Can You Solve Ants On A Cube Puzzle?

Can you solve Aunts puzzle.

Question. If Eight ants start at different corners of a cube. Suddenly each ant moves to an adjacent corner at random. That is, each ant walks along one of its three adjacent edges with equal chance.

Then What is the probability that none of the ants collide?

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Answer To Ants On A Cube Puzzle

Each of the 8 ants can move in 3 directions, so the total number of possible paths the ants could take is the product:

3 × 3 × … 3 = 38 = 6,561

In how many ways will the ants not collide?

The ants will collide if some pair of ants swap corners. To avoid collisions, we need the ants to be coordinated so they are all moving in the same direction. If we trace the movement of the ants, we should be able to form a closed loop.

There are two possible ways we can form closed loops on a cube. We can have two closed loops of opposite faces. Or we can have a single closed loop that spans all eight corners (also known as a Hamiltonian cycle). Let us count the number of closed loops in each case.

Case 1: opposite faces

Take two opposite faces of the cube, say left and right. On a particular face, the four ants can form a loop in two ways: they can all move either clockwise or counter-clockwise. The ants on the left face can make a closed loop in 2 ways, as can the ants on the right face. Thus, there are 2 × 2 = 4 ways the ants can move without colliding.

Furthermore, there are 3 possible pairs of opposite faces: left and right, top and bottom, and front and back. For each pair of opposite faces, there are 4 possible ways the ants can move.

Thus there are 3 × 4 = 12 possible ways the ants can move without colliding in this case.

Case 2: a closed loop of the entire cube

A path that goes through each node of a graph exactly once is called aHamiltonian cycle.

How many distinct directed Hamiltonian cycles are there on a cube? There are exactly 12 such paths.

Here is an illustration of the 6undirected Hamiltonian cycles.

(this is a “flattened” cubical graph–the small square = front face, large square = back face)

For each loop, the ants have two choices of moving in one direction or instead moving in the opposite direction. Thus, there are 6 × 2 = 12 possible directed loops.

Calculating the probability the ants do not collide

In total there are 12 ways (case 1) and 12 ways (case 2), making for a total of 24 ways the ants will not collide.

This means a 24/6561 = 8/2187 chance the ants do not intersect, or about a 0.37% chance the ants do not intersect.

Source-thanks to presh talwalker

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