Complete Solution:

Using the clues, there are six possible arrangements.

The actual US flag has 9 rows of stars: 4 rows of 5 stars, and 5 rows of 6 stars.

One way to approach this problem is to test all the possible patterns.Knowing that the numbers of stars in alternating rows differ by 1, determine which patterns allow an arrangement of 50 stars. For rows of 2 and 3 stars, you could have:

2 + 3 + 2 + 3 +…+ 2 + 3 = 50.

Ten rows of 2 stars and 10 rows of three stars equal 50 stars.

With rows of 3 stars and 4 stars, you have:

3 + 4 + 3 + 4 +…+ 3 + 4 = 49

The closest that you can get to 50 is 49, and 49 plus another row of either 3 or 4 stars will not make 50 stars. So rows of 3 and 4 will not work. Continuing to test possibilities in this way, you should find that there are six solutions that satisfy the clues given in the challenge. The actual American flag has 4 rows of 5 stars and 5 rows of 6 stars.

 Possible Combinations of 50 Stars No. of Stars in Row No. of Rows No. of Stars in Next Row No. of Rows Total No. of Stars 1 16 2 17 1×16 + 2×17 = 50 2 10 3 10 2×10 + 3×10 = 50 4 5 5 6 4×5 + 5×6 = 50 5 4 6 5 5×4 + 6×5 = 50 12 2 13 2 12×2 + 13×2 = 50 16 1 17 2 16×1 + 17×2 =50

Another way to think about this problem is to consider the sums of the pairs of numbers in which one number is 1 more than the other. Using 2 and 3, for example, the sum is 5. Since 10 sets of 5 make 50, there could be 10 rows of 2 stars and 10 rows of 3 stars. Using 3 and 4, the sum is 7. Seven sets of 7 is 49, and neither 3 nor 4 can be added to 49 to get 50. Therefore rows of 3 and 4 will not work. This process can be continued to find the rest of the possible solutions.

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