YGB #2: Aren t You a Square?

Transcription

1 YGB #2: Aren t You a Square? Problem Statement How can one mathematically determine the total number of squares on a chessboard? Counting them is certainly subject to error, so is it possible to know if one counted them correctly? Additionally, it is important to recognize that there are a number of possibilities for the size of the square, ranging from an individual black or white square to the entire chessboard itself. For this problem, it is assumed that I am working with a standard chessboard, which consists of a checkered pattern with squares of two alternating colors. With B representing a shaded (black) square and W representing a blank (white) square, it would appear as: c1 = {B, W, B, W, B, W, B}; c2 = {W, B, W, B, W, B, W}; c3 = {B, W, B, W, B, W, B}; c4 = {W, B, W, B, W, B, W}; c5 = {B, W, B, W, B, W, B}; c6 = {W, B, W, B, W, B, W}; c7 = {B, W, B, W, B, W, B}; c8 = {W, B, W, B, W, B, W}; list1 = Transpose[{c1, c2, c3, c4, c5, c6, c7, c8}]; Text[Grid[ Prepend[list1, {"W", "B", "W", "B", "W", "B", "W", "B"}], Dividers {All, All}]] Process Making Assumptions Before approaching this problem, it is necessary to make some assumptions regarding the chessboard. These assumptions are made based upon the chessboard diagram from the Problem Statement and are as follows: 1) The entire chessboard is one large square. Each side is thus equal in length. Page 1 of 9

2 2) Each row and column on the chessboard contains exactly eight small, individual squares. 3) The black and white checkered pattern is consistent throughout the chessboard. The third assumption can be further explained. Because the pattern is consistent, every other square along the same row or column has the same color. This also means that squares along the same diagonal line have the same color. If a given square is black, for example, then the squares on its left, right, top, and bottom are white whereas the squares on its top-left, top-right, bottom-left, and bottomright are black. Organized Counting It would appear that one of the easiest way to determine the total number of squares is to count them. However, knowing that counting will increase the likelihood of error, I decided to do so in an orderly system. To start, I determined how many small, individual squares there are on the chessboard. Earlier, I made the assumption that there are 8 of these squares per row and per column. In other words, there are a total of 8 rows of 8 squares. Therefore, I would mutiply 8 by 8 to find the number of small, individual squares. To more easily identify this type of square, I will begin to refer to them as 1x1 squares. Number of Squares 1 x1 = 8 * 8, or 8 2 Number of Squares 1 x1 = 64 There are 64 1x1 squares on the chessboard. The next smallest square size is 2x2, which would look like this: s21 = {B}; s22 = {W}; list2 = Transpose[{s21, s22}]; Text[Grid[Prepend[list2, {"W", "B"}], Dividers {All, All}]] W B B W There are 8 rows on the chessboard and each 2x2 square has 2 rows. I would thus divide 8 by 2 to calculate the number of 2x2 squares per column. Similarly, because there are 8 total columns and each 2x2 square has 2 columns, I would divide 8 by 2 again to calculate the number of squares per row. Number of Squares 2 x2 per Column = 8 / 2 = 4 Number of Squares 2 x2 per Row = 8 / 2 = 4 Number of Squares 2 x2 = per Column * per Row = 4 * 4 = 16 I would continue this process onto 3x3 squares, 4x4 squares, 5x5 squares, 6x6 squares, 7x7 squares, and finally, 8x8 squares (the largest possible square, which is the entire chessboard). When the dimensions of the square do not divide into the 8x8 chessboard evenly, I would subtract the remainder to determine the maximum number of complete squares. The remainder of the quotient only represents a fraction of the square. I would eventually reach the total sum of 92 squares. This answer is incorrect! The strategy Page 2 of 9

3 described above fails to take a large number of squares into consideration. The series of diagrams and explanations below illustrate the issue: Pretend that the entire chessboard is only a 4x4 square: brd441 = {B, W, B}; brd442 = {W, B, W}; brd443 = {B, W, B}; brd444 = {W, B, W}; list3 = Transpose[{brd441, brd442, brd443, brd444}]; Text[Grid[Prepend[list3, {"W", "B", "W", "B"}], Dividers {All, All}]] W B W B B W B W W B W B B W B W Also, assume that I am trying to determine the number of 2x2 squares. Using the current strategy, I would conclude that there are 4 2x2 squares. The diagram below displays their locations. Subscripts of the same value represent one square. brd4412 = {B 1, W 3, B 3 }; brd4422 = {W 1, B 3, W 3 }; brd4432 = {B 2, W 4, B 4 }; brd4442 = {W 2, B 4, W 4 }; list3 = Transpose[{brd4412, brd4422, brd4432, brd4442}]; Text[Grid[Prepend[list3, {"W 1 ", "B 1 ", "W 2 ", "B 2 "}], Dividers {All, All}]] W 1 B 1 W 2 B 2 B 1 W 1 B 2 W 2 W 3 B 3 W 4 B 4 B 3 W 3 B 4 W 4 But what about these 2x2 squares? brd4413 = {B 6, W 6, B}; brd4423 = {W 5,6,9, B 6,8,9, W 8 }; brd4433 = {B 5,7,9, W 7,8,9, B 8 }; brd4443 = {W 7, B 7, W}; list3 = Transpose[{brd4413, brd4423, brd4433, brd4443}]; Text[Grid[Prepend[list3, {"W", "B 5 ", "W 5 ", "B"}], Dividers {All, All}]] W B 5 W 5 B B 6 W 5,6,9 B 5,7,9 W 7 W 6 B 6,8,9 W 7,8,9 B 7 B W 8 B 8 W Of course, it is possible to take these additional squares into account and still use a systematic method to count the total number of squares on a standard chessboard, but that would be time consuming and extremely prone to error. Therefore, counting is not at all a reliable method for this problem. Page 3 of 9

4 Taking Advantage of the Color Pattern Recall that this is a standard 8x8 chessboard: c1 = {B, W, B, W, B, W, B}; c2 = {W, B, W, B, W, B, W}; c3 = {B, W, B, W, B, W, B}; c4 = {W, B, W, B, W, B, W}; c5 = {B, W, B, W, B, W, B}; c6 = {W, B, W, B, W, B, W}; c7 = {B, W, B, W, B, W, B}; c8 = {W, B, W, B, W, B, W}; list1 = Transpose[{c1, c2, c3, c4, c5, c6, c7, c8}]; Text[Grid[ Prepend[list1, {"W", "B", "W", "B", "W", "B", "W", "B"}], Dividers {All, All}]] If anything, the previous method allowed me to determine that there are 64 1x1 squares and 1 8x8 square on the chessboard. Because I know those quantities for certain (there cannot be any that weren t taken into consideration), I can leave them aside for now and just focus on 2x2, 3x3, 4x4, 5x5, 6x6, and 7x7 squares. Before attempting to solve the problem using the color pattern, I must make one additional assumption about the chessboard: that there are an equal number of black and white squares (so 32 each). Because the colors consistently alternate along the same row/column and the first and last squares of row/column are of different colors, I can conclude that there is an equal number of each. First, I would like to work with the 2x2 squares. Referring back to the previous method, I would have found that there are 16 2x2 squares. Looking at the chessboard diagram above, I realized that each of these 16 squares would have an identical pattern of s21 = {B}; s22 = {W}; list2 = Transpose[{s21, s22}]; Text[Grid[Prepend[list2, {"W", "B"}], Dividers {All, All}]] W B B W But as I had determined, the squares can overlap one another. The top-left square (I can actually choose any square, but to make the concept easier to express, I will use the top-left square as the reference point throughout this section) of a 2x2 square does not necessarily have to be white. Each row/column on the chessboard has 4 white squares and 4 black squares, so if the top-left square of a 2x2 square can be either color, then there should be 8 total. Page 4 of 9

5 4 2x2 Squares with White Top-Left Square + 4 2x2 Squares with Black Top-Left Square = 8 2x2 Squares However, I cannot include the final square of that row/column (regardless of color) because if that were the top-left square of a 2x2 square, then not all of the 2x2 square would be on the chessboard. Therefore, there are actually 7 rows of 7 2x2 squares, or 49 2x2 squares in total. The same concept applies to 3x3 squares. If they did not overlap, there would be 2 rows of 2 squares, or 4 in total. Similar to a 2x2 square, the top-left can be either black or white. There are 4 of each per row/column, which means a total of 8 3x3 squares per row/column. With the 2x2 squares, I subtracted 1 from the total because one of the squares fell off the chessboard. With 3x3 squares, I have to subtract 2 from 8, which means 6 squares per row/column. Multiplying 6 by 6, I will find that there are 36 3x3 squares on the entire chessboard. The pattern continues. Logically, I will always find an intial total of 8 squares per row/column. It then becomes a matter of subtracting away the number of squares that would not fit on the chessboard. The following table displays the concept, which is consistent from 1x1 squares all the way to 8x8 squares. Both the Initial Total and the Actual Total are for just one row/column. name = {"1x1", "2x2", "3x3", "4x4", "5x5", "6x6", "7x7", "8x8"}; initial = {8, 8, 8, 8, 8, 8, 8, 8}; sub = {0, 1, 2, 3, 4, 5, 6, 7}; total = {8, 7, 6, 5, 4, 3, 2, 1}; list4 = Transpose[{name, initial, sub, total}]; Text[ Grid[Prepend[list4, {"Dimensions", "Initial Total", "Subtract", "Actual Total"}], Dividers {All, All}]] Dimensions Initial Total Subtract Actual Total 1x x x x x x x x Now that I have obtained the total number of each type of square per row/column, I can easily calculate the total number of squares on the chessboard. This would be done by squaring the Actual Totals and then finding the sum of the results. Although this method allows me to find the answer, it is a bit difficult to express through words. Constructing an Equation The previous method allowed me to visualize exactly what is happening in the problem. In this part of the process, I will organize what I discovered and form a single equation that will directly solve the problem. For now, I am still limiting myself to a standard 8x8 chessboard. Using the information from the table that I constructed in the previous method, I can first write an expression for the total number of squares per row/column for each size: Page 5 of 9

6 For 1x1 squares: (8-0) For 2x2 squares: (8-1) For 3x3 squares: (8-2) For 4x4 squares: (8-3) For 5x5 squares: (8-4) For 6x6 squares: (8-5) For 7x7 squares: (8-6) For 8x8 squares: (8-7) Now, to find the total number of squares on the chessboard for each size, I can just square each of the previous expressions. I can do this because the chessboard itself is a square. If there are x nxn squares along one row, then there are also x nxn along one column. Mulitplying x by itself (or squaring it) calculates the total number of nxn squares on the chessboard. For 1x1 squares: (8-0) 2 For 2x2 squares: (8-1) 2 For 3x3 squares: (8-2) 2 For 4x4 squares: (8-3) 2 For 5x5 squares: (8-4) 2 For 6x6 squares: (8-5) 2 For 7x7 squares: (8-6) 2 For 8x8 squares: (8-7) 2 All that has to be done now is to add the expressions together to find the total number of squares on a standard chessboard. Total = (8-0) 2 + (8-1) 2 + (8-2) 2 + (8-3) 2 + (8-4) 2 + (8-5) 2 + (8-6) 2 + (8-7) 2 Solution I concluded at the end of my problem solving process that the follow equation would allow me to solve for the total number of squares on a standard 8x8 chessboard: Total = (8-0) 2 + (8-1) 2 + (8-2) 2 + (8-3) 2 + (8-4) 2 + (8-5) 2 + (8-6) 2 + (8-7) 2 Therefore, I will now solve the equation. Total = (8-0) 2 + (8-1) 2 + (8-2) 2 + (8-3) 2 + (8-4) 2 + (8-5) 2 + (8-6) 2 + (8-7) 2 Total = (8) 2 + (7) 2 + (6) 2 + (5) 2 + (4) 2 + (3) 2 + (2) 2 + (1) 2 Total = Total = 204 According to my calculations, there are 204 squares on a chessboard. Page 6 of 9

7 Generalizations The problem was developed based off of a chessboard, which certainly helped me visualize the different square sizes and dimensions that are possible. However, the equation that I wrote can be modified to calculate the number of squares in any larger square. An expression that represents the total number of squares in a special chessboard with dimensions s by s is: s n 2 I will now let Mathematica calculate the number of squares on a standard chessboard: 8 n What about a 4x4 chessboard? A 10x10 chessboard? A 1000x1000 chessboard? 4 30 n 2 10 n n It is also possible to determine the total number of rectangles on a chessboard of any size. For now, I will only work with the standard 8x8 chessboard. On the chessboard, a rectangle would be created by 2 horizontal lines and 2 vertical lines. How many of each are there? Again, I must recall that the 8x8 chessboard looks like this: Page 7 of 9

8 c1 = {B, W, B, W, B, W, B}; c2 = {W, B, W, B, W, B, W}; c3 = {B, W, B, W, B, W, B}; c4 = {W, B, W, B, W, B, W}; c5 = {B, W, B, W, B, W, B}; c6 = {W, B, W, B, W, B, W}; c7 = {B, W, B, W, B, W, B}; c8 = {W, B, W, B, W, B, W}; list1 = Transpose[{c1, c2, c3, c4, c5, c6, c7, c8}]; Text[Grid[ Prepend[list1, {"W", "B", "W", "B", "W", "B", "W", "B"}], Dividers {All, All}]] There are 9 horizontal lines and 9 vertical lines that make up the chessboard. Now, rather than treating the question as how many rectangles there are, I will treat it as how many possible combinations of horizontal and vertical lines there are if I were to pick two from each. Actually, because I am working with a square, I can just find the number of possible combinations in one direction and square that number for the answer. The question has been simplified to: how many possible pairs of lines are there if there are a total of 9 lines? This can be mathematically calculated: = 36 pairs Part of the reasoning behind the equation above is that I do not want any repetitions in the pairings. The first line can possibly pair with 8 other lines. The second line can pair with just 7 because its pairing with the first line has already been accounted for. The third line only has 6 possible pairings because its pairings with the first and second lines have already been accounted for. The same applies for the remaining lines. I now know that there are 36 possible pairs of lines in one direction. Like I had mentioned earlier, I will now square this number to find the number of possible pairings between horizontal pairs and vertical pairs, which is the number of rectangles on the chessboard: 36 2 = 1296 rectangles There are 1296 rectangles on a standard 8x8 chessboard. For a special chessboard of dimensions s by s, the concept is the exact same. There would be (s + 1) horizontal lines and (s + 1) vertical lines. For either direction, the number of pairs can be represented by: (s+1)-1 n Page 8 of 9

9 (s+1)-1 2 Then, the total number of rectangles is simply n. To test to see if the expression is correct or if there are indeed 1296 rectangles on a standard chessboard, I will allow Mathematica to perform the calculations when s is equal to 8. (8+1)-1 n (s+1)-1 If the chessboard was to be a rectangle, then I would perform n twice, once for the horizontal lines and once for the vertical lines. Then, I would multiply the two results together to find the total number of possible rectangles. Self-Assessment From this YGB problem, I learned to try to utilize as much given information as possible with a complex mathematics problem. Specifically, I find it somewhat interesting that I was able to use the color pattern of the chessboard to calculate the total number of squares it contains. A seemingly minute detail can have great potential if it is creatively interpreted. I worked very hard on this problem. I dedicated a lot of time and effort towards solving it and towards making any plausible generalizations. I also explored multiple approaches. The only source that I used was and it was to learn about the Sigma notation. I don t have too much experience using it and decided to try to incorporate it into this problem. Page 9 of 9