Solve tiling, probability, and nested-radical puzzles

You are given several independent math/coding puzzles. ## 1) Domino tiling a 2×N board You have a 2×N rectangle and an unlimited number of 1×2 and 2×1 dominoes. - **Task:** Determine the number of distinct ways to tile the 2×N rectangle exactly. - **Input:** An integer `N`. - **Output:** The number of tilings. - **Notes:** Tiles may be rotated; a tiling is different if the set of covered cell pairs differs. ## 2) Domino tiling a 3×N board You have a 3×N rectangle and the same unlimited 1×2 / 2×1 dominoes. - **Task:** Determine the number of distinct ways to tile the 3×N rectangle exactly. - **Input:** An integer `N`. - **Output:** The number of tilings. ## 3) Infinite nested square root Define \[ x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}} \] - **Task:** Compute the value of `x`. ## 4) Combinatorics probability (couples) There are 5 couples (10 people total). You randomly select 4 people uniformly from the 10. - **Task:** Compute the probability that among the 4 selected people, there are **exactly 2 complete couples** (i.e., you selected both partners from exactly two couples).