Maths Problem

Post Reply New Topic

This is the beginning of a weird maths puzzle story which I found in a notebook under my bed.

You are waiting for the train when you hear your mobile phone ringing. You look in your bag. You can't find your phone because your sandwiches have spread themselves everywhere. You remove the solid bits and look again, but you are distracted by finding another bag inside yours. It has several round glass beads of various diameters. You are not sure what to do with them. You don't even remember where you got the bag from. The label says, "85 fine round glass beads. Diameter unknown. 1.5cm long holes. Excellent quality. May be cursed."

Your beads look like this



David (you know him, don?t you?) also has a bag. He won?t let you touch it. He remembers that he was given them by a mysterious person who he is unable to describe. His label says, "175 excellent round glass beads. Diameter not known. 1cm long holes. Fine quality. Possibly cursed."

His beads look like this


You and David are discussing how to dispose of the beads when you are interrupted by a weird dwarf. He offers to buy the beads, and says he will pay by weight.

Assuming that the label on your bag is correct (you didn't count and measure them, unlike David), who will get the most money?

By the way, you missed the train. And you should answer your phone.

That?s ? funny I wonder how you solved it. The very fact that it?presumably?can be solved might be a good clue to get the right answer quickly in your head bypassing the calculations needed to actually work out each bead?s volume. I only realized this after doing it, though.

It seems like you?re doing justice to your nickname

EDIT ? Well, I?m assuming the density of David?s beads is the same as that of yours. Who knows if the difference between ?fine? and ?excellent? quality, or curses, mean a difference in density!

The glass is all the same density. I'm rewriting the story to make things like that clearer, and then I'm going to give it to my friends and watch them struggle.

Some useful formulae (only one is actually needed, but the others are useful anyway)

Volume of a sphere (radius R):

Volume of a cylinder (radius r, height h):

Volume of a spherical cap (height h) in a sphere of radius a:

Edit: Forgot to say what R, r and h mean.

^^^ You might want to define what r, R, and h are.

Thanks for pointing that out. I should probably get more sleep before posting anything else.

How does the possible curse fit into the equation?

I think it?s actually less cumbersome to calculate a bead?s volume as ?

http://pastebin.com/EDKkhZ6H (encrypted so it doesn?t spoil the problem).

Who ever has the most "tonnage" of beads gets the most money.

The problem is that you have a smaller number of heavier beads, but David has a bigger number of lighter beads.

Glass is glass (we can assume). So the density of the beads can be assumed to be the same. The curse is probably a red herring.

Thats the set up of the problem. Pretty simple.

But doing the calculations to answer it is quite complicated. You need to calculate the volume of the spheres, minus the volumes of the cylinders cut out of them, and the volume of the caps taken off the tops.

Just looking at the diagrams it looks like each of your beads has more than twice the volume of glass that each of David's have. So even though he has twice as many beads you would get the most money. But thats just a guesstimate.