Lattice Polygons, Part 2

Lattice polygons are polygons are formed by connecting dots on square dot paper.  (On a coordinate graph, the coordinates of the vertices are integers.). 

The lattice polygon below is a rectangle whose dimensions are 2 units by 4 units.  Its area is 8 square units.

(1)  Can you make a small change to the rectangle so that the area of the new polygon is 6 square units? There are lots of ways to do this.

(2)  How many different kinds of polygons can you make on a geoboard with an area of exactly 6 square units?  Can you make a triangle?  a quadrilateral?  a 5-sided figure?  6-sided?  7-sided?  8-sided?

It is convenient to use this online geoboard:  Geoboard

For the solution, click below:

Same Sum Circles, Part 2

The goal of this puzzle is to place the numbers 1 through 7 in the circles so that you get the same sum in each circle.  

Notice that the placement below is ALMOST a solution but not quite.  One circle has a sum of 17 and the other two have a sum of 19.  You can fix it by moving only one bean.  Which one?  Where would you move it?

Once you figure that out, see how many different sums you can make.  You can make quite a lot!

Click "read more" for the solution:

The Sock Problem #2

Silas reaches into his sock drawer in the dark, and he wants to make sure that he grabs a matching pair.  The drawer has 3 identical blue socks, 2 orange, 1 red, 1 purple, and 1 green sock. 

Question:  How many socks must Silas grab to ensure that he has at least one matching pair? 

For the solution, click "read more" below:

The Stripey Triangle Problem

A triangle is divided into 7 stripes, all the same width and all parallel to the base.  If the small purple triangle at the tip has an area of 3 square units, then what is the area of the largest triangle (the whole thing)?

For the solution, click "Read More" below.

The Consecutive Number Problem

Ginger writes down a sequence of numbers. Each is a positive whole number from 1 to 11, and she uses each number only once. Henrietta looks at the sequence and notices that for every pair of numbers next to each other in the sequence, one is divisible by the other. What is the maximum number of numbers that Ginger wrote down?

For the solution, click "Read More" below.

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.