The Brain Teaser here is about cutting a Perfect Cube [let's take the one with 10 Cms edge] into two equal parts and discussing about the CUT FACE or the newly created SURFACE of the Planar Cut.

Obviously for the remnants to be equal and identical the CUT must pass through the CENTRE of the Cube which is a sufficient criteria.

We are not bothered about the multi-cuts and their effects such as producung a perfect tetrahedron and four other equal pieces. 

Now we can see that this single cut can leave a CUT FACE of a Square exactly equal to one face of the Cube. Or can produce a CUT FACE of a Rectangle with one edge and one diagonal as its sides.

What is asked is :

1. What is the largest such Square CUT FACE which can be produced.

2. What is the Cut which will result in a HEXAGONAL CUT FACE leaving two pieces having 7 faces or shapes [one Hexagon and six others]

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PS :

This must be quite easy !

Edited November 16, 20205 yr by Commander
spelling

1.

Spoiler

10x10 is the only square possible? Or am I missing something?

2.

Spoiler

With the cube edges aligned with x y z axes, cut with a normal of (1, 1, 1).

Or in other words, align the cube with 2 opposite corners on a vertical axis, and make a horizontal cut. The cut should go through the midpoint of 2 adjacent edges on each of the cube's faces, separating each face into an area that is 7/8 of the original, and a corner cut off that is the other 1/8.

 

13 hours ago, md65536 said:

1.

  Reveal hidden contents

10x10 is the only square possible? Or am I missing something?

2.

  Reveal hidden contents

With the cube edges aligned with x y z axes, cut with a normal of (1, 1, 1).

Or in other words, align the cube with 2 opposite corners on a vertical axis, and make a horizontal cut. The cut should go through the midpoint of 2 adjacent edges on each of the cube's faces, separating each face into an area that is 7/8 of the original, and a corner cut off that is the other 1/8.

 

Yes, there is another Square possible.

On the second point you seem to be on the right track. However the exact cut must be spelt out on the 10 Cm edges.

1) The largest square that can be cut will have a side of 5√5 and will be an inclined cut along the line joining the mid point of a side to one of its opposite corners on one face to the corresponding  points on the opposite face, each having length:
√(100+25) = √25x5 = 5√5
 

2) A Hexagonal cut can be achieved by an inclined cut through mid points of adjacent sides on each face, in cyclic order. Side length: 5√2.

 

2 hours ago, Capt. Paul Abraham said:

1) The largest square that can be cut will have a side of 5√5 and will be an inclined cut along the line joining the mid point of a side to one of its opposite corners on one face to the corresponding  points on the opposite face, each having length:
√(100+25) = √25x5 = 5√5

Interesting shape! Am I understanding correctly that 2 opposite corners of the cut face are on the middles of edges (diagonal length 10√2) and the other two are on opposite corners of the cube (diagonal length 10√3)?

3 hours ago, Capt. Paul Abraham said:

1) The largest square that can be cut will have a side of 5√5 and will be an inclined cut along the line joining the mid point of a side to one of its opposite corners on one face to the corresponding  points on the opposite face, each having length:
√(100+25) = √25x5 = 5√5
 

2) A Hexagonal cut can be achieved by an inclined cut through mid points of adjacent sides on each face, in cyclic order. Side length: 5√2.

 

Yes the right answer ! Well done !!

Can someone help with diagramatic representation !

That's not a square, only a rhombus.

57 minutes ago, md65536 said:

That's not a square, only a rhombus.

Yes, you are right. Though the 4 sides are equal the diagonals are not and therefore it will not be a square.

I was mistaken and assumed that it will be a square.

Edited November 18, 20205 yr by Commander

There is a correction to this Brain teaser. The largest 4 equal sided CUT FACE expected as answer is not a Square but only a Rhombus. It has sides 5✔️5 on all the four sides but one diagonal is 10✔️2 and the other is 10✔️3

Edited November 19, 20205 yr by Commander