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Why does this work?

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The graph is the colored directed graph with vertices 0, 1, 2, 3, 4, 5, 6 and the following rules

• the white vertex is 0, all others are black
• the black arrows connect k to (k+1)%7
• the white arrows connect k to (10*k)%7

So each vertex has two outgoing arrows, one black, one white.

If you know the remainder of n modulo 7, this graph allows you to read off the remainders of n+1 and 10*n by following the black and white arrows emanating from the vertex n%7. An integer is divisible if and only if its remainder modulo 7 is 0.

In the example given on the page, you calculate the remainder of 325 as follows:

325 = ((3 *10 + 2) *10 + 5 = ((0+1+1+1) *10 +1+1) *10 +1+1+1+1+1.

Start on the white vertex, follow three times a black arrow, then once a white arrow, twice a black arrow, once a white arrow then five times a black arrow and you land on vertex 3 = 325%7.

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What makes 7 special? Could I construct a similar graph for 5 or 11?

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There’s nothing special about 7 here. Except that it’s the first number for which all divisibility tests in base 10 are kinda tricky, so it’s neat that you can visualize one such test in a planar graph.

You can define similar graphs G(b,m) for every base b>=2 and every modulus m>=1: the set of vertices is {0,...,m-1} and k is connected to (k+1)%m and (b*k)%m for each vertex k. Thus, the graph in the post is G(10,7).

The author lists the planar graphs for small values of b and m in this comment.

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I suspect the author is David Wilson of Wilson’s Algorithm

http://dbwilson.com/

https://en.wikipedia.org/wiki/Loop-erased_random_walk#Uniform_spanning_tree

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I didn’t see a graph unless I used a browser without adblock, so the image may be hosted on a network that adblockers don’t like.

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must be an odd rule then. no typical ad, analytics etc. keyword in the URL:

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Perhaps “visibility” is the culprit?

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This could well be. I’m using both Adblock and Ghostery and it’s very possible there’s some weird interaction there.