Difference between revisions of "2024 AMC 10A Problems/Problem 6"
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There are <math>6</math> moves needed to change <math>ABCD</math> to <math>DCBA</math>. | There are <math>6</math> moves needed to change <math>ABCD</math> to <math>DCBA</math>. | ||
| − | We see a pattern of <math>0,1,3,6,...</math>. We notice that the difference between consecutive terms is increasing by <math>1</math>, so in the same way, for <math>5</math> letters, we would need <math>10</math> moves, and for <math>6</math>, we would need <math>\boxed{15}</math> moves. | + | We see a pattern of <math>0,1,3,6,...</math>. We notice that the difference between consecutive terms is increasing by <math>1</math>, so in the same way, for <math>5</math> letters, we would need <math>10</math> moves, and for <math>6</math>, we would need <math>\boxed{\textbf{(D)} 15}</math> moves. |
Thinking why, when we start making these moves, we see that for a string of length <math>n</math>, it takes <math>n-1</math> moves to move the last letter to the front. After, we get a string that will be changed identically to a string of length <math>n-1</math>. This works in our pattern above and is another way to think about the problem! | Thinking why, when we start making these moves, we see that for a string of length <math>n</math>, it takes <math>n-1</math> moves to move the last letter to the front. After, we get a string that will be changed identically to a string of length <math>n-1</math>. This works in our pattern above and is another way to think about the problem! | ||
Revision as of 18:09, 8 November 2024
Contents
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 fifth spot, giving 
swaps for 
to move to the fourth spot, giving 
to move to the fourth spot, giving 
swaps for 
to move to the third 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 
to move to the second spot (so 
becomes the first spot), giving 
Together, the answer is 
~MRENTHUSIASM
Solution 2
Note: This is my first time writing a solution, so please feel free to edit this and change it, thank you in advance! (Work in progress)
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
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
