ML Interview Q Series: Calculating Ant Collision Probability on Polygons using Combinatorics
Browse all the Probability Interview Questions here.
6. Three ants are sitting at the corners of an equilateral triangle. Each ant randomly picks a direction and starts moving along the edge of the triangle. What is the probability that none of the ants collide? Now, what if it is k ants on all k corners of an equilateral polygon?
This problem was asked by Facebook.
In an equilateral triangle with three ants at the three corners, each ant independently chooses one of two directions: clockwise or counterclockwise. Once the ants start walking around the edges, a collision occurs when two or more ants meet at a point. For the three-ant scenario on a triangle, a particularly elegant insight is that no collisions occur precisely in the cases when all ants choose the same orientation. If every ant picks clockwise, they will all traverse the triangle in one “unison circle,” never catching each other. The same is true if all pick counterclockwise.
Below is a more detailed explanation of why this happens and some potential follow-up questions with thorough clarifications.
The Core Reasoning
When each ant moves along the perimeter, they keep going until they meet another ant or continue indefinitely. Ants collide if at least one of them picks a direction that makes their paths intersect at some point on an edge (or vertex). On a regular polygon, the only way to guarantee no collisions is that they all move “in lockstep,” traversing the perimeter in the same direction without ever catching up to each other from opposite sides.
In the triangle case, visualize labeling the corners A, B, and C, and the directions as “clockwise” or “counterclockwise.” A collision would occur if one ant is going clockwise from A to B at the same time another ant is traveling counterclockwise from A to C, or if ants in adjacent corners pick directions bringing them to meet in the middle. By symmetry, it turns out that collisions always happen unless all ants rotate in one uniform direction. This argument extends by symmetry to any regular k-sided polygon.
Connection to Combinatorics
Intuitive Geometry
An alternative geometric reasoning is that if even one ant chooses a direction different from the rest, collisions must happen. On an equilateral polygon, ants traveling along the perimeter in opposing directions eventually intersect on an edge or meet at a vertex.
Python Simulation Example
import random
import math
def simulate_ants(k, num_samples=10_000_00):
no_collision_count = 0
for _ in range(num_samples):
# Randomly assign each ant to clockwise (1) or counterclockwise (-1)
directions = [random.choice([1, -1]) for _ in range(k)]
# Check if all directions are the same
if all(d == directions[0] for d in directions):
no_collision_count += 1
return no_collision_count / num_samples
k = 3
prob_estimate = simulate_ants(k)
print(f"Estimated probability for k={k} ants not colliding: {prob_estimate}")
Implementation and Edge Cases
When implementing or analyzing this problem, remember that each ant is at a distinct vertex at the start. If we instead allowed ants to be placed at arbitrary points along the edges, the analysis would become more complicated. But for corners only, the math simplifies to counting uniform direction assignments. Another subtlety involves continuous movement around the polygon. Since the polygon is regular, distances between corners are uniform, so once any pair of ants moves in opposite directions along an edge, a collision must occur at some point.
Potential Pitfalls
A common mistake is to think that collisions might be avoided by partial directional mixes. But for a regular polygon, the symmetry ensures that once directions differ, a collision is unavoidable. Another subtlety is whether an “edge intersection” is truly considered a collision. By standard definitions, if two ants share the same position on an edge at any time, that is a collision.
Below are additional follow-up questions one might face in a thorough interview scenario, with elaborate clarifications.
For the three-ant triangle case, how exactly do we see that any mixed direction leads to collisions?
If one ant goes clockwise, while a second ant in an adjacent vertex goes counterclockwise, they start moving toward each other along the same edge between their two corners. This edge is of fixed length, and each ant must cover half the distance to collide in the middle (unless they have different speeds, but typically the puzzle assumes identical speeds). They inevitably meet and collide.
Moreover, if only two ants pick the same direction and the third picks the opposite direction, the third ant eventually collides with one of the others along some edge. Because the triangle is fully connected and distances are uniform, there is no “escape route” for a single ant that goes against the flow of the other two.
What if ants have different speeds? Could the probability of collision-free movement change?
Could collisions happen exactly at a vertex if they switch edges?
Yes, and in a standard puzzle scenario, colliding at a vertex is still considered a collision. If an ant arrives at a vertex at the same moment another arrives there from the adjacent edge, that counts as a collision. This is another reason uniform direction is the only way to avoid collisions, because any mismatch in directions around a shared vertex leads at some point to an encounter.
Is there a formal proof that these two collision-free assignments are the only ones for the k-sided polygon?
Yes, one can argue by contradiction. Suppose not all ants pick the same direction, meaning at least one pair of adjacent ants picks opposite directions. Because the polygon is equilateral, these two ants on an edge with opposite directions must collide at a point strictly between the corners or possibly at the shared vertex if one is slightly delayed. Therefore, the only possibility to avoid collisions is not to have a single adjacent pair with opposite directions, which is exactly when all ants’ directions are identical. That yields precisely two outcomes: all clockwise or all counterclockwise.
If the ants continue moving indefinitely, how do we confirm collisions happen at a finite time rather than never?
Under uniform nonzero speed, two ants traveling toward each other on the same edge will meet in finite time (a fraction of the time it takes to traverse that edge). For collisions at a vertex, it could occur the moment an ant arrives, and another ant arrives at the same vertex from a neighboring edge. The finite perimeter and steady motion guarantee that if collisions are destined to happen, they do so in finite time.
Do these probabilities change if the ants pick directions in continuous angles rather than strictly along the edges?
Yes, that is a different question. The puzzle is typically restricted to discrete movement around the perimeter. If instead ants can set any angle in a continuous 360° range, collisions become far more probable and the combinatorial structure of “2 directions per corner” no longer applies.
Could the puzzle mean something different, such as each ant picks a random direction in continuous space from its vertex?
If that were the case, you would have to interpret collisions in a continuous plane, not constrained to edges. That version is significantly more complicated. Usually, the puzzle is strictly about movement along the polygon edges.
If the question is truly about random directions in the plane from each corner of a polygon, then each ant can move along an infinite ray from its starting vertex, and one must analyze whether rays intersect. The standard puzzle is simpler, with each ant constrained to walk on the perimeter.
Would it matter if the ants are allowed to pass through each other without collision?
Yes, some interpretations say ants do not collide unless they specifically occupy the same point at the same time. A stricter puzzle says they collide if they ever share the same location. Another hypothetical variation might let them magically pass through each other with no collision. That would obviously change the puzzle’s question. In the standard puzzle, any time two ants meet at a point, they collide.
If we let the ants keep walking, do we have repeated collisions or just one?
If directions are not unanimous, they can have multiple collisions. But typically, the puzzle asks only about the existence of at least one collision event. Once a single collision is guaranteed, we consider that the event “ants collide.”
In conclusion, how do we summarize the probability for the triangle and the k-gon?
Below are additional follow-up questions
If the regular polygon is very large, does the size or scale of the shape change the collision probability?
To see why size is irrelevant in more detail, notice that collision events occur if and only if ants occupy the same point at the same time. Whether an edge is 1 meter or 100 kilometers long, two ants traveling toward each other from opposite corners along that edge will eventually meet if they have the same speed. The time to collision might be longer for larger edges, but a collision time that is merely delayed is still a collision. All the logic that leads to collisions or avoids them holds true at any scale. Thus, the result is fundamentally determined by the uniform or mixed direction choices, not by the physical magnitude of distances.
What if the ants choose their directions with probabilities different from 1/2?
The classical puzzle assumes that each ant chooses clockwise or counterclockwise with probability 1/2. If, however, the choice is biased—for instance, each ant chooses clockwise with probability p and counterclockwise with probability 1−p —then the probability of no collision changes slightly, but the collision-free assignments remain the same two (everyone going clockwise or everyone going counterclockwise).
A subtle pitfall here is to assume that the unbiased puzzle logic extends to biased scenarios without adaptation. One must specifically account for the fact that each ant now has a different chance to pick each direction. As long as the directions are chosen independently, though, the only collision-free outcomes remain the same. Hence, the difference is purely in how likely those collision-free outcomes become.
How does the analysis change if some edges are blocked or removed from the polygon?
Suppose we have a k-sided polygon, but some edges are removed or blocked, so the ants cannot traverse all edges. The fundamental collision condition changes because now certain paths do not exist. The puzzle’s original combinatorial analysis (all ants choose directions on a fully connected, cycle-like perimeter) no longer directly applies. We would have to inspect the actual graph structure formed by the remaining edges to see if collisions can still occur.
If, for instance, we remove one edge of the triangle, it becomes a path of two edges rather than a cycle of three edges. In that scenario, collisions might be more or less likely depending on how we define “directions” for each ant and how we allow them to move across corners. We do not have an immediate “all-or-none” condition like before. One would have to count how many ways ants can pick routes along the reduced graph and check in which cases collisions occur.
If the ants can reverse direction mid-walk, how does that affect collisions?
In the classical puzzle, each ant picks one direction at the start and continuously follows that edge path. If ants are allowed to reverse direction at arbitrary times, collisions might become even more likely. Indeed, if one ant flips direction randomly or reacts to the movements of others, that can lead to new intersection points that would not have occurred under fixed directions.
One might try to analyze this extended puzzle under certain assumptions:
Random direction flips: The ant might spontaneously decide to reverse with some probability per unit time. Over a long enough period, ants might wander back and forth, raising the chance that they eventually meet each other.
Strategic direction flips: If ants have some “avoid collision” strategy or some “seek collision” strategy, the probability can change drastically.
How does this extend if we have an n-sided shape that is not regular?
The puzzle’s essential condition is that collisions occur if two ants travel the same edge segment in opposite directions or possibly meet at a shared corner. Regularity of the polygon ensures a uniform distance between adjacent corners and symmetry of directions. If the polygon is not regular, the edges can have different lengths, angles can differ, and so on.
Even so, collisions remain unavoidable whenever ants pick directions that make them converge on the same edge. The only way to ensure no collisions is for all ants to pick directions that circumvent meeting each other. If we still only allow “clockwise or counterclockwise” around the perimeter, we must be consistent about how we define “clockwise” and “counterclockwise” in an irregular shape. Typically, one might define a perimeter path that traces the polygon’s boundary in a single loop and allow each ant to follow that loop in one direction or the other.
A subtlety arises if the “clockwise/counterclockwise” notion is ambiguous in a highly irregular shape with complicated interior angles. Usually, we define “clockwise” as traveling the perimeter in a fixed consistent orientation. If that orientation is well-defined for each corner, we can replicate the same combinatorial reasoning. Potential confusion could arise if, for instance, each ant has a local notion of clockwise, leading to partial mismatches. That complication typically is outside the puzzle’s standard scope.
Can collisions be avoided if ants start moving at different times?
In the standard setup, all ants start simultaneously. If we allow for staggered starts—where each ant might wait some random time before beginning to walk—this might create scenarios where an ant can pass a vertex before another ant even starts. However, it does not necessarily eliminate collisions unless all ants are extremely well-timed and oriented.
Specifically, if at any point two ants are traveling along the same edge in opposite directions, they will eventually meet. Delaying the start of one ant can sometimes cause it to “miss” the other ant if the latter has already moved on. But we must account for the fact that an ant traveling along a cycle can eventually come back to the same edge multiple times (if it continues indefinitely around the polygon).
A corner case is if one ant never starts (stays put forever). If it never moves, do we count that as a “collision possibility”? Typically, if it remains forever on its starting corner, any other ant passing that corner at the same time would create a collision. The detailed probability analysis might need to incorporate probabilities of start times. But for the standard puzzle, all ants do eventually move, so the direction mismatch is still the crucial factor.
How might real-world factors such as ants walking at slightly different speeds or taking curved paths affect collisions?
Real ants in nature do not perfectly travel along a straight edge. They might meander or have varying speeds. The theoretical puzzle idealizes ants as point-like entities with constant speed and perfect travel along polygon edges. In reality, each discrepancy from that ideal can slightly change the collision dynamics. A faster ant might catch a slower ant from behind, though the puzzle typically focuses on meeting face-to-face. Or an ant might wobble off the polygon’s boundary.
Despite these complexities, the fundamental reason collisions occur—two ants occupying the same point at the same time—remains. Any difference in speed or path might shift when and where collisions happen but does not necessarily prevent collisions if directions are mixed.
A subtle real-world pitfall is that collisions could be avoided if ants physically step around each other upon meeting, which actual ants might do. The puzzle calls any simultaneous occupation of the same point a collision, but in nature, ants might just dodge each other. The puzzle’s answer is purely theoretical. Interviewers sometimes expect the realization that real ants might behave differently, but the puzzle’s logic remains a valuable mental exercise in combinatorial probability and geometry.
What if collisions are only counted when ants approach each other head-on, ignoring “overtaking” collisions?
Sometimes, puzzles define a collision strictly as meeting head-on. If collisions are only head-on, then an ant catching up from behind might not count. This changes the analysis slightly if it is possible for an ant to start on an adjacent corner, pick the same direction, and eventually catch up to the ant in front due to different speeds. In the standard puzzle with identical speeds, overtaking cannot happen if both ants choose the same direction. They remain the same distance apart the entire time.
Does the puzzle change if the ants are placed on the midpoints of edges instead of the corners?
Yes. If we place one ant at the midpoint of each edge of a k-sided polygon, then each ant still must choose to move in one of two directions: toward one of the adjacent corners or the other. But now the collisions can happen in ways not considered in the corner-based puzzle. For instance, if an ant moves to the left midpoint while its neighbor also moves to the left from an adjacent midpoint, they might meet at the shared corner. Or two ants on adjacent edges might end up traveling the same edge from opposite corners.
Essentially, we no longer have a single ant per corner. Instead, we have to consider how k midpoints connect to each other via corners. The collision conditions become more intricate. We would need to do a separate combinatorial enumeration: each ant picks one of two corners to walk toward first. After reaching that corner, it continues around the polygon with the chosen orientation. We must then check all possible ways they could meet.
Could a collision occur among three or more ants simultaneously, and does it matter for the probability?
In theory, especially with identical speeds on a symmetric shape, three or more ants can converge at a point at the same time, typically at a shared vertex if they time their arrivals perfectly. But from the puzzle’s perspective, it does not matter whether collisions occur in pairs or in triplets. Any collision is enough to classify the outcome as “ants collided.” The probability question focuses on whether there is at least one collision event at any time.
In a real interview setting, the question of whether multi-ant collisions can happen might be used to check whether you understand the possibility of simultaneous events in uniform speed scenarios. While interesting, it does not alter the fundamental probability that eventually at least one collision happens unless all ants are rotating uniformly in the same direction.
Do boundary conditions change if the ants keep walking for just one lap versus infinitely many laps?
A small difference might arise if speeds are different and one ant can “finish its lap” and stop before encountering an oncoming ant. However, the standard puzzle assumes uniform speed or indefinite motion, so typically that nuance does not come into play. If uniform speed is assumed and the ants are allowed exactly one lap, collisions due to direction mismatches occur on the first cycle anyway. Thus, the boundary condition of “one lap” or “infinite laps” does not change the final collision-free probability for uniform speeds.
How does the outcome change if each ant has more than two possible choices (e.g., it can choose to skip corners or move diagonally in a polygon that has chords)?
For instance, in a square (k=4) with diagonals included as possible routes, an ant at one corner could have up to three directions to choose from: (1) the edge on one side, (2) the edge on the other side, (3) the diagonal. Another corner’s ant similarly picks among three. We must consider all combinations of these new route choices for the total number of possible assignments.
Collisions become even more numerous because two ants might intersect in the interior of the polygon if they choose intersecting chords or edges. The only guaranteed collision-free scenarios might be ones where each ant picks a path that does not intersect with the others in space or time, which is much harder to enumerate. Often, there might be more than two “safe” assignments or sometimes none at all, depending on the shape and chord arrangement. Consequently, the puzzle’s classical combinatorial formula does not hold. A thorough approach would be:
List all possible routes each ant might pick.
Check for every pair or group of ants whether they meet based on speeds and geometry.
Count how many route assignments produce no collisions.
If we interpret a collision as physically impossible (ants stepping aside each other), could the puzzle describe a topological phenomenon?
In some reinterpretations, especially in puzzle-like or advanced math contexts, collisions might be reimagined as points of intersection in paths on a planar graph. If collisions are physically impossible—ants pass around each other—some might see this as a topological puzzle about braids or permutations on a circle. Specifically, each ant’s path might define a line in a “space-time” diagram, and collisions occur if lines intersect.
One might then interpret the puzzle as counting certain permutations or certain braids that are “non-intersecting.” This leads to a deeper combinatorial or topological viewpoint: any mismatch in orientation will create at least one intersection of lines in the space-time diagram. The only non-intersecting braids come from identical orientation for all ants. However, these advanced interpretations might be tangential to a typical FANG interview question, which focuses on the simpler combinatorial probability approach.
Still, the interviewer could test your creativity and see if you know more abstract interpretations. The subtlety or pitfall is to keep the puzzle’s simple definition of “collision if two ants occupy the same point at the same time” separate from fancy topological frameworks, unless you are explicitly invited to discuss them.
