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This post is a variant on To cut a triangle into $n$ $p$-sided polygonal regions.

Question: Given a positive integer $n$, a triangular region is to be cut into $n$ convex pieces so that the sum over pieces of the number of sides is maximized (the pieces need not all have same number of sides). Will trying to maximize the number of edges of each piece while keeping them as equal to one another as possible always work? It is also possible that there may be no general algorithm and one might need to solve the problem for each value of n separately with proof of optimality.

Some examples: For $n = 3$, the best one can do appears to be to cut the triangle into $3$ quadrilaterals giving a total number of $12$ edges.

On Is it possible to divide a square into convex pentagons?, a partition of a triangle into $9$ convex pentagons is shown. I am not sure if $45$ is indeed the max value of the total number of edges when a triangle is cut into $9$ convex pieces.

Note: The same question could obviously be generalized from a triangle to a convex $m$-gon and also to higher dimensions.

Guess: If the way to cut some specific convex m-gon into n convex pieces with max total edges is found, we readily have such a partition of any convex m-gon. Moreover, the space of solutions is big enough to contain such a partition with each piece also having same area.

This post is a variant on To cut a triangle into $n$ $p$-sided polygonal regions.

Question: Given a positive integer $n$, a triangular region is to be cut into $n$ convex pieces so that the sum over pieces of the number of sides is maximized (the pieces need not all have same number of sides). Will trying to maximize the number of edges of each piece while keeping them as equal to one another as possible always work? It is also possible that there may be no general algorithm and one might need to solve the problem for each value of n separately with proof of optimality.

Some examples: For $n = 3$, the best one can do appears to be to cut the triangle into $3$ quadrilaterals giving a total number of $12$ edges.

On Is it possible to divide a square into convex pentagons?, a partition of a triangle into $9$ convex pentagons is shown. I am not sure if $45$ is indeed the max value of the total number of edges when a triangle is cut into $9$ convex pieces.

Note: The same question could obviously be generalized from a triangle to a convex $m$-gon and also to higher dimensions.

Guess: If the way to cut some specific convex m-gon into n convex pieces with max total edges is found, we readily have such a partition of any convex m-gon. Moreover, the space of solutions might always be big enough to contain such a partition with each piece also having same area.

This post is a variant on To cut a triangle into $n$ $p$-sided polygonal regions.

Question: Given a positive integer $n$, a triangular region is to be cut into $n$ convex pieces so that the sum over pieces of the number of sides is maximized (the pieces need not all have same number of sides). Will trying to maximize the number of edges of each piece while keeping them as equal to one another as possible always work? It is also possible that there may be no general algorithm and one might need to solve the problem for each value of n separately with proof of optimality.

Some examples: For $n = 3$, the best one can do appears to be to cut the triangle into $3$ quadrilaterals giving a total number of $12$ edges.

On Is it possible to divide a square into convex pentagons?, a partition of a triangle into $9$ convex pentagons is shown. I am not sure if $45$ is indeed the max value of the total number of edges when a triangle is cut into $9$ convex pieces.

Note: The same question could obviously be generalized from a triangle to a convex $m$-gon and also to higher dimensions.

This post is a variant on To cut a triangle into $n$ $p$-sided polygonal regions.

Question: Given a positive integer $n$, a triangular region is to be cut into $n$ convex pieces so that the sum over pieces of the number of sides is maximized (the pieces need not all have same number of sides). Will trying to maximize the number of edges of each piece while keeping them as equal to one another as possible always work? It is also possible that there may be no general algorithm and one might need to solve the problem for each value of n separately with proof of optimality.

Some examples: For $n = 3$, the best one can do appears to be to cut the triangle into $3$ quadrilaterals giving a total number of $12$ edges.

On Is it possible to divide a square into convex pentagons?, a partition of a triangle into $9$ convex pentagons is shown. I am not sure if $45$ is indeed the max value of the total number of edges when a triangle is cut into $9$ convex pieces.

Note: The same question could obviously be generalized from a triangle to a convex $m$-gon and also to higher dimensions.

Guess: If the way to cut some specific convex m-gon into n convex pieces with max total edges is found, we readily have such a partition of any convex m-gon. Moreover, the space of solutions is big enough to contain such a partition with each piece also having same area.

This post is a variant on To cut a triangle into $n$ $p$-sided polygonal regions

Question: Given a positive integer $n$, a triangular region is to be cut into $n$ convex pieces so that the sum over pieces of the number of sides is maximized (the pieces need not all have same number of sides). Will trying to maximize the number of edges of each piece while keeping them as equal to one another as possible always work? It is also possible that there may be no general algorithm and one might need to solve the problem for each value of n separately with proof of optimality.

Some examples: For $n = 3$, the best one can do appears to be to cut the triangle into $3$ quadrilaterals giving a total number of $12$ edges.

On https://puzzling.stackexchange.com/questions/92892/is-it-possible-to-divide-a-square-into-convex-pentagons, a partition of a triangle into $9$ convex pentagons is shown. I am not sure if $45$ is indeed the max value of the total number of edges when a triangle is cut into $9$ convex pieces.

Note: The same question could obviously be generalized from triangle to convex $m$-gon and also to higher dimensions.

This post is a variant on To cut a triangle into $n$ $p$-sided polygonal regions.

Question: Given a positive integer $n$, a triangular region is to be cut into $n$ convex pieces so that the sum over pieces of the number of sides is maximized (the pieces need not all have same number of sides). Will trying to maximize the number of edges of each piece while keeping them as equal to one another as possible always work? It is also possible that there may be no general algorithm and one might need to solve the problem for each value of n separately with proof of optimality.

Some examples: For $n = 3$, the best one can do appears to be to cut the triangle into $3$ quadrilaterals giving a total number of $12$ edges.

On Is it possible to divide a square into convex pentagons?, a partition of a triangle into $9$ convex pentagons is shown. I am not sure if $45$ is indeed the max value of the total number of edges when a triangle is cut into $9$ convex pieces.

Note: The same question could obviously be generalized from a triangle to a convex $m$-gon and also to higher dimensions.