19 replies to this topic

[bonanova]

A while ago we asked for a shape that fits snugly through three different holes: triangle, square and circle.

Actually, the answer is not unique. So now we ask:

What is the volume of the smallest shape that fits snugly through the holes?
The largest?

[xamdam]

Spoiler for Hmmm

Since you say "volume" and not "area" I get 0 for smallest and no limit for largest.

[bonanova]

Spoiler for Hmmm

Since you say "volume" and not "area" I get 0 for smallest and no limit for largest.

Spoiler for Consider

Zero volume [a point] will not fit any of the holes snugly.
A cube 2 inches on a side has finite volume [8 in³] but is too large to fit through the circular or triangular holes.

[plainglazed]

Have to assume you're starting with the same 2" x 2" x 2" cube and by snuggly you mean the shape that, at a maximum, fits thru the hole in two dimensions while being the smallest volume? Dont know if that made any sense but it's late and I've just returned home after a night out.

Spoiler for it seems

the three dimensional shape described in your previous post would have a volume of pi in² as would the two inch long volume of cross section equal to the incircle of the described triangle both of which may be the smallest depending on how you define "snuggly". A two inch long volume with an equilateral triangular cross section of side length sqrt(3) would have a volume of 6 sq. inches which would be my guess at the greatest. Still seems too many variables have been left open by the OP. Maybe bn has also just returned home after a night out? More likely I'm missing something...

[bonanova]

Have to assume you're starting with the same 2" x 2" x 2" cube and by snuggly you mean the shape that, at a maximum, fits thru the hole in two dimensions while being the smallest volume? Dont know if that made any sense but it's late and I've just returned home after a night out.

Spoiler for it seems

the three dimensional shape described in your previous post would have a volume of pi in² as would the two inch long volume of cross section equal to the incircle of the described triangle both of which may be the smallest depending on how you define "snuggly". A two inch long volume with an equilateral triangular cross section of side length sqrt(3) would have a volume of 6 sq. inches which would be my guess at the greatest. Still seems too many variables have been left open by the OP. Maybe bn has also just returned home after a night out? More likely I'm missing something...

Snugly means completely fills each of the hole cross sections at some moment as it passes through.
Of all the 3-dimensional shapes that do this, one has a maximum and another has a minimum volume.
The question is what are these volumes?

Been watching TV all evening, so you have the advantage on me.

[DeeGee]

A while ago we asked for a shape that fits snugly through three different holes: triangle, square and circle.

Actually, the answer is not unique. So now we ask:

What is the volume of the smallest shape that fits snugly through the holes?
The largest?

Spoiler for may be

If I understood it correctly, for the largest shapes, you could choose rods with the same cross-sectional area as the holes to be filled. So, for ex, for a triangular hole, you could have a rod with a triangular cross-section. The volume would then be the area of the cross-section multiplied by the depth of the hole.
For smallest shapes, you could have very thin rods or even thin cardboard or paper fitting snugly into the holes. The volumes would be very small.

[DeeGee]

Oops, I just realised it was rather easy... so I decided to make sure what i understood was correct and that led me to the link to the previous question. So, now that I have realised that what I had previously understood was way off, pls ignore my previous post.

Although, for the shape with the smallest volume, I would still stick to the previous answer! Although the material would be wood and the shape could still be made by chopping and chiseling it to a very small breadth while keeping the length as 2".

Edited by DeeGee, 07 September 2009 - 08:46 AM.

[DeeGee]

Spoiler for Post enlightenment - max volume

One option for the shape could be as below. Having triangular and circular shaped protrusions on either side of the cube. Naturally, the length of the cube is reduced by the amount of length of protrusions on either side.



In order to have maximum volume, the length of the square cross-section should be max and length of circle and triangle cross-sections should be small. That is the length of protrusions on either side must be smallest.

Let x be the length of circular cross-section part then volume here = pi.1².x = pi.x
Let y be the length of triangular cross-section then volume here = root(3)/4.4.y = root(3).y
Then volume of square crossection = 4.(2 - x - y)

Then volume of total shape = 8 - 2.27y - 0.86x

Now we need x and y to be smallest possible such that they can also fit snugly and restrict movement of the shape when inverted.
Lets say the minimum length of x and y must be 1/4", then volume of the shape is 7.2 cubic inches.
If the minimum length of x and y must be 1/8", then the volume would be 7.6 cubic inches.

So, the volume of the shape depends on how long (minimum) must be the protrusions on either side of the reduced cube.

[bonanova]

The shape[s] must pass completely through each of the holes.
Your cube will not pass through either the circular or triangular holes.

Consider them to be hole[s] in a wall separating two rooms.
The shape must pass from one room to the other, in turn, through
all three holes, fitting each hole snugly as it passes.

There are actually an infinity of 3-dimensional solids that will do this.
One has a largest and another has a smallest volume.

What are these volumes?

Refer to the figure in the referenced post for dimensions of the holes.
The circle has a 2" diameter; the square and triangle have 2" sides.
Since this puzzle asks a new question, it's fair game to look at the answers given there.

[plainglazed]

Spoiler for considering

the OP's above mentioned walls to be infinitesimally thin and the cut outs made with an infinitesimally thin blade, reserve the cut outs and glue the square perpendicular to the circle, cut the triangle in half with the same blade and glue it perpendicular to both, you have a volume that approaches zero as the smallest possible volume (8pi x thickness of the shapes in³). The largest volume is as described in the referenced post and is pi in³.

Edit: inches squared/inches cubed, what really is the difference?

Edited by plainglazed, 08 September 2009 - 10:43 AM.