Dodecahedron

This is an extremely difficult problem and may take up days of your time to figure out.

A regular dodecahedron has twelve pentagonal sides and twenty vertices. Assuming that one face is in the X-Y plane with an edge along (0,0,0) to (0,1,0), what are the coordinates of the remaining 18 vertices?




Image created using Robert Webb's Great Stella software: http://www.software3d.com/Stella.html

Pigs in a Pen

How do you put nine pigs in four pens so that each pen contains an odd number of pigs?

The 'Tree'mendous Apple Orchard

A certain farmer wanted to create a special orchard in the vacant field next to his house. He had been given ten special apple tree saplings: a Pink Lady, a CandyCrisp, a Fuji, a Cameo, a Granny Smith, a McIntosh, a Jonagold, a Red Delicious, a Senshu, and a Winesap. After careful consideration and planning it was decided that the apple trees would be planted so that there were five rows of trees with four trees in each row.

When he told his neighbors about his plan they all laughed and told him it was impossible to plant ten trees so that you have five rows of trees with four trees in each row.

The following year this is what his neighbors saw:

Click on the picture for a larger image


A few years later the same farmer was given six more apple trees: a Stayman, a Fortune, a Cortland, a Honeycrisp, a Macoun, and a Northern Spy. Six trees wasn't enough to create another star pattern of trees so he came up with a new plan. He changed the planned orchard and planted the six new trees so that all sixteen apple trees were placed in fifteen rows with four trees in each row.

What did the farmer's apple orchard look like when he was done?

Circling a Chess Board

On a standard chess board with individual squares that measure two inches on each side, what is the largest circle that can be drawn so that the circumference lies only on black squares?


Click on the image for a larger view

The Missing Tile

Here's another favorite of mine. I think this one, above most other puzzles, can truly be described as a mind bender.

In the diagram below (click on the image to get a larger version) you see two figures made up from the same pieces. In the upper figure the colored pieces are arranged so that the area of the figure is (13 x 5) ÷ 2 = 32.5 tiles. In the lower figure, the same colored pieces have been rearranged so that the area of the figure is (13 x 5) ÷ 2 - 1 = 31.5 tiles (the -1 is for the missing tile above the number 8.)



This is not an optical illusion; the grid lines are there to help demonstrate that all the squares are uniform. Feel free to print out a copy of the picture, cut out the colored pieces in one of the figures and lay them out on top of the other figure to prove that they are indeed the same size.

How can it be possible to disect a polygon, rearrange the order of the pieces and end up with less (or more) space than was used before?

Star Spangled Triangles

There are ten unique triangles in the figure below. The points, A, B, D, E, and F are obvious. You can also make triangles using opposing points along with the center pentagon; the combinations ACE, ACF, BCD, BCF, and DCE also are triangles.

Can you draw one line through the star to end up with more than 15 unique triangles?