This question is a follow up to a previous question:

Are circles required on the edge of the grid?

The image below is a puzzle from the FlowFree app.
Notice there are no colored circles on the edge of the grid AND surprisingly there are no colored circles in the two rightmost columns of the grid.

Puzzle grid

The image below is my solution to the above puzzle:

Solution grid

The rules stated in the app are:

Drag to connect matching colors with pipe, creating a flow.
Pair all colors, and cover the entire board with pipe to solve each puzzle.
But watch out, pipes will break if they cross or overlap!

My question is:

Does there exist a grid (not necessarily square) with a set of colored pairs of circles in the grid such that there are no colored circles in the top (and bottom) two rows of the grid and there are no colored circles in the leftmost (and rightmost) two columns of the grid and there is a unique solution that satisfies the app’s rules?

Clarification: If there is any discrepancy between the rules listed here and the FlowFree app’s rules, the rules listed here apply.


1 Answer 1


Yes these grids do exist. For example:


With the following unique solution:

flow2 solution

To improve this grid even further:

You can insert 2 new circles between the C's and the D's to create a unique grid that has 3 empty rows and columns all around.

If you compare this solution with the solution to the previous question, you'll see an interesting pattern:

To create a grid with N empty columns and rows, the grid width will be 4N + 3, the grid height will be 2N + 2, the number of circle pairs will be 2N + 2.

  • $\begingroup$ +1 A beautiful generalization! $\endgroup$ Commented May 26 at 19:29
  • 2
    $\begingroup$ @WillOctagonGibson, I had to find a generalization of it, because else tomorrow I would have need to answer a "Are circles required on the edge of the grid? Part 3" :D $\endgroup$
    – Lezzup
    Commented May 26 at 23:12
  • $\begingroup$ May I have your permission to share this puzzle with the creators of the FlowFree app? $\endgroup$ Commented Jun 1 at 8:26
  • $\begingroup$ @WillOctagonGibson, sure, no problem, go ahead! $\endgroup$
    – Lezzup
    Commented Jun 1 at 17:43
  • $\begingroup$ Thanks! I have emailed it to them. $\endgroup$ Commented Jun 1 at 18:27

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