Suppose you had five sticks of length 1, 2, 3, 4, and 5 inches. If you chose three at random, what is the likelihood tht the three sticks could be put together, tip to tip, so as to form a triangle?

Now suppose you had twenty sticks, of lengths 1 through 20 inches. If you picked three at random, what is the likelihood that the three could be put together, tip to tip, to form a right triangle?

(Assume that a triangle has to have some area)

The first question has been exhausted by previous posters. The second question concerns sticks in {1,2,3,...,20} forming a RIGHT triangle. To do so, they must consist of base a, height b, and hypotenuse c, such that a^2 + b^2 = c^2 (Pythagorean Theorem). If c is between 1 and 20, then c^2 must be in {25, 100, 169, 225, 289, 400} and c must be in {5, 10, 13, 15, 17, 20}.
5^2 = 3^2 + 4^2
10^2 = 6^2 + 8^2
13^2 = 5^2 + 12^2
15^2 = 9^2 + 12^2
17^2 = 8^2 + 15^2
20^2 = 12^2 + 16^2

(5,3,4) in any of 6 orders
(10,6,8) in any of 6 orders
(13,5,12) in any of 6 orders
(15,9,12) in any of 6 orders
(17,8,15) in any of 6 orders
(20,12,16) in any of 6 orders

This is a total of 36 possibilities, out of the total number of ways to pick 3 sticks out of 20, which is 20*19*18 = 6840. The likelihood is therefore 36/6840.
Edited on November 5, 2003, 4:29 pm

Posted by Dan on 2003-11-05 16:13:13