Question 327637
<pre><b>
Here is the 4-square train made with 13 toothpicks.

{{{drawing(400,200,-1,5,-1,2,

green(line(0,.1,0,.9)),
red(
line(0+.1,0,1-.1,0),
line(1+.1,0,2-.1,0),
line(2+.1,0,3-.1,0),
line(3+.1,0,4-.1,0),

line(0+.1,1,1-.1,1),
line(1+.1,1,2-.1,1),
line(2+.1,1,3-.1,1),
line(3+.1,1,4-.1,1),

 line(1,.1,1,.9), line(2,.1,2,.9), line(3,.1,3,.9),line(4,.1,4,.9)  
)

 )}}}

Notice that after the first toothpick, the green one below,

{{{drawing(400,200,-1,5,-1,2,

green(line(0,.1,0,.9))

 
 )}}}

You place 3 more (red) toothpicks to form the first square.

{{{drawing(400,200,-1,5,-1,2,

green(line(0,.1,0,.9)),

red( line(.1,0,.9,0), line(1,.1,1,.9), line(.1,1,.9,1))
 )}}}

Then you place 3 more red toothpicks to form the second square,

{{{drawing(400,200,-1,5,-1,2,

green(line(0,.1,0,.9)),

red( line(.1,0,.9,0), line(1,.1,1,.9), line(.1,1,.9,1),

line(1.1,0,1.9,0), line(2,.1,2,.9), line(1.1,1,1.9,1))




 )}}}

So after the first (green) toothpick, no matter how many squares
you have made, it always takes 3 more (red) toothpicks to make the 
next square

So the sum is 

{{{green(1)+red(3)+red(3)+"..."}}}{{{""=""}}}{{{400}}}toothpicks

If we have made n squares, we have added 3 a total of n times, so


{{{green(1) + red(3)n}}}{{{""=""}}}{{{400}}}

Subtract 1 from both sides:

{{{3n}}}{{{""=""}}}{{{399}}}

Divide by 3:

{{{n}}}{{{""=""}}}{{{133}}}

So we have mnade 133 squares, choice c. 

Edwin</pre>