# What are some strategies to the game of Flow?

Asked by

PhiNotPi (

12611)
October 3rd, 2012

Flow is an iDevice app that seems to be pretty popular recently. The game board is a square array of cells, with several pairs of colored endpoints. The goal is to connect all of the endpoints with lines, and the lines cannot overlap and must fill the entire board. Here and here are photos of the game near completion. These are examples on a small game board, but the boards can get up to 14×14 in size with more than a dozen colors of line.

This game seems as if it ought to have some sort of underlying strategy. What strategy could there be?

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## 4 Answers

Here is some stuff that might be relevant, but I left them out of the details to prevent cluttering up the question.

One of the first properties that I noticed was that the length of each line must have the same parity as the Manhattan distance between the endpoints. Also, the total length of all of the lines must have the same parity as the area of the board minus the number of different lines.

(L1 + L2 + L3 + ....) mod 2 = (Board Area – # of lines) mod 2

This also means that any solvable board can be made unsolvable by simply moving one endpoint over to an adjacent square. This affects the parity of that line, but does not affect the parity of any of the other lines or the area of the board.

I just work from the outside-in, and have gotten most of them “perfect” on the first try. I was doing them on the iPhone 4 – even the 14×14, because I figured the game play data wouldn’t carry over to the iPad… but now I have the 5, and I haven’t opened the app… I hope my progress wasn’t lost!

I dropped out of math in H.S. Algebra 2 because I was acing the tests but not doing homework, and therefore failing the class. Stupid. So I gave up on academic math, even though my brain seems to naturally ‘get’ it. As such, I don’t know the right terminology to describe my techniques, and am unqualified to discuss your theories… sorry.

I suggest to just keep playing it. The more levels you complete, the more you’ll figure out ways to figure them out.

Started writing a response, realized you aren’t talking about Flow.

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