Given an integers X. The task is to find the number of jumps to reach a point X in the number line starting from zero.
Note: The first jump made can be of length one unit and each successive jump will be exactly one unit longer than the previous jump in length. It is allowed to go either left or right in each jump.
Input : X = 8 Output : 4 Explanation : 0 -> -1 -> 1 -> 4-> 8 are possible stages. Input : X = 9 Output : 5 Explanation : 0 -> -1 -> -3 -> 0 -> 4-> 9 are possible stages
Approach : On observing carefully, it can be said easily that:
- If you have always jumped in the right direction then after n jumps you will be at the point p = 1 + 2 + 3 + 4 + … + n.
- If instead of jumping right, you jumped left in the kth jump, you would be at point p – 2k.
- Moreover, by carefully choosing which jumps to go left and which to go right, after n jumps, you can be at any point between n * (n + 1) / 2 and – (n * (n + 1) / 2) with the same parity as n * (n + 1) / 2.
Keeping the above points in mind, what you must do is simulate the jumping process, always jumping to the right, and if at some point, you’ve reached a point that has the same parity as X and is at or beyond X, you’ll have your answer.
Below is the implementation of the above approach:
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