## Tuesday, November 24, 2015

### Topologist's Map of the United States

To a topologist, two shapes are "the same" if one can be stretched, twisted, and distorted into the other shape without breaking or tearing it.  Shape, angle, size, distance, and position can be completely different, and yet the two shapes are topologically equivalent.

This is a topologist's map of the 48 contiguous United States.  While the states are not recognizable, you can work out which is which by how the states are connected to each other.  For example, which is the only state that borders exactly one state?  Can you find it on the topologist's map?  Can you identify all the states?  (One note:  Lake Michigan is included as one of the polygons, and the state of Michigan is represented as a single polygon, which was necessary to avoid some strange results.  So all together there are 49 polygons here.)

## Monday, February 2, 2015

### Stacked Dice

Solution: The sum of any two opposite faces of a die is 7.  So, excluding the top face, the sum of the numbers on the 3-dice tower is always equal to 7 x 2 x 3 = 42, and it doesn't matter how we arrange the dice.  The only thing that affects the total sum is the number showing on the top, which can be any number from 1 to 6.  So the minimum sum is 42 + 1 = 43 and the maximum sum is 42 + 6 = 48.  For 20 dice, the minimum sum is 7 x 2 x 20 + 1 = 281 and the the maximum sum is 7 x 2 x 20 + 6 = 286.  For n dice, the minimum sum is 7 x 2 x n + 1 and the maximum sum is 7 x 2 x n + 6.

## Tuesday, January 20, 2015

### Spilled Juice Problem

#### The Spilled Juice Problem

Henry had just finished his math homework when he spilled orange juice all over it.  Can you help Henry figure out what his homework problems (and solutions!) were?