Cross the stream from first bank to second by stepping on the 8 stones. Visit second bank twice and first bank once more. Cross in minimum number of steps.
Start from the first bank. Cross the stream by stepping on each of the 8 stones same number of times.
You can step on only the nearest stone from each bank. Visit the second bank twice and return to the first bank once more.
From the First bank cross the stream in least number of steps.
Recommended time to solve: 20 minutes.
Caution: Without a thoughtful approach, you may take too many steps to cross.
Solution to Stepping stone stream crossing riddle: Getting over the first hurdle
The stepping stones are numbered for ease of explanation.
From the first bank, only the stone 1 can be reached and from the second bank the stone 8 can only be reached. The second bank is to be visited twice and the first bank to be visited once more.
First, take a simple approach as a trial. Start from the first bank and cross over to the second bank. Return to the first bank again. Finally cross the stream a third time to the second bank.
You will cross the stream thrice taking 27 steps and using each stepping stone three times. The sequence of steps is:
First bank - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - Second bank - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 - First bank - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - Second bank.
This is a trivial result with the greatest number of steps taken. This must be reduced.
Raise an important question,
Question: Knowing that number of steps will double, why do you return all the way to the first bank after the first crossing?
Answer: You returned to the first bank to visit it once more required in the puzzle condition. And from First bank you will cross over to Second bank as required.
From this obvious answer, make the only conclusion possible,
Conclusion 1: From the Second bank, do not return to the First bank all the way. This is the only way to reduce the number of steps.
True, it makes sense. Returning all the way to the First bank will double the number of steps. You will triple it when you traverse the same path forward to the Second bank.
How to reduce the number of crossings? The only way is:
Conclusion 2: Cross the stream ONLY ONCE. From the first bank to the second.
But how? How to visit each bank twice?
You need an out-of-the-box idea for the breakthrough answer.
Reasoning: As you cannot return to the first bank from the second bank to reduce the number of steps, can you take one step forward after start and return to the first bank again? That will sure be equivalent to visiting the first bank twice!
This should solve the problem of visiting the first bank twice without increasing number of steps.
This is the first critical breakthrough.
Make the key conclusions using the breakthrough idea:
Conclusion 3: Start from the first bank, take one step forward and return to the first bank again.
Conclusion 4: At the end of the only crossing, step on the second bank and take one step back to stone 8. Turn around finally to end the crossing by stepping on the second bank again.
This must be the way the crossing will start and end:
The first two steps: First bank to Stone 1 back to First bank.
The last three steps: Stone 8 to Second bank, back to Stone 8. Back to second bank to end the crossing.
After the start, you are now on the First bank. Knowing last three steps, you need to plan how to reach stone 8 from the first bank.
Solution to the stepping stone stream crossing riddle: Getting over the second hurdle
The second challenge to get over is,
Second challenge: To reach stone 8 starting from the first bank and using each of the stones 2 to 7 twice.
Explore the simplest approach: Use each intermediate stone once to reach stone 8 from First bank:
First bank - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8.
This is 8 more steps. Total steps in this approach is 8 + 5 = 13, but will it be a valid crossing?
The stones 1 and 8 both will be used 2 times and the rest six stones will be used once only. This is an invalid sequence of steps.
Conclusion 5: The least we can do is to use the rest six steps once more. Total number will then be 13 + 6 = 19 and it must be the minimum number of steps to cross.
How to do this?
Known from trial: If you move only in the forward direction, the problem won’t be solved.
Only possible way is:
Conclusion 6: You must move forward a few steps and move backward again by a few steps so that resultant movement is at least one step forward.
This is the second critical breakthrough idea. But unfortunately, still too general an idea. Still to find out how many steps to move forward and how many to move backward.
The figure of steps with the banks repeated for convenience of analysis.
You are on the First bank. Goal is to reach stone 8.
As a trial, take two steps forward to stone 1 and then to stone 2. Take one step back to stone 1. Repeat the sequence. The result is:
First bank - 1 - 2 - back to 1 - 2 - 3 - back to 2…
This is an invalid sequence of steps. Stone 2 is used 3 times. This is because the last forward step in the repeated sequence is the next step to the first. In the next 2 steps forward, 1 step back sequence, you will use this last step twice more, total 3 times.
Final conclusion 7: The only way is to take three steps forward and one step backward and repeat the sequence. In each of this 3 step sequence, your net progress will be two steps forward.
The result will be:
First bank - 1 - 2 - 3 - back to 2 - 3 - 4 - 5 - back to 4 - 5 - 6 - 7 - back to 6 - 7 - 8.
Success! Each of the six stones 2 to 7 is used twice. Stone 1 and stone 8 used once each. At start and end, these two will be used once more.
Combining the three parts, the final sequence of steps is:
First bank - 1 - back to First bank - 1 - 2 - 3 - back to 2 - 3 - 4 - 5 - back to 4 - 5 - 6 - 7 - back to 6 - 7 - 8 - Second bank - back to 8 - Second bank.
Each dash between two places is one step. The stream is crossed taking 19 steps. Second bank visited twice and First bank returned to once. All 8 steps used 2 times each.
The solution figure identifies the total 19 steps including the backward steps.
This is not an easy puzzle. Being aware of its complexity, the approach adopted has been to identify the most important barrier and the only possible way to overcome the barrier.
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