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Even in the United Kingdom, we are all familiar with the flag of the United States, known as the "Stars and Stripes" or "Old Glory". The US flag has a long and interesting history which goes back to colonial days before the declaration of independence, and some of this history may be found in Smith (1975). In this article I suggest that there is sufficient interest in the changing arrangements of stars in the flag over the years to spark off a mathematical investigation into the various possible patterns. First a brief history of the origins of the flag of the United States. It must be remembered that in early days, flags were not so well defined as they are today, and variations in design and shape frequently existed. Up to 1707, the flag of the Honourable East India Company consisted of red and white stripes of equal width, usually thirteen in number, and with the red cross on a white ground of St George in a rectangular canton (the canton is the upper comer of the flag next to the staff, usually the upper left hand corner in pictures). The depth of this canton seems to have been the top five stripes. From 1676, following complaints by the famous diarist Samuel Pepys, the Com- pany's ships were only allowed to display this flag beyond the island of St Helena in the South Atlantic and in eastern waters; nearer home they were obliged to use the red ensign as prescribed in a proclamation of 1674. In 1707, following the union with Scotland, the cross of St George was replaced by the then new union flag which included the white saltire (diagonal cross) of St Andrew on a blue ground, but which did not then contain the red saltire of a St Patrick for Ireland. The union flag in the canton then seems to have usually had a depth equal to the first six of the thirteen stripes. In 1801, after the union with Ireland, the red saltire of St Patrick was added to the union flag to give it its present form and the canton of the East India Company's flag was altered accordingly (Wilson, 1986). By a curious coincidence, an almost identical flag to that of the East India Company was adopted unofficially by the American colonies in 1775, with the then current form of a the union flag indicating their status as British colonies (Fig l(a)). The one difference was that the canton, while still containing the union flag, now occupied the first seven of the thirteen stripes. There seems to be no particular a reason for the thirteen stripes in the East India Company's flag (and other numbers of stripes are found), but in the new flag, the stripes symbolised the thirteen colonies. The fact that there were two virtually identical flags during the years 1775 to 1777 seems to be just a curious coincidence. Since the East India flag was confined to practical distant waters, there was unlikely to be much confusion at first, but with the development of American trade around Cape Horn it could have been confusing in later years if both had remained the same. However the declaration of independence in 1776 meant that the British flag was no longer acceptable to the American states as part of their M: own, and the result was that in 1777 the union flag in the canton was replaced by thirteen white stars on a blue ground (Fig l(b)). The reason for the thirteen stars, as for the thirteen stripes, was to symbolise the thirteen states in 1aa the original union. Problems soon arose with the elevation to statehood of Vermont (1791) and Kentucky (1792). The flag was not in fact changed until 1795, when one with fifteen stars and fifteen stripes was substituted, the canton now occupying the first eight stripes (Fig l(c)). In spite of the creation of five more states (in 1796, 1803, 1812, 1816 and 1817), no 111t0 further official changes were made until a flag with 20 stars but only thirteen stripes to commemorate the original colonies was instituted from 4 July 1818 (Fig 4). One is by Keith Selkirk, tempted to speculate that the return to thirteen stripes University of Nottingham might have been because American seamstresses objected to sewing yet more stripes! Since then extra stars have 42 Mathematics in School, March 1992 (a)1775-1777 (b) 1777-1795 (c)1795-1818 Fig. 1 Flags of the United States: (Wilson, p. 74) been added on the 4 July following the creation of each has been found with staggered rows of 3/2/3/2/3 (the new state, but the stripes have remained at the original commonest arrangement, Fig 1(b) or 4/5/4, or as twelve thirteen. stars in a circle with the thirteenth at the centre or as The mathematical interest in all this lies in the arrange- thirteen stars in a circle (Fig. 2(b)), as well as at least two ment of stars to produce a pattern. There are various other arrangements (Figs 2(a) and 2(c)). (The circular possibilities for symmetrical designs, but in most cases the arrangement has also been adopted by the European Com- stars have been arranged in rows, which is the most obvious munity and by Malaysia, which has a very similar flag to possibility when their number becomes large. These rows the American one). can either be in rectangular formation or can be staggered, Those of us who are older might remember the forty- that is in triangular formation. The 13 star flag, for example eight star flag which was in use until 3 July 1959 had six (a) unofficial variant 1777 (b) infrequent variant (c) Massachusetts flag c. 1780 Fig. 2 Var. of. the. thirteen star flag (smith pp. 41, 41, 32 Mathematics in School, March 1992 43 It is interesting to note that the designers of the curren, flag preferred to use a staggered pattern of stars rather than have five rows of ten stars which does not fit so well into the required space. Perhaps the maximum difference between the numbers of stars in the rows and in the columns might be three, or possibly the number of columns should not exceed the number of rows by more than 50%. Such numbers could be called "near-square" numbers, a near-square number being one with two factors which do not differ by more than three or with two factors of which one is not 50%/ more than the other, according to which of the two choices is made. This constraint is an example of the need in design to overcome both mathematical and artistic difficulties. With staggered rows there are rather more possibilities, Fig. 3 Vermont flag of 1804-1837 with 17 stars and 17 stripes (recon- structed). (Smith; p. 225) and for odd numbers of rows they come in pairs according to whether the first row is a long one or a short one, for example 32 can be produced by rows of 6/7/6/7/6 and 33 by 7/6/7/6/7 (Compare figs 3 and 6). The numbers which occur in these patterns are of the type pq + (p+ 1)(q- 1) or rows of eight stars placed in a rectangular pattern. Similar pq+(p-1)(q-1), where p and q are sufficiently large arrangements occurred several times in earlier years positive integers and have certain artistic constraints. (Figs 4(b) and 5(a). For one year there were then forty- Where there are odd numbers of rows, these patterns have nine stars, and from 4 July 1960 the flag has had fifty stars the advantage of both horizontal and vertical reflective in staggered rows of 6/5/6/5/6/5/6/5/6 (Fig 6). symmetry. Between 1818 and 1959 there have been many variations There is also, of course the possibility of staggering with on these themes. It would serve little purpose to detail all the same number in each row, for example 30 could be the patterns through the years, even if I was able to find illustrations of them, but there have at various times been 6/6/6/6/6 (Figs l(c) and 4(a)). This design lacks vertical reflective symmetry (Fig l(c)), and for even numbers of flags with the numbers of stars as follows, each from the rows horizontal reflective symmetry as well (Fig 4(a)), relevant fourth of July: although the latter case has rotational symmetry of order 21 (1819) 30 (1848) 38 (1877) 2. It is really a variation on the rectangular pattern, but it 23 (1820) 31 (1851) 43 (1890) does allow greater flexibility to the overall rectangular 24 (1822) 32 (1858) 44 (1891) shape of the block of stars. 25 (1836) 33 (1859) 45 (1896) It is difficult to find suitable arrangements for some 26 (1837) 34 (1861) 46 (1908) numbers, for example between 20 and 60 the really difficult 27 (1845) 35 (1863) 48 (1912) ones are 21, 29, 34, 47 and 57. Some ingenuity will be 28 (1846) 36 (1865) 49 (1959) needed to devise suitable arrangements for these numbers. 29 (1847) 37 (1867) 50 (1960) Some other numbers such as 31 require rather more rows than looks correct, in this case 4/5/4/5/4/5/4. Curiously the It is much more interesting to set a project to design only one of the really difficult numbers which must have suitable patterns for these numbers of stars, and this could been solved in practice is 21 (though this might have been easily be linked with a history project on the development done by three rows of seven).

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