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.)

For a printable version click here.

For the solution, click "Read More" below.

Geometric Progressions

For the solution, click "Read More" below.

Four Paths Problem

For the solution, click "Read More" below.

Coincidence? Why are Primes Clustered Around Multiples of 6?

For the solution, click "Read More" below.

The Colored Blocks Problem

For the solution, click "Read More" below.

Which Net?

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Flower Problem

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Colored Blocks Problem #2

For the solution, click "Read More" below.

Folded Polyhedra

For the solution, click "Read More" below.

The Box Problem

For the solution, click "Read More" below.

Common Factor Puzzles

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Floating Blocks Problem

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Four Cards Problem

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Five Siblings Problem

For the solution, click "Read More" below:

Six Roosters Problem

For the solution, click "read more" below:

Keyhole Problem

For the solution, click "read more" below:

Transparent Cube Problem

For the solution, click "read more" below:

Drafter's Puzzles

For the solution, click "read more" below:

Hole Punching Problems

For the solution, click "read more" below:

The Dog Problem

For the solution, click "read more" below:

Flattened Polyhedra

For the solution, click "read more" below:

Handshake Problem

For the solution, click "read more" below:

Three Boxes Problem

For the solution, click "read more" below:

Alice's Party

For the solution, click here.

Strange Symbols

Solution to puzzle 1:

Solution to puzzle 2:

For the full solution, click here.

Museum Guard Problem

For the solution, click here.

For more museum guard problems (and solutions), click here. 

Decorated Cube Problem

For the solution, click "read more" below.

The Cupcake Problem

For the solution, click "Read More," below.

The Horse Puzzle

For the solution, click "read more," below.

Origami Square Folding Problem

This problem has surprisingly many solutions.  For the solution, click here.

Dots and Polygons Game

For printable game boards, click here.

The Shaded Triangles Problem

For the solution, click here.

Toothpick Perimeter Problem

It is possible to make an area as small as 2 square units (and perhaps smaller?) with 12 toothpicks!  For the solution, click here.

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.

Toothpick Triangles

For the solution, click here.

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?

For the solution, click here.

Six Toothpick Problem

For the solution, click "Read More" below.

The Two Mangoes Problem

Solution:  

(1)  If one mango and one banana cost $4.70, then two bananas and two mangoes cost twice as much:  $9.40.
(2)  From there we can use the middle bag to find the cost of a single banana:  $13.00 (2 mangoes and 5 bananas) - $9.40 (2 mangoes and 2 bananas) = $3.60 (3 bananas).   If 3 bananas cost $3.60, then each banana costs $1.20.
(3)  From the first bag then we can see that a single mango costs $3.50 ($4.70-$1.20), and so the bag of 2 mangoes costs $7.00.