1st August 2010, 10:31 pm

by David Wilson

I was somewhat taken aback by the popularity of my earlier essay “Divisibility by 7 is a Walk on a Graph.” Tanya tells me it got a good number of hits. The graph in that article is rather crude, and takes a bit of care to use, because the arrows go off in random directions from each node. So taking a hint from a commenter on the first graph, I redrew the graph, sacrificing planarity in favor of ease of use. Specifically, I arranged the black arrows in a counterclockwise circle, which makes them easy to follow.

The graph is used in the same way as the first graph. To find the remainder on dividing a number by 7, start at node 0, for each digit D of the number, move along D black arrows (for digit 0 do not move at all), and as you pass from one digit to the next, move along a single white arrow.

For example, let n = 325. Start at node 0, move along 3 black arrows (to node 3), then 1 white arrow (to node 2), then 2 black arrows (to node 4), then 1 white arrow (to node 5), and finally 5 black arrows (to node 3). Finishing at node 3 shows that the remainder on dividing 325 by 7 is 3.

I fancy it to be a little animal face.





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