Mariolys is creating a numbered list of distinct integers. Luigis sees that there are N numbers in the list, and states that he can find either a subsequence of length 13 with increasing terms or a subsequence of length 17 with decreasing terms. What is the minimum value of N for Luigis' claim to be true?
Details and assumptions:
Luigis doesn't know the numbers on Mariolys list.
The chosen terms of the subsequence need not be consecutive. In the list {1,45,23,56,25,58,8,2}, we have an increasing subsequence of length 4 formed by 1,23,25,58 (which are the 1st, 3rd, 5th and 7th terms of the original list). We also have a decreasing subsequence of length 4 formed by 56,25,8,2 (which are the 4th, 5th, 7th and 9th terms of the original list).
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To find the minimum value of N for Luigis' claim to be true, we need to determine the longest possible increasing subsequence and the longest possible decreasing subsequence that can be formed from a list of N distinct integers.
Let's start by finding the longest possible increasing subsequence of length 13. To do this, we can use the concept of the "patience sorting" algorithm.
In the patience sorting algorithm, imagine that we have N stacks initially empty, and we go through the list of numbers one by one, placing each number in the leftmost stack where it fits according to the increasing order. If a number does not fit in any existing stack, we create a new stack and put it there. After going through the entire list, the number of stacks required will be the length of the longest increasing subsequence.
In this problem, we want to find the longest increasing subsequence of length 13. So we start with 13 empty stacks and go through the list of numbers. If we can fit any number in any existing stack, we place it there. If a number cannot fit in any existing stack, we create a new stack and put it there. After going through the entire list, we count the number of stacks required.
Now let's find the longest possible decreasing subsequence of length 17. We can apply the same patience sorting algorithm with a small modification. In this case, instead of sorting the numbers in increasing order, we sort them in decreasing order. So we start with 17 empty stacks and go through the list of numbers. If we can fit any number in any existing stack, we place it there. If a number cannot fit in any existing stack, we create a new stack and put it there. After going through the entire list, we count the number of stacks required.
To find the minimum value of N for Luigis' claim to be true, we need to find the larger of these two numbers: the number of stacks required for the longest increasing subsequence of length 13 and the number of stacks required for the longest decreasing subsequence of length 17. Let's call these numbers A and B, respectively.
The minimum value of N for Luigis' claim to be true is obtained by taking the maximum between A and B. So N = max(A, B).
By applying this approach, you can calculate N by implementing the patience sorting algorithm twice: once for the longest increasing subsequence of length 13 and once for the longest decreasing subsequence of length 17. The larger of the two values obtained will be the minimum value of N for Luigis' claim to be true.