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Omni-MATH / A white equilateral triangle is split into n² equal smaller triangles by lines that are parallel to the sides of the triangle. Denote a [i]line of triangles[/i] to be all triangles…

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problem

A white equilateral triangle is split into n2n^2 equal smaller triangles by lines that are parallel to the sides of the triangle. Denote a [i]line of triangles[/i] to be all triangles that are placed between two adjacent parallel lines that forms the grid. In particular, a triangle in a corner is also considered to be a line of triangles. We are to paint all triangles black by a sequence of operations of the following kind: choose a line of triangles that contains at least one white triangle and paint this line black (a possible situation with n=6n=6 after four operations is shown in Figure 1; arrows show possible next operations in this situation). Find the smallest and largest possible number of operations.
Plain-text mathematical notation (without MathML)
A white equilateral triangle is split into n² equal smaller triangles by lines that are parallel to the sides of the triangle. Denote a [i]line of triangles[/i] to be all triangles that are placed between two adjacent parallel lines that forms the grid. In particular, a triangle in a corner is also considered to be a line of triangles.

We are to paint all triangles black by a sequence of operations of the following kind: choose a line of triangles that contains at least one white triangle and paint this line black (a possible situation with n=6 after four operations is shown in Figure 1; arrows show possible next operations in this situation). Find the smallest and largest possible number of operations.
Original LaTeX notation
A white equilateral triangle is split into $n^2$ equal smaller triangles by lines that are parallel to the sides of the triangle. Denote a [i]line of triangles[/i] to be all triangles that are placed between two adjacent parallel lines that forms the grid. In particular, a triangle in a corner is also considered to be a line of triangles.

We are to paint all triangles black by a sequence of operations of the following kind: choose a line of triangles that contains at least one white triangle and paint this line black (a possible situation with $n=6$ after four operations is shown in Figure 1; arrows show possible next operations in this situation). Find the smallest and largest possible number of operations.

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Source and history

task_id
f1a6021f-4eaa-5a72-8536-d848e46fb58f
task_key
test--f1a6021f-4eaa-5a72-8536-d848e46fb58f
task_revision_id
1
config / split
/ test

Official source

initial import