Combien y a-t-il de palindromes à n chiffres ?

Summary

This video explores the fascinating concept of palindromes, focusing on the number of palindromes for different digit lengths. It explains how palindromes are numbers or words that read the same forwards and backwards. The discussion progresses from simple single-digit palindromes to more complex ones with multiple digits, revealing patterns in their occurrence. Through step-by-step calculations, the video introduces a general formula to compute the number of palindromes for any given digit count, distinguishing between even and odd numbers of digits. Ultimately, a unified formula is provided for all cases, offering a clear and concise solution to this intriguing mathematical question.

Takeaways

• 😀 A palindrome is a word or sequence that reads the same backward and forward, like 'Abas' or '121'.
• 😀 The goal of the script is to explore how many palindromes exist for a given number of digits, starting from one-digit numbers.
• 😀 For single-digit palindromes, there are 10 possibilities (0-9), but we focus on strictly positive palindromes, making the count 9.
• 😀 Two-digit palindromes are fewer, with 9 possibilities (from 11 to 99), excluding numbers that start with 0.
• 😀 Three-digit palindromes can be formed by taking a two-digit palindrome and adding any digit in the middle, resulting in 90 three-digit palindromes.
• 😀 Four-digit palindromes follow a similar rule, leading to 90 palindromes as well, due to the symmetry of the digits.
• 😀 A pattern emerges: 9 palindromes at one digit, 9 at two digits, 90 at three digits, 90 at four digits, and so on.
• 😀 The conjecture is that the number of palindromes increases by a factor of 10 for every two digits added.
• 😀 The script then transitions into proving this conjecture using a general formula, considering both even and odd numbers of digits.
• 😀 For even-digit palindromes, the number of possibilities is calculated as 9 times 10 raised to the power of (n/2 - 1). For odd-digit palindromes, it is 9 times 10 raised to the power of (n-1/2).

Q & A

What is a palindrome as defined in the script?

A palindrome is a word or sequence of words that can be read the same way forwards and backwards. In the context of numbers, it refers to numbers like 121, 9339, or sequences like 1234567654321, which read the same from left to right and right to left.

How are palindromes classified based on the number of digits?

Palindromes are classified based on the number of digits they have. For example, there are palindromes with 1 digit (like 1, 2, 3), 2 digits (like 11, 22), 3 digits (like 121, 131), and so on. The script focuses on finding the number of palindromes for each number of digits.

How many palindromes exist with 1 digit?

There are 9 palindromes with 1 digit, specifically the digits 1 through 9, as 0 is excluded from being a valid single-digit palindrome.

How many palindromes exist with 2 digits?

There are 9 palindromes with 2 digits. This excludes 00, and includes palindromes like 11, 22, 33, etc.

What is the process to calculate the number of palindromes with 3 digits?

To calculate the number of 3-digit palindromes, you take any 2-digit palindrome and add any digit between 0 and 9 in the middle. This gives a total of 90 palindromes with 3 digits.

How are palindromes with 4 digits calculated?

To calculate 4-digit palindromes, the process is similar to that for 3 digits. You double the middle digit, which results in 90 palindromes with 4 digits.

What pattern emerges in the number of palindromes as the number of digits increases?

A clear pattern emerges where the number of palindromes increases by a factor of 10 with each additional pair of digits. For example, there are 10 palindromes with 1 digit, 9 with 2 digits, 90 with 3 digits, 90 with 4 digits, and so on.

What is the conjecture about the number of palindromes with n digits?

The conjecture is that the number of palindromes increases by a factor of 10 as the number of digits increases by 2. For example, there are 900 palindromes with 5 digits, 900 with 6 digits, 9000 with 7 digits, and so on.

What is the formula for the number of palindromes for even n?

For even n, the number of palindromes is given by the formula 9 × 10^(n/2 - 1), where n is the number of digits.

What is the formula for the number of palindromes for odd n?

For odd n, the number of palindromes is given by the formula 9 × 10^((n-1)/2), where n is the number of digits.

What is the significance of the floor function in the final formula for palindromes?

The floor function in the final formula, 9 × 10^⌊(n-1)/2⌋, combines both the cases of even and odd n into one formula. It ensures that when n is even, it results in the formula for even n, and when n is odd, it results in the formula for odd n.