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leetcode/problem/alice-and-bob-playing-flower-game.html
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leetcode/problem/alice-and-bob-playing-flower-game.html
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<p>Alice and Bob are playing a turn-based game on a circular field surrounded by flowers. The circle represents the field, and there are <code>x</code> flowers in the clockwise direction between Alice and Bob, and <code>y</code> flowers in the anti-clockwise direction between them.</p>
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<p>The game proceeds as follows:</p>
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<ol>
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<li>Alice takes the first turn.</li>
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<li>In each turn, a player must choose either the clockwise or anti-clockwise direction and pick one flower from that side.</li>
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<li>At the end of the turn, if there are no flowers left at all, the <strong>current</strong> player captures their opponent and wins the game.</li>
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</ol>
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<p>Given two integers, <code>n</code> and <code>m</code>, the task is to compute the number of possible pairs <code>(x, y)</code> that satisfy the conditions:</p>
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<ul>
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<li>Alice must win the game according to the described rules.</li>
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<li>The number of flowers <code>x</code> in the clockwise direction must be in the range <code>[1,n]</code>.</li>
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<li>The number of flowers <code>y</code> in the anti-clockwise direction must be in the range <code>[1,m]</code>.</li>
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</ul>
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<p>Return <em>the number of possible pairs</em> <code>(x, y)</code> <em>that satisfy the conditions mentioned in the statement</em>.</p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<pre>
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<strong>Input:</strong> n = 3, m = 2
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<strong>Output:</strong> 3
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<strong>Explanation:</strong> The following pairs satisfy conditions described in the statement: (1,2), (3,2), (2,1).
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</pre>
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<p><strong class="example">Example 2:</strong></p>
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<pre>
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<strong>Input:</strong> n = 1, m = 1
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<strong>Output:</strong> 0
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<strong>Explanation:</strong> No pairs satisfy the conditions described in the statement.
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</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>1 <= n, m <= 10<sup>5</sup></code></li>
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</ul>
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<p>You are given an array of <strong>positive</strong> integers <code>nums</code>.</p>
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<p>You need to select a <span data-keyword="subset">subset</span> of <code>nums</code> which satisfies the following condition:</p>
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<ul>
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<li>You can place the selected elements in a <strong>0-indexed</strong> array such that it follows the pattern: <code>[x, x<sup>2</sup>, x<sup>4</sup>, ..., x<sup>k/2</sup>, x<sup>k</sup>, x<sup>k/2</sup>, ..., x<sup>4</sup>, x<sup>2</sup>, x]</code> (<strong>Note</strong> that <code>k</code> can be be any <strong>non-negative</strong> power of <code>2</code>). For example, <code>[2, 4, 16, 4, 2]</code> and <code>[3, 9, 3]</code> follow the pattern while <code>[2, 4, 8, 4, 2]</code> does not.</li>
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</ul>
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<p>Return <em>the <strong>maximum</strong> number of elements in a subset that satisfies these conditions.</em></p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<pre>
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<strong>Input:</strong> nums = [5,4,1,2,2]
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<strong>Output:</strong> 3
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<strong>Explanation:</strong> We can select the subset {4,2,2}, which can be placed in the array as [2,4,2] which follows the pattern and 2<sup>2</sup> == 4. Hence the answer is 3.
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</pre>
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<p><strong class="example">Example 2:</strong></p>
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<pre>
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<strong>Input:</strong> nums = [1,3,2,4]
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<strong>Output:</strong> 1
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<strong>Explanation:</strong> We can select the subset {1}, which can be placed in the array as [1] which follows the pattern. Hence the answer is 1. Note that we could have also selected the subsets {2}, {4}, or {3}, there may be multiple subsets which provide the same answer.
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</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>2 <= nums.length <= 10<sup>5</sup></code></li>
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<li><code>1 <= nums[i] <= 10<sup>9</sup></code></li>
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</ul>
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<p>You are given a <strong>0-indexed</strong> integer array <code>nums</code> and an integer <code>k</code>.</p>
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<p>In one operation, you can pick any index <code>i</code> of <code>nums</code> such that <code>0 <= i < nums.length - 1</code> and replace <code>nums[i]</code> and <code>nums[i + 1]</code> with a single occurrence of <code>nums[i] & nums[i + 1]</code>, where <code>&</code> represents the bitwise <code>AND</code> operator.</p>
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<p>Return <em>the <strong>minimum</strong> possible value of the bitwise </em><code>OR</code><em> of the remaining elements of</em> <code>nums</code> <em>after applying <strong>at most</strong></em> <code>k</code> <em>operations</em>.</p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<pre>
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<strong>Input:</strong> nums = [3,5,3,2,7], k = 2
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<strong>Output:</strong> 3
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<strong>Explanation:</strong> Let's do the following operations:
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1. Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [1,3,2,7].
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2. Replace nums[2] and nums[3] with (nums[2] & nums[3]) so that nums becomes equal to [1,3,2].
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The bitwise-or of the final array is 3.
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It can be shown that 3 is the minimum possible value of the bitwise OR of the remaining elements of nums after applying at most k operations.</pre>
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<p><strong class="example">Example 2:</strong></p>
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<pre>
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<strong>Input:</strong> nums = [7,3,15,14,2,8], k = 4
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<strong>Output:</strong> 2
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<strong>Explanation:</strong> Let's do the following operations:
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1. Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [3,15,14,2,8].
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2. Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [3,14,2,8].
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3. Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [2,2,8].
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4. Replace nums[1] and nums[2] with (nums[1] & nums[2]) so that nums becomes equal to [2,0].
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The bitwise-or of the final array is 2.
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It can be shown that 2 is the minimum possible value of the bitwise OR of the remaining elements of nums after applying at most k operations.
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</pre>
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<p><strong class="example">Example 3:</strong></p>
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<pre>
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<strong>Input:</strong> nums = [10,7,10,3,9,14,9,4], k = 1
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<strong>Output:</strong> 15
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<strong>Explanation:</strong> Without applying any operations, the bitwise-or of nums is 15.
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It can be shown that 15 is the minimum possible value of the bitwise OR of the remaining elements of nums after applying at most k operations.
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</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>1 <= nums.length <= 10<sup>5</sup></code></li>
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<li><code>0 <= nums[i] < 2<sup>30</sup></code></li>
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<li><code>0 <= k < nums.length</code></li>
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</ul>
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leetcode/problem/number-of-changing-keys.html
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leetcode/problem/number-of-changing-keys.html
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<p>You are given a <strong>0-indexed </strong>string <code>s</code> typed by a user. Changing a key is defined as using a key different from the last used key. For example, <code>s = "ab"</code> has a change of a key while <code>s = "bBBb"</code> does not have any.</p>
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<p>Return <em>the number of times the user had to change the key. </em></p>
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<p><strong>Note: </strong>Modifiers like <code>shift</code> or <code>caps lock</code> won't be counted in changing the key that is if a user typed the letter <code>'a'</code> and then the letter <code>'A'</code> then it will not be considered as a changing of key.</p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<pre>
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<strong>Input:</strong> s = "aAbBcC"
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<strong>Output:</strong> 2
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<strong>Explanation:</strong>
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From s[0] = 'a' to s[1] = 'A', there is no change of key as caps lock or shift is not counted.
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From s[1] = 'A' to s[2] = 'b', there is a change of key.
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From s[2] = 'b' to s[3] = 'B', there is no change of key as caps lock or shift is not counted.
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From s[3] = 'B' to s[4] = 'c', there is a change of key.
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From s[4] = 'c' to s[5] = 'C', there is no change of key as caps lock or shift is not counted.
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</pre>
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<p><strong class="example">Example 2:</strong></p>
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<pre>
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<strong>Input:</strong> s = "AaAaAaaA"
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<strong>Output:</strong> 0
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<strong>Explanation:</strong> There is no change of key since only the letters 'a' and 'A' are<!-- notionvc: 8849fe75-f31e-41dc-a2e0-b7d33d8427d2 --> pressed which does not require change of key.
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</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>1 <= s.length <= 100</code></li>
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<li><code>s</code> consists of only upper case and lower case English letters.</li>
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</ul>
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