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 

Solutions 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.)

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.

Solution below:

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?

Solution below:

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.

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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 ?

Solution is below:

Solutions:

In general, this number system is a base 2 system. The puzzle is more interesting if you’ve never studied the base 2 number system and have to work the system out logically from the statements above. Even if you have studied the base 2 number system, this may give you a new way to think about it.

Using this information alone:

• How would you count from one to four?  

1, 10, 11, 100

10 equals two because 1 + 1 = 10

11 equals three because 10 + 1 = 11

100 equals four because 10 + 10 = 100

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

Four divided by two equals two, so it would say 100 10 = 10, just like a “normal” calculator.

Extending the pattern:

• Can you extend the pattern to count up to twenty?

1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100 

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

Doubling a number scoots the “decimal” point over one, so 

0.1 + 0.1 = 1 and 0.1 has the value of one half.

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

Multiplying by “10” is doubling in this number system, which means scooting the “decimal” point over once.  So:

0.1 + 0.1   =    0.1 x 10    =     1

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

Doubling a number scoots the decimal point over one, so 111 + 111 = 1110.

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

Multiplying by “10” is doubling in this number system, which means scooting the “decimal” point over once.  So:

111 + 111   =    111 x 10    =     1110

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: