Shape Problem

danielw.brady · #1 $Old$ October 29th, 2009, 12:42 PM

We have been set this problem as part of a problem solving excersise at University, apparently it has not ever been solved correctly by students.

Draw four identical shapes that fit together to create a larger shape with no holes in it. Each shape must have at least one border with the other three.

(The shapes must be identical in shape and size, they should only be rotated, not reflected, sides must touch, points don't count).

Anyone?

Soroban · #2 $Old$ October 29th, 2009, 03:48 PM

Hello, danielw.brady!

The instructions are not clear.
I will guess what you meant.

Quote:

Draw four identical shapes that fit together to create a larger similar shape with no holes in it.
Each shape must have at least one common border with another shape.

(The shapes must be identical in shape and size.
They should only be rotated, not reflected.
Sides must touch, points don't count).

There is an infinite number of solutions.

Any triangle will do this . . .

Code:

              *
             *  *
            *     *
           *        *
          * * * * * * *
         *  *        *  *
        *     *     *     *
       *        *  *        *
      * * * * * * * * * * * * *

And any parallelogram . . .

Code:

            *---------*---------*
           /         /         /
          *---------*---------*
         /         /         /
        *---------*---------*

And many more interesting shapes . . .

Code:

      *-------*
      |       |
      |   *---*
      |   |   |
      *---*   *---*---*
      |   |       |   |
      |   *-------*   |
      |       |       |
      *-------*-------*

Code:

              *
             / \
            /   \
           / x   \
          /   *---*-------*
         /   / \         / \
        *---*   *   *---*   \
       /         \ /         \
      *-------*---*-----------*

This last one is interesting . . . but wrong!
. . The piece marked "x" must reflected.

ux0 · #3 $Old$ October 29th, 2009, 03:52 PM

Can the four shapes be in 2 dimensions and connect to make a 3 dimensional shape? I think a regular tetrahedron would work... its made up of four regular triangles. Other than that I can't think of anything

danielw.brady · #4 $Old$ October 29th, 2009, 05:10 PM

Thank you Soroban the soloutions however do not satisfy the problem, the problem states that each shape must be in contact with all the other ones, they must touch on a side, a point or a corner touching does not count.

If they did as you say four squares works quite simply.

The shape and shapes must be 2-D, and drawn on a 2-D surface, i've already asked the lecturer about drawing it on a sphere for example which works quite easily but he laughed and said no!