# Sprout’s game plays a significant role in Jordan’s curve.

Post-graduation, the professor used to teach the proof of the Jordan curve theorem, which is related to topology and complex analysis. And the pupils used to copy-paste from the green board and practice five times, during which they would vomit.

Neither the professor had the time to explain nor did we have the time to think about how to prove the theorem and how it works, as we had around 1,000 theorems in mathematics. As a result, we never received a clear picture of this.We gained insight into how to convey this theorem when we became mathematics educators. As a result, the sprout game is a very effective strategy for teaching the Jordan curve.

They are like sisters.Sprouts is a two-player pencil and paper game that is both easy and delectably subtle. The game is initiated by marking a series of locations on a page. Each player begins by drawing a curve that connects two locations or loops around a point, without crossing any previously drawn lines, and then marking a new spot on the curve.

A site may have a maximum of three lines connected to it, and any spot with three lines is deemed dead, as it no longer plays a function in the game. Eventually, no other moves are permitted, and the player who draws the final line wins the game.

Despite the straightforward rules, the game’s analysis provides several difficulties, and no general winning strategy is known. It is quite straightforward to demonstrate that if there are n starting positions, the game must have at least 2n moves and must end in no more than 3n–1. Thus, with eight starting positions, there will be between sixteen and twenty-three moves.

Sprouts’ mathematics, which we can refer to as sproutology, is based on topology, a branch of geometry that emphasizes continuity and connectivity but ignores distances and shapes. Because a figure created on an elastic sheet retains its topological qualities when stretched but not ripped, topology is frequently referred to as rubber-sheet geometry.

Sprouts are topological in nature, as the actual locations of the spots are irrelevant; what matters is the pattern of connections between them. The game makes use of theJordan curve theorem, according to which simple closed curves divide the plane into two parts.This seemingly self-evident conclusion is really rather difficult to establish. The Sprouts one-spot game is trivial: the first player must connect the spot to itself and draw another; the second player then connects the two spots, winning the game.Games with a limited number of beginning positions have been thoroughly researched, and a pattern emerges: if the remaining after dividing n by 6 is 3, 4, or 5, the first player can force a win (assuming faultless play); otherwise, the second player has a winning strategy. This ‘Sprouts hypothesis’ has not been established.

Sprouts can be checked manually for up to seven spots to begin, but as the number of spots increases, the process quickly becomes too complex, necessitating computer analysis.

Conclusion:Thus, if each topic in mathematics is taught through games, the notions become incredibly instinctual and easy to grasp.