Honza had 100 identical preserving jars, with which he built triangular pyramids. The highest floor of the pyramid always has one jar, the second floor from the top represents an equilateral triangle, the side of which consists of two jars, and so on. An example of a three-story pyramid construction is shown in the image.

![](https://cdn.mathpix.com/cropped/2024_04_17_46b1947d893b2b50fba1g-2.jpg?height=317&width=349&top_left_y=1892&top_left_x=405)

1st floor

![](https://cdn.mathpix.com/cropped/2024_04_17_46b1947d893b2b50fba1g-2.jpg?height=314&width=331&top_left_y=1893&top_left_x=837)

1st and 2nd floors

![](https://cdn.mathpix.com/cropped/2024_04_17_46b1947d893b2b50fba1g-2.jpg?height=303&width=345&top_left_y=1893&top_left_x=1255)

three-story pyramid

1. How many jars did Honza need for a five-story pyramid?
2. How many floors did the pyramid have, for which the most of Honza's jars were used?

(K. Jasenčáková)

Hint. How do the numbers of jars in adjacent floors differ?
1. We will count the glasses by floors starting from the top. From the problem statement and the guiding images, we know that in the fifth (highest) floor, there is 1 glass, in the fourth floor, there are 3 glasses, and in the third floor, there are 6 glasses. Each subsequent (lower) floor can be imagined as adding one row of glasses to the previous (higher) floor:
![](https://cdn.mathpix.com/cropped/2024_04_17_46b1947d893b2b50fba1g-3.jpg?height=282&width=1332&top_left_y=440&top_left_x=361)

For a five-story pyramid, Honza needed

$$
1+\underbrace{1+2}_{3}+\underbrace{1+2+3}_{6}+\underbrace{1+2+3+4}_{10}+\underbrace{1+2+3+4+5}_{15}=35 \text { glasses. }
$$

2. We will continue working with the same idea as in the previous paragraph until we exhaust the maximum of one hundred usable glasses: for a six-story pyramid, it is necessary to have

$$
35+\underbrace{15+6}_{21}=56 \text { glasses, }
$$

for a seven-story pyramid, it is necessary to have

$$
56+\underbrace{21+7}_{28}=84 \text { glasses, }
$$

for an eight-story pyramid, it is necessary to have

$$
84+\underbrace{28+8}_{36}=120 \text { glasses. }
$$

With one hundred glasses, the highest pyramid that can be built is a seven-story pyramid.