Question: Karan bought a little box of midget matches, each one inch in length. He found that he could arrange them all in the form of a triangle whose area was just as many square inches as there were matches. He then used up six of the matches, and found that with the remainder he could again construct another triangle whose area was just as many square inches as there were matches.
And using another six matches he could again do precisely the same.
How many matches were there in the box originally?
Note that the match-box can hold maximum of 50 matches.
Answer:
Initially, there were 42 or 36 matches in the match-box.
There are 42 matches in the box with which he could form a triangle 20, 15, 7, with an area of 42 square inches. After 6 matches had been used, the remaining 36 matches would form a triangle 17, 10, 9, with an area of 36 square inches. After using another 6 matches, the remaining 30 matches would form a triangle 13, 12, 5, with an area of 30 square inches.
After using another 6, the 24 remaining would form a triangle 10, 8, 6, with an area of 24 square inches.
Thus, there are two possible answers. There were either 42 or 36 matches in the match-box.
Also it is interesting to know that there are just 5 such triangles for which the perimeter and the area is the same (assuming all sides are integers) and they are :
1. 24 (10, 8, 6)
2. 30 (13, 12, 5)
3. 36 (17, 10, 9)
4. 42 (20, 15, 7)
5. 60 (29, 25, 6)