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Omni-MATH / Let N be a positive integer. A collection of 4N² unit tiles with two segments drawn on them as shown is assembled into a 2N×2N board. Tiles can be rotated. [asy]size(1.5cm);draw((0…

Problem

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problem

Let NN be a positive integer. A collection of 4N24N^2 unit tiles with two segments drawn on them as shown is assembled into a 2N×2N2N\times2N board. Tiles can be rotated. [asy]size(1.5cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);draw((0,0.5)--(0.5,0),red);draw((0.5,1)--(1,0.5),red);[/asy] The segments on the tiles define paths on the board. Determine the least possible number and the largest possible number of such paths.
Plain-text mathematical notation (without MathML)
Let N be a positive integer. A collection of 4N² unit tiles with two segments drawn on them as shown is assembled into a 2N×2N board. Tiles can be rotated.
[asy]size(1.5cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);draw((0,0.5)--(0.5,0),red);draw((0.5,1)--(1,0.5),red);[/asy]
The  segments  on  the  tiles  define  paths  on  the  board. Determine  the  least  possible  number  and  the largest possible number of such paths.
Original LaTeX notation
Let $N$ be a positive integer. A collection of $4N^2$ unit tiles with two segments drawn on them as shown is assembled into a $2N\times2N$ board. Tiles can be rotated.
[asy]size(1.5cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);draw((0,0.5)--(0.5,0),red);draw((0.5,1)--(1,0.5),red);[/asy]
The  segments  on  the  tiles  define  paths  on  the  board. Determine  the  least  possible  number  and  the largest possible number of such paths.

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Source and history

task_id
2b695675-3789-5497-8eb1-436993f59565
task_key
test--2b695675-3789-5497-8eb1-436993f59565
task_revision_id
1
config / split
/ test

Official source

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