Flag Numbers

To have stars in each of the four corners, the grid must have odd length and odd width, say $2n + 1$ and $2m + 1$ . The alternating pattern fills half of the grid, rounded upward. Thus

$2N - 1 = (2n + 1)(2m + 1).$

The question becomes how many odd numbers less than 200 can be factored non-trivially.

This is pretty much the same question as how many odd numbers less than 200 are not prime , except that we must also exclude 1.

There are 46 prime numbers less than 200. One of them is 2, which is of no interest for us; but we replace this by 1.

Therefore the answer is $100 - 46 = \boxed{54}$ .

---

Not all flag numbers make for good flags. If the US were to lose two states, its flag pattern with 48 stars, its flag pattern would have dimensions 19 x 5: three rows of 10 stars, alternated with two rows of 9 stars. We will therefore call a flag number good if the length of the pattern is less than twice its width. In the table below, the 20 good flag numbers below 100 are marked in boldface.

---

Here is a list of flag numbers below 100 and the grid on which they can be formed:

$\begin{array}{ccccc} \mathbf 5 & 3\times 3 \\ \mathbf 8 & 5\times 3 \\ 11 & 7\times 3 \\ \mathbf{13} & 5\times 5 \\ 14 & 9\times 3 \\ 17 & 11\times 3 \\ \mathbf{18} & 7\times 5 \\ 20 & 13\times 3 \\ \mathbf{23} & 9\times 5 & 15\times 3 \\ \mathbf{25} & 7\times 7 \\ 26 & 17 \times 3 \\ 28 & 11 \times 5 \\ 29 & 19 \times 3 \\ \mathbf{32} & 9 \times 7 & 21 \times 3 \\ 33 & 13 \times 5 \\ 35 & 23 \times 3 \\ 38 & 15 \times 5 & 25 \times 3 \\ \mathbf{39} & 11 \times 7 \\ \mathbf{41} & 9 \times 9 & 27 \times 3 \\ 43 & 17 \times 5 \\ 44 & 29 \times 3 \\ \mathbf{46} & 13 \times 7 \\ 47 & 31 \times 3 \\ 48 & 19 \times 5 \\ \mathbf{50} & 11 \times 9 & 33 \times 3 \\ 53 & 15 \times 7 & 21 \times 5 & 35 \times 3 \\ 56 & 37 \times 3 \\ 58 & 23 \times 5 \\ \mathbf{59} & 13 \times 9 & 39 \times 3 \\ 60 & 17 \times 7 \\ \mathbf{61} & 11 \times 11 \\ 62 & 41 \times 3 \\ 63 & 25 \times 5 \\ 65 & 43 \times 3 \\ 67 & 19 \times 7 \\ \mathbf{68} & 15 \times 9 & 27 \times 5 \\ 71 & 47 \times 3 \\ \mathbf{72} & 13 \times 11 \\ 73 & 29 \times 5 \\ 74 & 21 \times 7 & 49 \times 3 \\ \mathbf{77} & 17 \times 9 & 51 \times 3 \\ 78 & 31 \times 5 \\ 80 & 53 \times 3 \\ 81 & 23 \times 7 \\ \mathbf{83} & 15 \times 11 & 33 \times 5 & 55 \times 3 \\ \mathbf{85} & 13 \times 13 \\ 86 & 19 \times 9 & 57 \times 3 \\ 88 & 25 \times 7 & 35 \times 5 \\ 89 & 59 \times 3 \\ 92 & 61 \times 3 \\ 93 & 37 \times 5 \\ \mathbf{94} & 17 \times 11 \\ 95 & 21 \times 9 & 27 \times 7 & 63 \times 3 \\ \mathbf{98} & 15 \times 13 & 39 \times 5 & 65 \times 3 \end{array}$

---

Follow-up question : What is the smallest flag numbers that corresponds to two distinct "good" flag patterns?