## Task 2 - 340612

The tile pattern in the figure is composed of 14 white and 10 patterned triangular tiles. One can find several squares and triangles, each made up of more than one tile.

How many such squares can be found in total?

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0292.jpg?height=274&width=262&top_left_y=1391&top_left_x=1428)

There are exactly 4 squares made up of 2 tiles each, 5 squares made up of 4 tiles each, 4 squares made up of 7 tiles each, 1 square made up of 8 tiles (see examples in Fig. a, b, c, d), making a total of 14 squares of the desired type.
a)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0292.jpg?height=105&width=117&top_left_y=1935&top_left_x=655)
b)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0292.jpg?height=106&width=111&top_left_y=1940&top_left_x=824)
c)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0292.jpg?height=197&width=194&top_left_y=1849&top_left_x=997)
d)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0292.jpg?height=195&width=194&top_left_y=1850&top_left_x=1251)
e)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0292.jpg?height=102&width=106&top_left_y=2142&top_left_x=467)
f)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0292.jpg?height=103&width=186&top_left_y=2147&top_left_x=632)
g)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0292.jpg?height=103&width=168&top_left_y=2144&top_left_x=887)
h)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0292.jpg?height=188&width=174&top_left_y=2062&top_left_x=1135)
i)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0292.jpg?height=148&width=260&top_left_y=2102&top_left_x=1366)

There are exactly 20 triangles made up of 2 tiles each, 8 triangles made up of 3 tiles each, 8 triangles made up of 4 tiles each, 4 triangles made up of 8 tiles each, 4 triangles made up of 9 tiles each (see examples in Fig. e, f, g, h, i), making a total of 44 triangles of the desired type.
From them, exactly 22 bought at least Cola, exactly 14 bought at least a Hamburger, and exactly 13 bought at least Popcorn. Exactly 10 participants bought at least Cola and a Hamburger, exactly 4 bought at least Cola and Popcorn, and exactly 5 bought at least a Hamburger and Popcorn. All three items were bought simultaneously by only 2 participants.

Show that the number of participants is uniquely determined by these statements! Calculate this number!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0293.jpg?height=317&width=320&top_left_y=618&top_left_x=320)

For the total number \( t \) of participants and the numbers represented in the diagram, the following equations hold according to the given information:

\[ a + b + c + d + e + f + g = t, \quad (1) \]

\[ a + b + c + e = 22, \quad (2) \]

\[ a + b + d + f = 4, \quad (3) \]

\[ a + c + d + g = 13, \quad (4) \]

\[ a + b = 10, \quad (5) \]

\[ a + c = 4, \quad (6) \]

\[ a + d = 5, \quad (7) \]

\[ a = 2. \quad (8) \]

From (5) and (8), it follows that \( b = 8 \), from (6) and (8), it follows that \( c = 2 \), and from (7) and (8), it follows that \( d = 3 \).

From (2) and (8), (9), (10), it follows that \( e = 10 \), from (3) and (8), (9), (11), it follows that \( f = 1 \), and from (4) and (8), (10), (11), it follows that \( g = 6 \).

Thus, from (1) and (8) - (14), it follows that the number \( t \) is uniquely determined; it is \( t = 32 \).