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# Hard maths problem watch

“There are several circular cylindrical non-touching caps and two small bread crumbs on the table. The crumbs are at a distance d from each other. An ant is planning to travel from one bread crumb to the other along the table avoiding the caps. Explain why it can always make his journey such that the length of the trip is less than 1.6d.”

I know that the answer is because the greatest possible distance would be if there was one large cap (with a diameter of d) separating the 2 crumbs, in which case the ant would have to travel half the circumference of the cap i.e. pi*d/2 which equals 1.57 which is less than 1.6. However, I need help in theoretically explaining (using algebra) why this is the maximum possible distance and why several smaller caps would result in a smaller distance travelled.
2. (Original post by Billy_\1100)

“There are several circular cylindrical non-touching caps and two small bread crumbs on the table. The crumbs are at a distance d from each other. An ant is planning to travel from one bread crumb to the other along the table avoiding the caps. Explain why it can always make his journey such that the length of the trip is less than 1.6d.”

I know that the answer is because the greatest possible distance would be if there was one large cap (with a diameter of d) separating the 2 crumbs, in which case the ant would have to travel half the circumference of the cap i.e. pi*d/2 which equals 1.57 which is less than 1.6. However, I need help in theoretically explaining (using algebra) why this is the maximum possible distance and why several smaller caps would result in a smaller distance travelled.
No matter how many caps you between the two crumbs, the ant will always have to walk less than or equal to half the circumference along each cap. The sum of the diameters of the caps have to be less than d (else they wouldn't fit between the two crumbs)

Edit: this assumes the centers of the caps lie on the straight line between the two crumbs, but the argument still works if you extend it to sum of diameters to arc-length of relevant sectors. (I think, at least - haven't thought too hard)

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