# Problem

Discussion in 'Mathematics' started by AsstHead, Jul 11, 2012.

From an English Dept. colleague based on a novel that they are studying with a cohort of students.
"A circular dry lake has radius 5 miles. A group of boys are digging for burried treasure and each circular hole has radius 5 ft. What is the smallest number of holes that the boys can dig to be sure of finding the treasure?"
Any ideas?

2. ### pipipiNew commenter

Holes?
Well the answer is 1. Where the X marks the spot. Otherwise...

What is the area of a circle?

Change all the units so that they are all the same.
Work out the area of the lake.
Work out the area of the hole.

3. ### emilyisobel

It depends on the size of the treasure and it is also a circle packing problem - the holes would have to overlap to be certain as circles don't pack efficiently.

4. ### pipipiNew commenter

I think the answer should be a whole number.

6. ### DMNew commenter

Answer: don't dig circular holes. Circles don't tessellate.

7. ### Stato

Are you sure you're not really researching questions for the UK maths challenge? Was the book "Holes" by Louis Sachar by any chance? That's quite a hard problem and I'm sure the boys in the story would have probably lost all hope if they'd realised how long it might take if they were really unlucky! I love that book but had never thought to ask that question....I'm on it

9. ### Woostarite

I would guess that you'd need to find n for which r(n) = 5ft/5miles = 0.0001894 by the definition set out in the 'disk covering problem': http://en.wikipedia.org/wiki/Disk_covering_problem
It's the opposite of what is usually done: the number of circles being fixed and the radius of the smaller circles the constant to be determined.

10. ### McCaheyNew commenter

use a metal detector!

11. ### fieldextensionNew commenter

I am not interested enough to work on the problem, but you might begin by finding an upper and lower bound for the answer.
A lower bound for the number of holes needed could be found by supposing the holes dug are square and of side 10 feet (or more precisely, have a square cross section).
An upper bound for the number of holes needed could be found by supposing
the holes dug are square and of side 5root2 feet (i.e. the largest square that would fit completely inside the holes actually dug).
Since squares tessellate, they will be easier to work with.

12. ### valed

I know a couple of squares and I can assure you that neither Bob nor Tom know how to tessellate