The classic 8 coin puzzle solution presents a logical challenge: how to identify a single counterfeit coin, known to be lighter, from a group of eight identical-looking coins using a balance scale in the fewest possible weighings. This article details the method to achieve this feat in a minimum of two weighings, providing a practical and straightforward approach.
The Premise of the 8 Coin Puzzle Solution
You are given eight coins that appear identical. One of these coins is a counterfeit, and it is specifically lighter than the genuine coins. Your tool is a balance scale, which can only tell you if two groups of coins are equal in weight, or if one group is lighter or heavier than the other. The objective of the 8 coin puzzle solution is to pinpoint the lighter, counterfeit coin in just two uses of the balance scale.
This puzzle is a fundamental exercise in logical deduction and resource optimization. It demonstrates how strategic division and analysis of outcomes can lead to a quick resolution even with limited information.
Understanding the Balance Scale
A balance scale operates on a simple principle:
- If the items on both sides are of equal weight, the scale remains balanced.
- If the items on one side are lighter, that side will rise.
- If the items on one side are heavier, that side will descend.
For the 8 coin puzzle solution, we are specifically looking for the side that rises, indicating the presence of the lighter, counterfeit coin.
The Strategy for the 8 Coin Puzzle Solution
The core strategy for the 8 coin puzzle solution involves dividing the coins into specific groups and interpreting the results of each weighing to narrow down the possibilities. The key is to eliminate as many genuine coins as possible with each weighing.
Weighing 1: Initial Division and Testing
The first step in finding the 8 coin puzzle solution is to divide the eight coins into three distinct groups. This division is crucial for effective elimination.
1. Group A: Three coins
- Group B: Three coins
- Group C: Two coins (these are the coins not placed on the scale initially)
Now, place Group A on one side of the balance scale and Group B on the other side.
Scenario 1: The Scale Balances
If the scale remains balanced when Group A and Group B are weighed against each other, this provides significant information for the 8 coin puzzle solution.
- Implication: All coins in Group A and all coins in Group B must be genuine. If any of these six coins were the counterfeit (lighter) one, the scale would not have balanced.
- Conclusion: The counterfeit coin must be within Group C. Group C consists of the two coins that were not placed on the scale.
Scenario 2: The Scale Does Not Balance
If the scale does not balance, one side will be lighter than the other.
- Implication: The lighter side contains the counterfeit coin.
- Specificity:
- If Group A is lighter, the counterfeit coin is one of the three coins in Group A.
- If Group B is lighter, the counterfeit coin is one of the three coins in Group B.
- Conclusion: The counterfeit coin is located within the lighter group of three coins. The two coins in Group C are confirmed genuine, as are the coins on the heavier side of the scale.
Weighing 2: Pinpointing the Counterfeit Coin
The second weighing is used to isolate the counterfeit coin based on the outcome of Weighing 1.
If the Scale Balanced in Weighing 1 (Scenario 1)
Recall that if the scale balanced in Weighing 1, the counterfeit coin is known to be one of the two coins in Group C.
- Action: Take these two coins and weigh them against each other on the balance scale.
- Outcome:
- One coin will be lighter than the other. The lighter coin is the counterfeit one.
- There is no possibility of them balancing, as one of them must be the counterfeit.
This completes the 8 coin puzzle solution for this scenario.
If the Scale Did Not Balance in Weighing 1 (Scenario 2)
Recall that if the scale did not balance in Weighing 1, the counterfeit coin is known to be one of the three coins from the lighter group (either Group A or Group B). Let’s call this identified group “Group L” (Lighter Group).
- Action: Take any two of the three coins from Group L. Weigh these two coins against each other on the balance scale.
- Outcome:
- If one of the two coins on the scale is lighter: That specific coin is the counterfeit one. The other coin on the scale and the third coin from Group L are genuine.
- If the two coins on the scale balance: This means both coins on the scale are genuine. Therefore, the third coin from Group L (the one you did not weigh) must be the counterfeit one.
This completes the 8 coin puzzle solution for this scenario.
Step-by-Step Example of the 8 Coin Puzzle Solution
To illustrate the 8 coin puzzle solution, let’s follow a hypothetical example.
Assume our coins are numbered 1 through 8, and coin #5 is the lighter, counterfeit one.
Initial Setup:
Coins: C1, C2, C3, C4, C5, C6, C7, C8 (C5 is counterfeit)
Weighing 1:
- Group A: C1, C2, C3
- Group B: C4, C5, C6
- Group C: C7, C8
Place C1, C2, C3 on the left side of the scale.
Place C4, C5, C6 on the right side of the scale.
- Result: The right side (C4, C5, C6) goes up, indicating it is lighter.
- Deduction: The counterfeit coin is among C4, C5, C6. Coins C1, C2, C3, C7, C8 are genuine.
Weighing 2:
We know the counterfeit is in the group {C4, C5, C6}.
- Action: Pick any two coins from this group. Let’s pick C4 and C5.
- Placement: Place C4 on the left side of the scale, C5 on the right side.
- Result: The right side (C5) goes up, indicating it is lighter.
- Conclusion: C5 is the counterfeit coin.
Let’s consider another outcome for Weighing 1:
Weighing 1 (Alternative Scenario):
Coins: C1, C2, C3, C4, C5, C6, C7, C8 (C7 is counterfeit)
- Group A: C1, C2, C3
- Group B: C4, C5, C6
- Group C: C7, C8
Place C1, C2, C3 on the left side of the scale.
Place C4, C5, C6 on the right side of the scale.
- Result: The scale balances.
- Deduction: The counterfeit coin is among C7, C8. Coins C1, C2, C3, C4, C5, C6 are genuine.
Weighing 2 (Alternative Scenario):
We know the counterfeit is in the group {C7, C8}.
- Action: Place C7 on the left side of the scale, C8 on the right side.
- Result: The left side (C7) goes up, indicating it is lighter.
- Conclusion: C7 is the counterfeit coin.
Why Two Weighings are Sufficient
The efficiency of the 8 coin puzzle solution lies in the power of trinary logic, or rather, the ability of each weighing to yield one of three distinct outcomes: left side lighter, right side lighter, or balanced.
- Weighing 1: With 8 coins, we divide them into groups of 3, 3, and 2.
- If the scale balances, we isolate the problem to 2 coins.
- If the scale tips, we isolate the problem to 3 coins.
In either case, the possibilities are significantly reduced. From 8 initial possibilities, we reduce to either 2 or 3 possibilities.
- Weighing 2:
- If 2 possibilities remain: Weighing them against each other directly identifies the lighter coin.
- If 3 possibilities remain: Weighing two of them against each other identifies the lighter coin (either one of the two on the scale or the one left off).
This systematic reduction ensures that in just two steps, the single lighter coin is identified. The 8 coin puzzle solution is a classic example of optimal problem-solving under constraints.
Generalization and Limitations
The 8 coin puzzle solution is a specific instance of a broader class of balance scale problems.
- A single weighing on a balance scale can distinguish between 3 possibilities (left lighter, right lighter, balanced).
- Two weighings can distinguish between 3 x 3 = 9 possibilities. This is why 8 coins (or up to 9 coins) can be solved in two weighings.
- Three weighings can distinguish between 3 x 3 x 3 = 27 possibilities.
Limitations:
- This specific 8 coin puzzle solution assumes you know the counterfeit coin is lighter. If the counterfeit coin could be either lighter or heavier, the problem becomes more complex and might require additional weighings or a different strategy.
- The method assumes only one counterfeit coin exists.
The 8 coin puzzle solution remains a foundational problem in recreational mathematics and logic, demonstrating the power of structured thinking and efficient division of possibilities.
How to find the one odd coin out of 8 identical looking ones which may be heavier or lighter than the rest using a balance scale in 3 weighings?
I can help with that. Solution 1
If the first weighing does not yield a balance, the lighter fake is among the three lighter coins. Take any two of them and put them on the opposite cups of the scale. If they weigh the same, it is the third coin in the lighter group that is fake; if they do not weigh the same, the lighter one is the fake.
How to solve the coin puzzle?
Thanks for asking. Points uh upwards. And the challenge with this puzzle is to move uh three of those coins. And make the triangle point in the opposite. Direction.
How many outcomes are there when tossing 8 coins?
I can help with that. Hence, the total number of outcomes is equal to 256.
What is the algorithm for the fake coin problem?
Thanks for asking. Here is an algorithm:
If more than one coins are there then divide the coins into piles of A = ceiling(n/3), B=ceiling(n/3), and C= n-2* ceiling(n/3). Weigh A and B, if scale balances then repeat from the first step with total number of coins equalling number of coins in C.