51 Star Flag (continued)

A previous post on this blog discussed the mathematics of grid and checkerboard flags. These are suitable for most numbers of stars, including 51. Occasionally, however, these techniques are not sufficient. For instance, there are no suitable grid/checkerboard flags for 62, 79 or 89 stars. You may also just get bored of checkerboard and grid flags and want something different. As it turns out, there are plenty of options. Without further ado:

Alternative Flag Styles

The first five styles were actually used for US flags. They are named after the state that whose joining caused that flag. The other styles have never been used, but are valuable mathematically.

Michigan style is a simple grid flag with an extra star in the top and bottom rows.
Number of stars = rows x columns + 2

Oregon style is a simple grid flag with two stars removed from the center row. (though it only looks good if the number of rows is odd, so that the the modified row can be centered)
Number of stars = rows x columns - 2

Kansas is like Oregon except that it only has one star removed.
Number of stars = rows x columns - 1

Nevada is the most complicated design so far. The best way to look at it is to mentally combine the left and right columns and make a simple grid flag with one extra star. (But only if the number of rows is odd)
Number of stars = rows x columns + 1

Mathematical trick for Nevada

Colorado is a simple grid with a star removed from the top and bottom rows. This gives it the same math as Oregon, but with no odd rows restriction.
Number of stars = rows x columns - 2

No Corners is the heaviest modification so far. It is a grid with four stars removed.
Number of stars = rows x columns - 4

Modified Checkerboard can be best thought of as a checkerboard flag with an extra star added to the left and right columns. This design only works if it is based on an odd x odd checkerboard pattern with corners black (see previous post for checkerboard flag analysis).
Number of stars = (rows x columns - 1) / 2 + 2 
                        (Checkerboard with black corners, plus 2)

This gives us an army of new formulas with which to do our flag hunting. Solving them all for rows x columns we get:

rows x columns = (Number of stars) - 2       Michigan
rows x columns = (Number of stars) - 1       Nevada
rows x columns = (Number of stars) + 1       Kansas
rows x columns = (Number of stars) + 2       Oregon or Colorado
rows x columns = (Number of stars) + 4       No Corners
rows x columns = (Number of stars) x 2 - 3   Modified Checkerboard

From the last post:
rows x columns = (Number of stars) x 2       Even checkerboard flags
rows x columns = (Number of stars) x 2 - 1   Odd checkerboard, white corners
rows x columns = (Number of stars) x 2 + 1   Odd checkerboards, black corners
rows x columns = (Number of stars)           Simple grid flags

The procedure for finding a flag with a given number of stars is to use these formulas to find possible values of rows x columns for your given number of stars. Then, try to factor rows x columns into appropriate values for rows and columns.

For 62, this gives us:
Michigan:                  62 - 2 = 60 = 6 x 10
Nevada:                    62 - 1 = 61 = PRIME
Kansas:                    62 + 1 = 63 = 7 x 9
Oregon/Colorado:           62 + 2 = 64 = 8 x 8 
                              (Oregon requires odd rows, though)
No Corners:                62 + 4 = 66 = 6 x 11
Mod. Check:                62 x 2 - 3 = 121 = 11 x 11
Even Check:                62 x 2 = 124 = 4 x 31 UGLY
Odd Check, White Corners:  62 x 2 - 1 = 123 = 3 x 41 UGLY
Odd Check, Black Corners:  62 x 2 + 1 = 125 = 5 x 25 UGLY
Simple Grid:               62 = 2 x 31 UGLY

This gives us five options for a 62 star flag:

There are other flag designs that could be imagined, but this collection gives us plenty of formulas to work with, and all of the patterns are aesthetically pleasing.

The last post mentioned that 62, 79 and 89 are hard to find flag patterns for.  62 has been done just now.  Here are suitable patterns for 79 and 89, using the mathematics described above.

A nice article from Slate (www.slate.com) and a fun widget from PopSci (www.popsci.com) about this topic do not have solutions for 29, 69 and 87 stars.  With the mathematics described above, we can find patterns for all of them.

With this I bring you my final set of options for a 51 star flag:

The mathematics I have so far described gives us Checkerboard, Michigan and Modified Checkerboard styles.  The Checkerboard style is the currently accepted design if Puerto Rico should become a state.  I add two other styles here.  One is the Special Circular.  51 happens to be an excellent number to arrange in a circle.  The circular pattern is a popular alternative to the accepted checkerboard and it was not designed by this blog.  The 'Special' pattern is my own invention, and it has a lot of interesting mathematics behind it.  It is actually a further modification of the idea of a checkerboard flag.  But those mathematics are for another post :-)