*September 22, 2015*

The Collatz conjecture is one of my favorite things from math. It says that if you take any number and repeatedly divide it by two when even, and multiply by 3 and add one when odd, that you will eventually reach 1. Despite appearing simple, this conjecture has never been proven – though it has been shown to be true for every number tested so far.

In one XKCD comic, the sequence of numbers is drawn as a graph with edges from each number in the sequence to the next. I found several versions of this graph, but wanted to see an even bigger one. So I wrote the following program which generates such a graph for all numbers up to the starting number given. Here is the program code (which uses the awesome graphviz utility for producing the images):

```
#include <stdio.h>
#include <string.h>
#include <stdlib.h>
/* compute the next number in the Collatz sequence of value */
unsigned collatz(unsigned value) {
if (value % 2) {
return 3 * value + 1;
} else {
return value / 2;
}
}
/* a node in the graph */
struct Node {
unsigned value;
unsigned next;;
};
/* test if there is a path from node s to node t in the graph */
int find(int s, int t, struct Node* graph, int n) {
/* if node s IS node t, we've done it */
if (s == t) {
return 1;
}
/* if node s is bigger than n, then there is no path -
* well there is one, but we wont' show it! */
if (s > n) {
return 0;
}
/* if the node is zero, then no path - this can happen
* when we have already removed the node s would travel
* trhough from the graph */
if (s == 0) {
return 0;
}
/* now we recurse on the link from s */
return find(graph[s].next, t, graph, n);
}
int main(int argc, char** argv) {
/* check they passed a number */
if (argc != 2) {
printf("usage: %s N\n", argv[0]);
return -1;
}
unsigned n = atoi(argv[1]);
if (n == 0) {
printf("usage: %s N\n", argv[0]);
return -1;
}
/* an array of nodes (one extra for n itself */
struct Node graph[n + 1];
graph[1].value = 1;
graph[1].next = 0;
/* for each number we are looking at */
unsigned i;
for (i = 2; i <= n; i++) {
/* create the node for it */
graph[i].value = i;
/* compute its next collatz step */
graph[i].next = collatz(i);
}
/* go through the graph and remove ones which don't reach 1, this
* ensures the graph which is printed is complete */
for (i = 2; i <= n; i++) {
if (!find(i, 1, graph, n)) {
graph[i].value = 0;
graph[i].next = 0;
}
}
/* open the file and dump the preamble */
FILE* f = fopen("collatz.gv", "w");
fprintf(f, "digraph G {\n");
/* dump all the nodes */
for (i = 1; i<= n; i++) {
/* print the node */
if (graph[i].value) {
fprintf(f, " n%d[label=\"%d\"];\n", graph[i].value, graph[i].value);
}
/* print the link */
if (graph[i].next) {
fprintf(f, " n%d -> n%d;\n", graph[i].value, graph[i].next);
}
}
/* dump the postamble and close */
fprintf(f, "}\n");
fclose(f);
/* generate the image */
system("dot -Tpng collatz.gv -o collatz.png");
return 0;
}
```

And here is the output for a size of 32 (the program only shows the portion of the graph which actually connects to node 1):

This link shows the very large (2447 * 4763) image for all numbers up to 1,000.

Copyright © 2018 Ian Finlayson | Licensed under a Creative Commons Attribution 4.0 International License.