## Task 3 - 020733

Hans has written a one and is in high spirits. When he gets home, he therefore runs up the 20 steps to his apartment on the 1st floor in such a way that he always climbs 3 steps up and 2 steps down again, without skipping any step.

Klaus, who lives in the same building on the 4th floor, says: "If you keep going like that, I'll be in front of my door before you are in front of yours."

They agree that they will both climb in the same rhythm, and that the one who reaches the landing in front of their apartment first wins. (To the 4th floor, there are 4 times 20 steps.)

a) Who wins?

b) Who would win if there were only 10 steps to the 1st floor and the other 3 staircases each had 20 steps?

c) How many steps would the bottom staircase need to have for both boys to arrive at the same time? (Here, the other 3 staircases should also have 20 steps each.)

Justify your answers!
a) The number of steps Klaus needs is equal to the number of stairs, which is 80. Hans manages only $3-2=1$ stair in $3+2=5$ steps. However, it's important to note that Hans, starting from the 17th stair, can directly reach his stick (with three steps) without taking the two steps back.

So Hans needs $17 \cdot 5 + 3 = 88$ steps, which means Klaus wins.

b) Klaus needs $10 + 60 = 70$ steps; Hans only needs $7 \cdot 5 + 3 = 38$ and wins in this case.

c) Equality in the number of steps for $s$ stairs: $(s-3) \cdot 5 + 3 = s + 60$, so $5s - s = 60 - 3 + 15$ and further $s = 18$.