2024 AMC 10A Problems/Problem 6
Contents
[hide]Problem
What is the minimum number of successive swaps of adjacent letters in the string that are needed to change the string to (For example, swaps are required to change to one such sequence of swaps is )
Solution
Procedurally, it takes:
- swaps for to move to the sixth spot, giving
- swaps for to move to the fifth spot, giving
- swaps for to move to the fourth spot, giving
- swaps for to move to the third spot, giving
- swap for to move to the second spot (so becomes the first spot), giving
Together, the answer is
~MRENTHUSIASM
Solution 2 (Recursive Approach)
Note: This is my first time writing a solution, so please feel free to edit this and change it, thank you in advance!
We can proceed by a recursive tactic on the number of letters in the string.
Looking at the string , there are moves needed to change it to the string
Then, there is move to change to .
Similarly, there is moves needed for three letters (said in the problem).
There are moves needed to change to .
We see a pattern of . We notice that the difference between consecutive terms is increasing by , so in the same way, for letters, we would need moves, and for , we would need moves.
Thinking why, when we start making these moves, we see that for a string of length , it takes moves to move the last letter to the front. After, we get a string that will be changed identically to a string of length . This works in our pattern above and is another way to think about the problem!
~world123
Video Solution 1 by Power Solve
https://youtu.be/j-37jvqzhrg?si=ieBRx0-CUihcKttE&t=616
Video Solution by Daily Dose of Math
~Thesmartgreekmathdude
See also
2024 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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