Mr. Ticháček had three plaster gnomes in his garden: the largest he called Mašík, the middle one Jířa, and the smallest Fanýnek. Because he liked to play with them, over time he found out that when he placed Fanýnek on Jířa, they were as tall as Mašík. When he placed Fanýnek on Mašík, together they measured $34 \mathrm{~cm}$ more than Jířa. And when he placed Jířa on Mašík, they were $72 \mathrm{~cm}$ taller than Fanýnek.

How tall are Mr. Ticháček's gnomes?

(M. Petrová)

Hint. Visualize all three situations.
We will visualize all three situations as follows:
![](https://cdn.mathpix.com/cropped/2024_04_17_e836f4ee23919875e85ag-3.jpg?height=556&width=836&top_left_y=1732&top_left_x=610)

Since Jiřa with Fanýnek on his head measures the same as Mašík, we can replace Mašík with this pair in the second and third situations:

![](https://cdn.mathpix.com/cropped/2024_04_17_e836f4ee23919875e85ag-4.jpg?height=565&width=546&top_left_y=180&top_left_x=755)

Now everything is clear: Two Fanýnks on top of each other measure $34 \mathrm{~cm}$, so one Fanýnek measures $17 \mathrm{~cm}$. Two Jiřas on top of each other measure $72 \mathrm{~cm}$, so one Jiřa measures $36 \mathrm{~cm}$. Mašík then measures the sum of Fanýnek and Jiřa, which is $17+36=53(\mathrm{~cm})$.