Mind-Bending Shapes
Geometric forms abound in the sculptures and drawings of pioneering conceptual artist Sol LeWitt. His systematic and playful arrangements of shapes often embody visual riddles, so we've devised a series of exercises, based on LeWitt's artwork, for you to wrap your mind around.


Cubes Each of the 122 sculptures in LeWitt's Variations of Incomplete Open Cubes is a shape with a series of contiguous lines using at least three— but not more than 11— of the edges of a cube. Sixteen possible variations appear in the diagram above, with each two-dimensional rendering representing a three-dimensional structure. Figure 4 of the diagram, for instance, represents the shape at right. Lines that appear to meet at the center of a figure in the diagram do not necessarily meet in the actual three-dimensional figure.


1. [Easy] The first three figures in the diagram show all the incomplete open cubes you can make with just three edges. Can you make six different shapes that each contain four edges? The edges in each shape must all connect to form a single figure. Mirror images of the same shape are considered different, but rotations are not.

2. [Challenging] Which pairs of figures represent mirror images and/or rotated versions of the same three-dimensional shape? Hint: Only two are not part of mirror-image or rotation pairs.

3. [Difficult] How many distinct shapes can you make using five connected edges of a cube? Again, mirror images of the same shape are considered different, but rotations are not.



Arcs The diagram above is based on LeWitt's Wall Drawing #358. In this work, each square tile contains a quarter arc in one of four possible orientations.

1. [Easy] By joining two adjacent tiles, you can make a "domino"— a two-part figure containing two of the quarter arcs. Which domino occurs four times in this figure, either horizontally or vertically? A single square may be used in two separate dominoes, and dominoes may be rotated. The two dominoes below, for example, are considered the same:



2. [Challenging] Which of the 10 possible dominoes does not occur at all?

3. [Difficult] Rotate just one tile so that all nine horizontal dominoes and all eight vertical dominoes are different. Horizontal dominoes may be the same as vertical dominoes.



Lines LeWitt's etching Straight Lines in Four Directions and All Their Possible Combinations contains all the possible ways to overlay horizontal, vertical, or 45-degree diagonal lines that pass through the middle of a square. Notice that the number of lines per tile increases as you scan the tiles from left to right and top to bottom.

1. [Easy] Consider the overall pattern created by the black lines across all 16 tiles in the diagram above. Notice that the black lines form five complete squares. Can you increase the number of complete squares to six by interchanging two tiles without rotating them?

2. [Challenging] Can you rearrange the tiles, again without rotating any of them, to make 10 complete squares? Hint: All squares use diagonal lines.

3. [Difficult] Rearrange the 16 squares— still keeping them in a 4-by-4 grid and not rotating any of them— so that all the lines go all the way from one edge of the grid to another. This puzzle, created by mathematics writer Barry Cipra, has three distinct solutions. We've filled in one solution below. Can you complete the other two? Hint: In each solution, the "asterisk" goes in the upper left-hand corner.

4. [Hard] Try the opposite of Cipra's linear challenge: Rearrange the 16 squares, keeping them in a 4-by-4 grid and not rotating any of them, so that no line continues from any one square to an adjacent square horizontally, vertically, or diagonally.






Solution

Want to see the solution to this puzzle?

Let us know what you think of Bogglers: E-mail alternate solutions, comments, and suggestions to bogglers@discover.com.

Got new solutions for the puzzle? Want to see other people's solutions? Talk to the puzzle master in his discussion forum at www.scottkim.com.



View an installation photograph of All Variations of Incomplete Open Cubes: www.library.upenn.edu/ finearts/slide/287/28745013.htm.



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