How to enclose maximum number of triangles by 9 matchsticks
In the puzzle figure, 7 triangles are enclosed by the 9 matchsticks. Can you enclose maximum number of triangles by 9 matchsticks?
In the following figure, 9 matchsticks enclose 7 small triangles of the triangular grid. How many maximum number of small triangles of the grid can you enclose by suitable placement of the 9 matchsticks on the triangular grid?
Recommended time to solve is 15 minutes.
With the right approach, the puzzle should not prove to be too difficult. Do give it a try before you look through the solution.
Solution to the puzzle of enclosing maximum number of triangles on the triangular grid by 9 matches: Find the wasteful pattern
How many triangles the grid has? It is 16. Seven of which are enclosed by the 9 matches at the base of the grid, and nine are left.
When you look closely at the matchsticks and number of triangles they enclose, you discover with surprise that triangle 1 is enclosed by two dedicated sticks, whereas each of the five triangles 2 to 6 is enclosed just by 1 stick.
Conclusion 1: Enclosing a triangle by two corner sticks is wasteful. To increase the number of triangles enclosed, each pair of existing corner matchsticks must be used in enclosing at least 2 triangles, not 1.
This is the key pattern that will lead us to the solution.
Solution to the puzzle of enclosing maximum number of triangles on the triangular grid by 9 matches: Eliminate corner triangles
On this cue, you decide to give up enclosing triangle 1. With the two sticks freed you find you can reach up to the layer of triangles above on the left of the grid. Simultaneously, using the other three upper side sticks you can form an enclosure of 11 triangles on the grid. A huge improvement! The figure is shown.
It is the same group of 9 matchsticks, but now they enclose 11 triangles on the grid instead of a paltry 7 triangles earlier. This has been possible just by eliminating one wasteful pair of corner matchsticks.
Can we enclose more? Sure we can. The triangle 7 is still enclosed by the second pair of wasteful matchsticks. Give up triangle 7 and reuse the two corner matchsticks. You will have your solution.
With the same 9 matchsticks, 13 triangles of the grid are now enclosed. This is the maximum. Only three corner triangles are left. If you try to enclose any of these three, you will have to give up more triangles than you gain because enclosing a corner triangle needs two dedicated sticks instead of one.
In the solution, by elimination of the corner triangles, on an average more than one triangle is enclosed by a single stick.
This is the highest triangle enclosure efficiency that gives us the maximum number of enclosed triangles.
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