Mensa Math Brain Teasers- Solution (1)

 1) A straight wooden rod (think of it like a line segment) is cut at two arbitrary points to form three smaller pieces. What is the probability that these three pieces can form a triangle?

Solution:  

Let the length of the original rod = 1

Let x be the location of the first break for which :

0 <    x   <    1

Let y be the location of the second break such that:

0 <    y   <    1

But x   ≠   y  because there must be 3 pieces

If x < y  the length of the pieces are:

x,  y - x and 1 - y

If y < x the length of the pieces are: y, x - y and 1 - x

To be able to form a triangle any two pieces must be greater than or equal to the third. 

Thus:

x + (y - x) >   1 - y ®  y >  1  ®  2y >  1  y  >   ½

x +  (1 - y)  > y - x ®  2x + 1 >   2y ®   x + ½  >  y

And:

(y - x)  + (1 - y)   >    x  ®   1  >   2x ®   x    <   ½

All three conditions are satisfied in the R2 region of the graph  below:  

The probability of forming a triangle is the sum of the areas of the two regions, R1 and R2.

1/8 + 1/8 = 1/4