make a habit of identifying curious patterns all around you and thresh out the reasons behind the patterns being so.
If you are more interested
In any range of numbers of the coin flip puzzle, only the coins in positions that are perfect squares like 1, 4, 9, 16, 25, 36 and so on will remain heads down. All the rest of the coins will end as heads up.
This is a mathematical truth.
This will mean, as the range becomes longer, the number of perfect squares being fewer and fewer, the percentage of coins ending as face up increases gradually. In the 10 numbers of 10-step ranges, the percentage of heads up coins will reach a value of 90% as shown in the graph.
But, when the upper limit of the range further increases beyond 100 to 110, 120...200 and so on, the percentage increases further tending towards 100% but never reaching the limit of 100% (rather reaching 100% only at infinitely large value of upper limit of the range). This is called an asymptotic curve.
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