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3
votes
1answer
356 views

Counting Total Number of Non-Equivalent Configurations in a 2-D Grid

This is a challenging question I've been trying (unsuccessfully) to solve via programming, math or both. Suppose you're given a 2D grid, whose width and height, $w$ and $h$, can each range from $1$ ...
5
votes
1answer
95 views

How to show all possible implied parenthesis?

Can I use recursion to find out the possible parenthesis we can add to this expression: 2*3-4*5 ? (2*(3-(4*5))) = -34 ((2*3)-(4*5)) = -14 ((2*(3-4))*5) = -10 (2*((3-4)*5)) = -10 (((2*3)-4)*5) = ...
1
vote
1answer
128 views

How many ways to find a sum totalling n using only certain Integers?

Using an infinite supply of integers of a set S, how many ways are there to reach a sum of n? Clarification: The Integers are arbitrary, positive, and may not include 1. At first I thought it was ...
2
votes
2answers
87 views

Finding number of numbers <= N, containing atleast one of the digits 2,4,6,8

Given an integer $N$, I want to find the number of numbers $\le N$, that contain at least one of the digits from the set $\{2, 4, 6, 8\}$. How do I go about solving this problem? I was thinking of ...
2
votes
0answers
52 views

Smarter recursion to compute #tilings of $m \times n$ board with small shapes that fit in $2 \times 2$ square?

This is a generalization of another question I posted because I wasn't clear that I cared about more than $2 \times 1$ dominoes (it's just a special case), and there is an explicit tractable formula ...
2
votes
1answer
76 views

Smarter recursion to compute #tilings of $m \times n$ board with $2 \times 1$ dominoes?

So I was thinking about how to computationally (e.g., with recursion) obtain the number of tilings of an $m \times n$ board with $2 \times 1$ dominoes. If $m \leq n$, then we can use recursion on $n$ ...
2
votes
2answers
993 views

How Dynamic programming can be used for Coin Change problem?

As far as I can unserstand Dynamic programming stands simply for memoization (which is a fancy name for lazy evaluation or plain "caching"). Now, I read that there is we can reduce complexity of coin-...
1
vote
2answers
160 views

What are efficient ways to compute the derivatives of iterated functions?

The derivatives of iterated functions at a fixed point $z_0$ are useful in constructing a Taylors series of iterated analytic functions - in other words, the Taylors series of a dynamical system $f^t(...
11
votes
1answer
1k views

Towers of Hanoi but with arbitrary initial and final configuration

Recently, I came across this problem, a variation of towers of hanoi. Problem statement: Consider the folowing variation of the well know problem Towers of Hanoi: We are given $n$ towers ...
0
votes
3answers
885 views

Recursive function calculating number of ways to sum $a + 2 b + 3 c = x$

Using python need to code a recursive function with one input and no global integers that calculates the number of options to get $x$ using $a*1+b*2+c*3$. Say $x=3$, there are four options: $\lbrace (...
4
votes
2answers
7k views

A Recursive Formula For Generalized Josephus problem

The Josephus Problem asks where to start taking out every kth person in the circle consisted of n people, such that you are the last "survivor". The following recursive formula is given: $$\begin{...