# Tag Info

Accepted

### Cardinality of the set of algorithms

An algorithm is informally described as a finite sequence of written instructions for accomplishing some task. More formally, they're identified as Turing machines, though you could equally well ...
• 82.1k

### Represent a 5 card poker hand

If we have a set of size $n$, you can represent an element of the set using $\lceil \lg n \rceil$ bits. You say that there are 2598960 possible 5-card hands. That means that a 5-card hand can be ...
• 164k
Accepted

• 278k

• 278k

### Upper bound of of fib(n+2)

Lee Gao's answer is excellent. Here is a different account. Consider the following automaton: This is an unambiguous finite automaton (UFA) without $\epsilon$ transitions: an NFA such that each word ...
• 278k

### Suitable choice for moderate-size square matrix multiplication?

Dumas and Pan recently wrote a non-asymptotic survey of fast matrix multiplication in practice, which hopefully answers your questions. They concentrate on matrices of order at most a million, and ...
• 278k
Accepted

### Why is the number of digits (bits) in the binary representation of a positive integer $n$ is the integral part of $1 + \log_2 n$?

If you have $n$ bits available you can represent the numbers $0$ to $2^n-1$ in binary. Therefore, if you want to represent a number $x$ in binary, you need a number of bits $n$ that is large enough so ...
• 1,058

### Cardinality of the set of algorithms

See Gödel Numbering, it is a basic fact in computer science that algorithms are countable, as are by extension recursively enumerable sets. Algorithms being countable, it is easy to show that there ...
• 171

### Number of combinations without given pairs

Don't expect an efficient algorithm. This is the problem of counting the number of independent sets in an undirected graph. This problem is #P-complete [1,2], so there is unlikely to be any ...
• 164k