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. |