Problem 43 - Discrete Mathematics with Applications

Question

Suppose de Bruijn Hotel has 16 guest rooms. Each guest receives a three bit code to enter his room. Each door has a keypad with two push buttons, one for 0 and the other for \(1 .\) When the correct sequence of the four bits is entered, regardless of what bit was entered earlier, the door will open. Suppose a burglar wishes to enter a room. Find the minimum number of bits he needs to enter to be certain that the door will open.

Step-by-Step Solution

Answer

The burglar needs to enter a minimum of \(9\) bits to be certain that the door will open.

Step 1: List all possible 3-bit codes

List all 3-bit codes as follows: 000, 001, 010, 011, 100, 101, 110, 111

Step 2: Find the longest sequence that includes all 3-bit codes

A De Bruijn sequence of order 3 over {0,1} will have a length of 8 since the order is equal to the number of unique 3-bit codes. Start by finding a suitable sequence: 00010111 Notice that every possible 3-bit code can be found in this sequence and its cyclic shifts, e.g., 000, 001, 010, 011, 100, 101, 110, 111

Step 3: Account for the 4-bit inputs

Since the door opens when the correct sequence of 4 bits is entered, we need to make sure that our sequence is resistant to the possible extra bits that the burglar may input. We can do this by adding one more bit to the sequence: 000101110 Now, even if the burglar enters an extra bit before the correct 3-bit code, the door will still open (for instance, if the first bit entered was wrong: 100010111).

Step 4: Find the minimum number of bits to enter

The minimum number of bits the burglar needs to enter to be certain that the door will open is equal to the length of the sequence we obtained in Step 3: 000101110 The length of this sequence is 9 bits. Thus, the burglar needs to enter a minimum of \(9\) bits to be certain that the door will open.

Key Concepts

3-bit codes

A 3-bit code is a sequence composed of exactly three binary digits: either 0s or 1s. These codes are crucial in various applications, particularly in digital electronics and computing systems, where binary numbers serve as the foundation. In a 3-bit system, we can generate a total of 8 different combinations. These combinations range from 000 to 111.
Each of these combinations represents a unique code and can be used in systems requiring identifiers, such as hotel room keypads. The concept of 3-bit codes is foundational for understanding more complex sequences, like De Bruijn sequences.

• Examples of 3-bit codes: 000, 001, 010, 011, 100, 101, 110, 111
• Total number of combinations: There are 2^3 = 8 possible sequences.

In many cases, these codes must be encoded in a way that allows for easy search and verification, like in the given problem where the correct room code needs to be entered.

bit sequences

Bit sequences are strings of binary digits, which are the smallest units of data in computers, comprising 0s and 1s. These sequences form the basis for various operations in computing, including the functioning of security systems like keypads.
In the context of the De Bruijn sequence, a specific type of bit sequence is used that contains every possible n-length subsequence exactly once. For a 3-bit code, these sequences play a vital role in ensuring that all possible combinations can be tested with minimal effort.

• Purpose of bit sequences: They allow for structured data representation, providing a way to encode information.
• Using in security systems: By correctly designing bit sequences, we can ensure a robust entry mechanism where the correct code will unlock the door.

Moreover, the design focuses on minimizing the number of button presses needed to test all combinations, as seen in the key problem solution using the sequence 000101110, which ensures simplicity and efficiency.

cyclic shifts

Cyclic shifts refer to the transformation of a sequence of bits by moving elements from one end of the sequence to the other. They are highly valuable in identifying repeated patterns within long sequences, such as De Bruijn sequences.
When generating a sequence that contains all possible sub-sequences of a given length, cyclic shifts help illustrate that every combination can be found within the larger sequence. This concept is applied in the security system context where every potential entry codes are embedded in one continuous sequence.

• Understanding cyclic shifts: A cyclic shift of a sequence like 00010111 can result in 00101110 or 10111000, depending on the direction of the shift.
• Application: In our example, having cyclic shifts ensures that all 3-bit combinations are represented in one stretch of sequence, improving efficiency.

By utilizing these shifts, the system doesn’t need to reset or repeat testing sequences entirely; instead, it covers all possibilities in a streamlined manner, making it efficient for tasks like entering security codes.