Ants on a triangle Surely the explanation doesn't need all those permutations to confuse us! How about this: Every ant has two choices of direction to walk in - call them left arm or right arm. The only way for them not to collide is if they all choose left or all choose right. The probablility that they all go left is 1/2 * 1/2 *1/2 = 1/8. The probability that they all go right is the same, so the total is 1/8 + 1/8 = 1/4. Max Monday, July 5, 2004 Great simplification of the original solution. I think I can simplify it even more: the ants will avoid each other if and only if ants 2 and 3 go the same direction (clockwise or counter-clockwise) as ant 1. Doesn't matter which direction ant 1 chooses. The chance of each of the two other ants going the same direction as 1 is 1/2. 1/2 * 1/2 = 1/4. Brad Corbin Monday, July 5, 2004 No dudes, i think ya all are wrong !! The probability that ants WILL collide is inversely proportional to length of triangle. Therefore prob of the not colliding is 1-1/L ashish Monday, July 12, 2004 Every Ant can move in 2 possible directions , backward and forward ( B or F ) , so the whole sample space consists of 8 choices, out of which 2 , namely BBB and FFF are no collision choices. so probability of not colliding is 2 /8 = 1/4. F F B F F F F B B F B F B F B B F F B B B B B F Rajeev Jha Tuesday, August 10, 2004   Fog Creek Home