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.)

Solution:

If we stick to rectilinear figures (figures whose sides meet at right angles), we can make areas as small as 5 and areas as large as 9, including every whole number in between.

Notice that a 3-4-5 triangle also has a perimeter of 12, and its area is 6.

Notice also that we can also use the strategy of starting with a small figure, such as the one in the bottom left of the figure below, which has an area of 4, and "bending in" one of the sides.  As long as we "bend out" the other side in exactly the same way, we haven't changed the area.  So that figure has an area of 4.  Using that strategy, you can create figures with areas even smaller than 4 with the same 12 toothpicks.

The Chocolate Problem

Solution: 

These are the numbers of chocolate bars that each party guest has at the time she sits down:

 

A=3, B=2, C=1 1/2, D=1, E=1, F=1, G= 3/4, H=2/3, I=3/5, J=1/2, K=1/2, L=1/2, . . . .

 

Notice that when D sits down, she can sit at any table and would get 1 chocolate bar.  So while different people may have different table arrangements, these numbers should be the same for everyone.  

 

Can you find a pattern?  If you can, then you could start to explore larger numbers, such as where does the 100th person sit?

 

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. Click HERE for the solution.

Spatial thinking challenge

The figures below are all cut out of identical cubes.

The diagram below shows the front view of the figures in two dimensions.

Question:  What would the figures look like from the top in two dimensions?  Strangely, four of them look the same from above.  Which one is different?

For the solution, click HERE.

The Funny Calculator

A calculator has these buttons:

Just like a “normal” calculator,  

When you type 1 x 10, it says 10.

When you type 1 + 10, it says 11.

But…

When you type 1 + 1, it says 10.

When you type 10 + 10, it says 100.

--------------------------------------------------------------------------------------------

Using this information alone:

How would you count from one to four?  

What would the calculator say if you typed in 100 / 10 ?
    

Extending the pattern:


 Can you extend the pattern to count up to twenty?

What would the calculator say if you typed in 0.1 + 0.1?

What would the calculator say if you typed in 0.1 x 10 ?

What would the calculator say if you typed in 111 + 111 ?

What would the calculator say if you typed in 111 x 10 ?

For the solution, click HERE.

Counting triangles!

How many triangles are there in this picture?

It's not so easy.....

Did you say 7?  If so, you're right!

Watch this interactive video to practice organizing your thoughts when you solve counting problems:

Digital Clock Problem

The clock below uses columns of dots to represent the time, in minutes and hours.  What time does the clock read?

Use the clues below to decode the number system and read the time on the clock above.

Clues:

Here is the solution:

Each column represents a different power of 4.  And the number of dots tells you how many 1s, 4s, and 16s you have.  One way to see this is to start with the 8:04 clock and focus on the minutes.  It takes only one dot to make the number :04, so the single dot must have that value.  Then look at the 9:20 clock and again focus on the minutes.  It takes 2 dots to make :20.  One of those dots, we now know, is worth 4, so the other must be worth 16.  

All together the dots on the mystery clock represent:

4 + 4 + 1 + 1 hours, and

16 + 1 + 1 + 1 minutes, so

10:19