Stars and Bars

The first official flag of the Confederate States of America. Flown over confederate and state buildings, but not usually carried into battle (see the Confederate Battle Flag) because the pattern and colors of this flag did not distinguish it from the Stars and Stripes of the Union.

The most catastrophic consequence of the similarity of flags may have been the death of Stonewall Jackson whose party was carrying this flag when they were attacked in error by Southern troops at the Battle of Chancellorsville.

The flag had two horizontal red stripes separated by a white stripe (top third red, middle third white, bottom third red). Superimosed in the top left corner (the Canton: 1/2 height, 1/3 width) was a blue square with seven or more white stars arranged in a circle. The stars represented the seven original Confederate states, plus more to represent later states or sympathetic states. A 12 star variation was used by Nathan Bedford Forest, who swore not to include the star for Georgia, "as long as a Yankee remains on Georgia's soil." Also of interest, the Cherokee Brigade's flag had 13 white stars in a circle around 5 red ones for the "five civilized nations" - the five native american tribes that joined the Confederacy.

See Flags of the Confederacy and also rebel flag.

How many ways can k indistinguishable balls be placed in n distinguishable boxes?

To answer this, you take a 'star' for each of the k balls, and n - 1 'bars' denoting the separators between the n boxes. Then each arrangement of the balls corresponds to a permutation of these k + n - 1 objects. For instance, when k = 6, n = 4, one of the arrangements might be written (3, 0, 2, 1); that is, 3 balls in the first box, etc. This arrangement corresponds to the string

* * * | | * * | *

of 6 stars and 3 bars. But now that we have only two kinds of objects, it is easy to count the number of arrangements: there is one arrangement for every choice of which k out of the k + n - 1 slots receive a star. Thus the answer is the binomial coefficient C(k + n - 1, k) = C(k + n - 1, n - 1).

Among other things, this number counts the multi-indices of dimension n and weight k, which is the dimension of the space Sym^k(Rⁿ) of symmetric tensors of type (k, 0) or (0, k) on Rⁿ. This space comes up when you formulate multivariable calculus in a general context, as the kth derivative of a function Rⁿ → R is a symmetric (k, 0) tensor.

A good place to learn more about this kind of problem is Concrete mathematics by Ronald Graham, Donald Knuth and Oren Patashnik. Richard Stanley's two-volume masterwork Enumerative combinatorics is a more advanced reference; the stars-and-bars problem is one component of what he calls the Twelvefold Way of basic counting.

A drink at Austin Grill, a tex-mex restaurant in the Washington D.C. metro area. It is described as "The frozen flag of Texas."

It is a red, white, and blue margarita made with Strawberry and Lime frozen margaritas topped with Blue Curacao.

This drink can easily be made at home, it would make an excellent addition to a Fourth of July party. Ingredients needed:

• 1 can frozen strawberry daiquiri mixer
• 1 can frozen margarita mixer
• Tequila
• Blue Curacao
• Ice

1. Following the directions on the package, mix up the strawberry portion using tequila instead of rum. Pour even amounts into 4 glasses. Cylindrically shaped glasses, will give you the best effect.
2. Following the directions on the package, mix up the lime portion and gently pour over the strawberry portion creating a layered effect.
3. Add a shot of blue curacao to top off your Stars and Bars margarita. Garnish with lime or even a sparkler if your feeling festive.