Split a pile of coins numbered 1 to 8 into odd numbered and even numbered piles of coins in two of the five slots without placing a coin on a smaller one.
In the following figure, a single pile of 8 coins numbered 1 to 8 is shown in slot E. The coins are in increasing number from top to bottom. Split the pile of 8 coins into two piles of odd numbered and even numbered coins in slot A and slot B in the smallest number of moves.
Each of the two piles should have the coins in the sequence of increasing number from top to bottom. At no stage of coin moves using the five slots, a coin can be moved on top of a coin with a smaller number.
Recommended time to solve: 20 minutes.
To solve the puzzle without confusion, use End state analysis in a systematic, reasoned approach of problem solving.
Solution to the Odd and even numbered coin piles riddle
In the first four steps, coins 1, 2, 3 and 4 are placed in four vacant slots A, B, C and D.
When you move coin 3 to slot C, a new possibility of moving coin 1 on top of coin 3 is created in the fourth move. Instead, if you continue to expose coin 5, you will have the flexibility of another new move of coin 2 on coin 4.
This is a more flexible approach exposing coin 5 in first four moves straightway. It also exploits the resources of the four empty clots A, B, C and D fully.
Following is this State 1 achieved in four coin moves.
After the first four moves, the coins may be moved in several valid ways. But instead of thinking in this start to end forward direction, we’ll now analyze the state of coin placements near the end.
Near the end, coin 8 is to be exposed and before that coins 6 and 7 are also to be exposed. None of these two coins 6 and 7 can be placed on any existing coin pile as all the coins moved before coin 6 are of smaller number than 6. It follows,
Conclusion 1: To expose coin 8, coins 6 and 7 must be moved and placed into two empty slots.
Conclusion 2: Rest 5 coins 1 to 5 must be placed in the other two remaining slots.
Challenge to meet: how should these 5 coins be distributed among the two slots? Will it be 4 coins in first, 1 in second, or 3 coins in first and 2 in second?
This is the time to form the guiding principle in formation of intermediate coin piles. An intermediate coin pile, such as the pile 1-2-3, needs 3 moves to form as well as 3 more moves to dismantle, 6 moves in all.
As an intermediate coin pile must be dismantled to move the coins to the odd and even pile, larger the intermediate pile, more the wasted moves.
So the guiding principle of creating an intermediate coin pile is:
Conclusion 3: To achieve the solution in the least number of moves, the intermediate coin piles have to be few and short.
Applying this principle then, for least move solution,
Conclusion 4: Coins 1 to 5 are to be placed in two slots as two piles of 3 coins and 2 coins, and not 4 coins and 1 coin. Now, coin 8 is exposed by moving coins 6 and 7 to the two empty slots.
Following figure shows the near-end-state scheme,
Coin 7 is shown in slot A intentionally because it will be its final destination. Once coin 7 occupies slot A, the odd pile formation could start.
But what will be the composition of the 3 coin and 2 coin piles?
The only combination that will work is,
Conclusion 5: 1-3-5 as the 3 coin pile and 2-4 as the 2 coin pile. This is because coin 6 has been exposed and placed in slot C after the 3 coin pile has been formed and so pile 2-4-6 pile couldn’t have been formed.
Which slots should the two piles occupy?
The 1-3-5 three coin pile in slot B is the natural choice. These three would be moved to slot A to complete the odd numbered pile and in the process vacate slot B for coin 8 to occupy. Other possibility of 2-4 pile in slot B would create more wasteful moves.
This is the use of the powerful problem solving technique of end state analysis.
Following is this best choice for the near-end-state:
From State 1 configuration, State 3 configuration is to be reached in the smallest number of moves. State 1 configuration is shown again for convenience.
First obvious move is coin 2 to slot D on coin 4. Slot B falls vacant. Move coin 5 to slot B to start an intermediate pile of 1-3-5. Move coin 3 to slot B on coin 5, move coin 1 on coin 3 in slot B. Slots A and C fall vacant. Move coin 6 to slot C and coin 7 to slot A. State 3 configuration is reached in 6 more moves besides first 4.
Immediate goal is:
Immediate goal: Starting from State 3 configuration, form the odd numbered pile 1-3-5-7 in slot A in the least number of moves.
The State 3 configuration is shown again for convenience.
Move coin 1 to slot A on coin 7. Vacate slot C by moving coin 6 to slot E on coin 8. Move coin 3 to vacated slot C. Move coin 1 to slot C on coin 3. Move coin 5 exposed to slot A on coin 7. Move coin 1 to slot B. Move coin 3 to slot A on coin 5. Move coin 1 to slot A on coin 3. Odd numbered pile of 4 coins 1-3-5-7 formed in 8 moves besides 10 earlier ones.
Following shows this last but one stage configuration.
It is now easy to take the last few steps to form the even numbered pile.
Move coin 6 to slot C for temporary parking. Move coin 8 to its final destination of slot B. Move coin 6 to slot B on coin 8. Move coin 2 to slot C for temporary parking. Move coin 4 to slot B on coin 6. Finally move coin 2 to slot B on coin 4 to complete the even numbered pile of coins 2-4-6-8 as well.
Solution comprises the least number of 24 moves.
Final solution is:
Combine the moves from start to end. The moves to form the two piles are:
Stage 1: Exposing coin 5:
Move coin 1 from slot E to slot A. Move coin 2 from slot E to slot B. Move coin 3 from slot E to slot C. Move coin 4 from slot E to slot D: total 4 moves.
Stage 2: Forming two piles 1-3-5 and 2-4 and exposing coin 8:
Move coin 2 from slot B to slot D on coin 4. Move coin 5 from slot E to slot B.
Move coin 3 from slot C to slot B on coin 5. Move coin 1 from slot A to slot B on coin 3.
Move coin 6 to from slot E to slot C. Move coin 7 from slot E to slot A: total 6, cumulative 10 moves.
Stage 3: Forming odd numbered pile:
Move coin 1 from slot B to slot A on coin 7. Move coin 6 from slot C to slot E on coin 8.
Move coin 3 from slot B to slot C. Move coin 1 from slot A to slot C on coin 3.
Move coin 5 from slot B to slot A on coin 7. Move coin 1 from slot C to slot B.
Move coin 3 from slot C to slot A on coin 5. Move coin 1 from slot B to slot A on coin 3: total 8, cumulative 18 moves.
Final stage: Forming even numbered pile:
Move coin 6 from slot E to slot C. Move coin 8 from slot E to slot B.
Move coin 6 from slot C to slot B on coin 8. Move coin 2 from slot D to slot C.
Move coin 4 from slot D to slot B on coin 6. Move coin 2 from slot C to slot B on coin 4: total 6, cumulative 24 smallest number of moves.
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