Ctirad programmed two cooperative drawing robots, Miky and Niki. Miky can construct squares, regular pentagons, and regular hexagons. However, in one day, he draws only congruent polygons. Niki adds all the diagonals to all of Miky's polygons.

1. On Monday, Miky constructed the same number of segments as Niki. Which polygons were they drawing?
2. On Tuesday, Miky constructed 18 segments. How many did Niki construct?
3. On Wednesday, Miky and Niki constructed a total of 70 segments. How many polygons did Ctirad give them to draw?

(M. Petrová)

Hint. How many segments do Miky and Niki construct for a square, pentagon, and hexagon, respectively?
A square has two diagonals, a pentagon has five, and a hexagon has nine:
![](https://cdn.mathpix.com/cropped/2024_04_17_f3fae1a08c67cff3918eg-3.jpg?height=382&width=1422&top_left_y=1528&top_left_x=321)

1. Only a pentagon has the same number of sides and diagonals. On Monday, the robots drew pentagons.
2. Miky constructed 18 sides, so he did not construct any squares or pentagons (18 is not divisible by 4 or 5), he constructed three hexagons $(18: 6=3)$. In three hexagons, there are 27 diagonals $(3 \cdot 9=27)$. On Tuesday, Niky constructed 27 segments.
3. The total number of segments in one square is 6, in one pentagon is 10, and in one hexagon is 15. Miky and Niky constructed a total of 70 segments, so they did not construct any squares or hexagons (70 is not divisible by 6 or 15), they constructed 7 pentagons $(70: 10=7)$. On Wednesday, they were given the task to draw 7 pentagons.