Arrange four circular coins touching each other to accommodate a fifth coin
Place four perfectly circular and similar one rupee coins on a flat table top. The coins must touch each other in such a way that if you place a fifth similar 1 rupee coin as shown in the figure below, it will touch all the other four.
You may not do any marking, measuring or use any extra object.
Time to solve: 15 minutes.
Solution to the riddle of four coins touching
Soon you realize that if you depend on guesstimating coin positions with your eyes, you can never be sure of the perfect positioning of the coins.
Alright, this elusive puzzle looks simple, but how to go ahead with confidence, you wonder!
Yes, of course you can place two coins on the flat table top and slide the two towards each other so that the two touch each other. This is the least you can do.
This mutually touching pair of coins form a part of the required configuration. You have solved at least a part of the problem with little effort.
Realize at this point that,
Conclusion 1: The riddle at the core is on two coins touching each other.
What is the special property of the mutually touching and circular pair of coins? How to place a third coin touching these two with no help of an extra tool?
Analyzing the figure of the two touching coins for a few minutes, you discover the trick. It is the key pattern.
Conclusion 2: When two similar circular coins touch each other, a hollow is created between the two. You can easily place another coin in this hollow, taking no help with any extra tool or marking so that the new coin touch the existing two coins.
Now add the 3rd coin touching the pair of mutually touching coins you have formed. Following shows the mutually touching three coins. For convenience, the coins are numbered.
A hollow is shown between coins 2 and 3. Any more hollows in the configuration? Yes, two more—between coins 1, 3 and coins 1, 2.
Place the fourth coin in the hollow between coins 1 and 3. Placing it instead between coins 2 and 3 will be equivalent. Both new configurations will have a fair similarity with the desired configuration.
The hollow between coins 1 and 2 is ignored as placing the 4th coin in this position will make the resulting figure too different from the desired configuration.
The 4 coin new figure shown.
Coins 4, 1 and 2 together form three-fourth part of the desired solution. If only you could have moved the coin 3 in the faded out desired place! No way to do it directly in one step from this position by moving one coin.
Conclusion 3: There must at least be one more step before the solution.
Continue to analyze coin movement options.
With no help from a hollow joint between two coins, you cannot possibly move coin 3.
On the other hand, out of the three coins 4, 1 and 2, you can leave a hollow, though not a touching hollow but still a hollow between coins 4 and 2, if you move coin 1 to the hollow between coins 2 and 3.
And if you do that, moved coin 1 will touch coin 3 and more importantly coin 2. You realize that this new place for coin 1 will be the right touching place for the fourth coin in the final configuration.
Conclusion 4: Moving coin 1 to the hollow between coins 2 and 3 touching the two will not only place it in the fourth place of the desired solution configuration, it will leave a second different hollow between coin 4 and 2. In the last step you will then move coin 3 in this new hollow.
The second four-coin touching configuration shown.
Now you know what you have to do.
Conclusion 5: You will just move coin 3 in the gap between the two coins 4 and 2 so that it touches the two coins WITHOUT DISTURBING THE TWO.
To be absolutely safe,
Precise action 1: first you pick up coin 3 without disturbing the other three coins and place it in between coins 4 and 2. Now you slide the coin deeper into the gap. This intermediate stage formation is shown.
Facing the last step, you take the precise second action to finally solve the puzzle.
Precise action 2: You press on the two coins 4 and 2 with your two fingers and SLIDE coin 3 deeper into the gap carefully till you can't slide it any more. The coin 3 is now wedged between coin 4 and 2, touching both.
Coins 1, 2, 3, 4 together form your solution. The ghost of the 5th coin that you never had takes its appointed place touching the other four.
Following shows the solution formation.
Systematic problem solving is what you have done. You have taken a reasonable but small step, analyzed the outcome and taken the next step—always towards the solution.
Solve the six coins touching puzzle, extended from the four coins touching riddle
Place six coins touching in such a way that if a seventh coin is placed at the center, it will touch all six. Formation shown,
If you have understood how the four coins touching riddle is solved, it should not be difficult for you to solve this puzzle that is nothing but an extended form of the first puzzle.
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