Largest cube passing through a unit square hole?

Question

What is the largest cube that can fit through a 1 foot by 1 foot square hole in a wall? Contrary to intuition, the cube is bigger than a 1 by 1 by 1 cube.
If you know the answer, please also briefly tell me how you calculated it.

Answer 1

Sorry, not possible. Intuition wins this one, it's 1' x 1' x 1'.

Answer 2

Since it's only a unit square hole, you only need to treat the problem as 2-D. The largest square that can fit through the hole has a diagonal that is equal to the diameter of the hole. If the area of the hole is 1 square foot, the diameter of the hole is 1.128 ft, which translate into 0.798 ft for the side of the square.

Yet, I think you meant the hole has 1 foot radius, or 2 foot diameter, and the sides of the square is 1.414 ft. A cube with sides of 1.414 ft is much bigger than a cube of 1 ft sides.


XR

Answer 3

the most confortable way to think of this is this one: you have to imagine yourself passing the cube through the hole so that the face that would be perpendicular to the floor in the 1x1x1 case now forms an angle of 45 degrees with the floor. Now the hypotenuse of a face will be equal to the square's length. That is x^2+x^2 =1 hence x = sqrt2/2 .
the dimensions of the cube are sqrt2/2 x sqrt2/2 x sqrt2/2
for and only for an angle of 45 deg.

Answer 4

If you take a cube and look along the body diagonal (straight through
from one corner to the opposite corner), you see the profile of a
hexagon. You can put a square in this hexagon whose side length is
greater than the original side length of the cube. This means that
you can pass a larger cube completely through a smaller one.

To verify this, you can draw a regular hexagon and try to maximize the
area of a square that is inscribed within it.

For directions on constructing a demonstration of this very
interesting mathematical fact, check out the following page:

Gabriel Nivasch -- Square Perforation on a Cube
http://yucs.org/~gnivasch/cube/

Note that while the hexagon construction above does not give the
_largest_ cube that can pass through another one. That cube is called
Prince Rupert's Cube:

Eric Weisstein's World of Mathematics -- Prince Rupert's Cube
http://mathworld.wolfram.com/PrinceRupertsCube.html

Answer 5

Your teacher is daft. If he's asking about a 1x1x1 cube and a 1x1 square opening just get out your old toy where you had the different shaped blocks that you put through the similarly shaped holes.

Blue Sky's answer is dealing with 2 cubes. Taking a cube of one size and sliding a larger cube through it, by tilting the smaller cube and cutting it so the larger cube is sliding through a hexagon shape. Is that what your teacher is really asking?

Answer 6

If you turn the cube through 45° in the square hole.
It will form 4 right angle triangles ..1 in each corner.
Each triangle will be 6" by 6" by 'x'.
'x' = √(6² + 6²) = √72
This will be a hypotenuse of 8.48".
Easily fit through the hole...?

Answer 7

Cube is same on all dimnesions therfore 1x1x1

if lenght can vary: 1 by 1 by any length

Or infintely thin, squre root of 2 x square root of 2

Answer 8

i have self assurance in case you employ an algebraic equation to make your cubes its available. i'm no longer certain how yet by technique of creating the tremendous dice versatile i have self assurance it may take position. strong success on pulling this off, this may have me questioning for a lengthy time period.

Answer 9

let the side of the dsired cube be a,
then asqrt(3 =1sqrt(2)
a=sqrt(3/2)foot=1.2 foot aproximately.

Answer 10

Paradox.