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The problem presented asks us to determine two things based on building a card tower (château de cartes) using a specific pattern shown in the image:

1. **How many cards are used to build a ten-level tower?**
2. **How many levels can be built with 1,000 cards, and how many cards would remain?**

### Solution:

**1. Finding the number of cards in a ten-level tower:**

Each level of the tower requires cards in a specific pattern. 

- For each level, we need:
  - 2 cards to form a single triangle (side-by-side).
  - Additionally, 1 card is placed flat on top of the triangles from the level below (except the first level).
  
The number of cards used in level \( n \) can be determined by the following formula:
\[ C_n = 3n - 1 \]
Where:
- \( C_n \) is the number of cards in level \( n \).

Thus, the total number of cards used in a tower with \( N \) levels is the sum of the cards needed for each level:
\[
T(N) = \sum_{n=1}^{N} (3n - 1)
\]

For a ten-level tower, we calculate this sum for \( N = 10 \).

**2. Finding how many levels can be built with 1,000 cards:**

We need to find the largest \( N \) such that:
\[
T(N) \leq 1000
\]

Once \( N \) is determined, the remaining number of cards will be:
\[
R = 1000 - T(N)
\]

Let's compute both parts now.### Results:

1. **For a ten-level tower**, a total of **155 cards** are used.
   
2. **With 1,000 cards**, you can build **25 levels** of the tower, and **50 cards** will remain unused.

Would you like more details or have any questions? Here are some related problems for further practice:

1. How many cards are required to build a 15-level tower?
2. What is the formula for determining the number of cards in a \(N\)-level tower?
3. If the pattern changes to using 4 cards per triangle instead of 3, how would that affect the number of levels with 1,000 cards?
4. How many cards are needed for a 5-level tower?
5. What is the difference in card usage between a 20-level tower and a 25-level tower?

**Tip:** When dealing with summations or sequences, recognizing the pattern can help you derive a general formula for calculations.

On se propose de construire un château de cartes selon le modèle ci-contre. 1. Combien de cartes sont utilisées si on construit ainsi dix étages ? 2. Combien d'étages peut-on construire avec 1000 cartes ? Combien restera-t-il de cartes ?

Découvrez comment calculer le nombre de cartes nécessaires pour construire un château de cartes avec 10 étages. Cette solution inclut des concepts arithmétiques de séquences et de sommes, idéal pour les élèves de collège.

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