The Rolling Cube Problem

Cubes that are freshly painted with blue and yellow paint roll along a table and leave paint behind on the table.

Which of the paint trails above could have been made by a cube whose faces were painted with blue and yellow paint?

For the solution, click "Read More" below.

The Birthday Party Problem

 

Betty’s friends threw her a birthday party. Everyone (including Betty) met exactly 3 new people at the party. 

When Betty told her mom about the party later, her mom said, “That sounds like fun! How many people were there at the party?” 

“I think there were 14, or maybe 15, including me,” answered Betty.

Question: If Betty was right that there were either 14 or 15 people at the party, then which number is correct? 

For the solution, click "Read More" below.

The Isosceles Triangle Problem

The lines below are parallel. 

If we place a point on the top line, we can connect that point to points A and B to create an isosceles triangle:  (Remember -- an isosceles triangle has two sides of the same length.)

Question:  How many other ways can this be done?  How many distinct isosceles triangles can you make that include points A and B as two of the three vertices?

For the solution, click "Read More" below.

Four fours!

Warm up:

Place operation symbols between the ones to make the equations true.  (Use the four operations and parentheses.)

1    1    1    1    =     0

1    1    1    1    =     1

1    1    1    1    =     2

1    1    1    1    =     3

1    1    1    1    =     4

Work out:

4   4   4   4  =  0 

4   4   4   4  =  1 

4   4   4   4  =  2 

4   4   4   4  =  3 

4   4   4   4  =  4 

4   4   4   4  =  5 

4   4   4   4  =  6 

4   4   4   4  =  7 

4   4   4   4  =  8 

4   4   4   4  =  9 

4   4   4   4  =  10 

For the solution, click "Read More" below.

Toothpick Puzzle!

This is a repost of a problem from several years ago.  This time I am including an interactive document that you can use to move your toothpicks around and create your figures. First, make sure you are signed in with a Google/gmail account - otherwise you will be able to see the document but can't make your own copy.  Open the document and then under the "file" menu, click "make a copy" to make your own copy:  Toothpick Perimeter document

Here's the problem:

Notice that you can make figures with toothpicks.  We can think of each toothpick as a unit of length.  Below I made a figure with a perimeter of 12 and an area of 9:

Question:  How many different areas can be made with the same perimeter of 12?  (Notice that we can make a perimeter of 8 simply by "bending in" a corner of the figure. The area is smaller but the perimeter is the same.)  What is the smallest area you can make with 12 toothpicks?  The largest?  Can you make every whole number in between?

(This activity was inspired by a puzzle in Kordemsky's The Moscow Puzzles.)

For the solution, click "Read More" below.

The Chocolate Problem

For the solution, click "Read More" below.

Drafter's Puzzle #2

A solid figure cut out of a cube looks like this in 2 dimensions when viewed from the front, back, and top.  Notice that it has a horizontal edge visible from the front but not from the top.  What could the solid look like?

Notice that it cannot look like this.  Otherwise we would see an edge from the top too!

What does the solid look like?  There are various possibilities.

For the solution, click "Read More" below.