Guest Posted October 30, 2009 Report Share Posted October 30, 2009 A 2x2 grid can be achieved by drawing three squares on a sheet. How many squares (in minimum) are needed to get a 8x8 grid? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 (edited) A 2x2 grid can be achieved by drawing three squares on a sheet. How many squares (in minimum) are needed to get a 8x8 grid? First guess. Assuming they have to be squares, not rectangles. 15 The squares corner to corner on a 8x8 grid (0 based) (0,0)(8,8) (0,0)(7,7) (0,0)(6,6) (0,0)(5,5) (0,0)(4,4) (0,0)(3,3) (0,0)(2,2) (0,0)(1,1) (1,1)(8,8) (2,2)(8,8) (3,3)(8,8) (4,4)(8,8) (5,5)(8,8) (6,6)(8,8) (7,7)(8,8) Edited October 30, 2009 by Wede Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 (edited) Second guess assuming rectangles are allowed. 9 rectangles on the 8x8 0 based grid. (0,0)(8,8) (0,0)(8,1) (0,2)(8,3) (0,4)(8,5) (0,6)(8,7) (1,0)(2,8) (3,0)(4,8) (5,0)(6,8) (7,0)(8,8) Edited October 30, 2009 by Wede Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 (edited) Second guess assuming rectangles are allowed. 9 rectangles on the 8x8 0 based grid. (0,0)(8,8) (0,0)(8,1) (0,2)(8,3) (0,4)(8,5) (0,6)(8,7) (1,0)(2,8) (3,0)(4,8) (5,0)(6,8) (7,0)(8,8) Yeah but I think the problem wording says squares Edited October 30, 2009 by Silkman Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 8 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 A question: isn't it possible to get a 2x2 grid by drawing only two squares, each one based in opposite corners? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 A question: isn't it possible to get a 2x2 grid by drawing only two squares, each one based in opposite corners? Never mind, I read the premise more thoroughly. You honor, I retract my question. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 15 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 (edited) 15 Just looked back at some of the other answers and maybe I'm not off the mark! Edited October 30, 2009 by cazz_mm Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 15 Quote Link to comment Share on other sites More sharing options...
0 plainglazed Posted October 30, 2009 Report Share Posted October 30, 2009 (edited) EDIT: nevermind, misread the OP. going to finish my coffee and rework... yeah, think 15, alsoslurpppp.... Edited October 30, 2009 by plainglazed Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 43 because you use the 2 by 2 and put 2 squares in each to give a 4 by 4 square. 4 times 2 + 3=11. Then, in each of the 4 by 4 cells put 2 squares in. 11 + (16*2) = 43 Hopefully, Yay!Might be right. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 (edited) never mind I should read better ;-( Edited October 30, 2009 by well... Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 I came up with 15 as my answer as well. I discovered that if you start from one corner and a 1x1 square you have a 1x1 grid. (1 square for 1x1 grid) To build the 2x2 grid, you need a 2x2 square on the outside; but that leaves two borders unfilled, so add another 1x1 square. (3 squares for 2x2 grid) Now to build upon this we will again start from one corner adding progressively larger squares until we reach the NxN size for an NxN grid. At this point it would follow that many borders have not been covered since all squares so far share two sides and a corner. To fill in these borders, another NxN square would do us no good, so add progressively smaller squares starting from the opposite corner from the existing 1x1 square. You will find that the last 2 borders will require again a 1x1 square just as it did for the 2x2 grid. So, if you were to label the squares with the starting position in the bottom left corner with a 1x1 square increasing to the top left and decreasing again to the top right, you would have the following: 8 7 6 5 4 3 2 1 x x x x x x x x 8 7 6 5 4 3 2 1 7 x 7 6 x 6 5 x 5 4 x 4 3 x 3 2 x 2 1 x 1 Noticing the symmetry, it would make sense that a formula exists. Since you only require one of the largest squares, but two of every other square smaller; the number of squares required for an NxN grid comes out to (2*(N-1))+1 which also means that there will always be an odd number of squares. The result for an 8x8 grid is (2*7)+1 = 15 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 33 squares to be drawn. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 (edited) Now to build upon this we will again start from one corner adding progressively larger squares until we reach the NxN size for an NxN grid. At this point it would follow that many borders have not been covered since all squares so far share two sides and a corner. To fill in these borders, another NxN square would do us no good, so add progressively smaller squares starting from the opposite corner from the existing 1x1 square. You will find that the last 2 borders will require again a 1x1 square just as it did for the 2x2 grid. Now to build upon this we will again start from one corner adding progressively larger squares until we reach the NxN size for an NxN grid. At this point it would follow that 2*(N-1) of the 4N+SUM(4*((N-1)..1)) borders have not been covered (the ones that would be like spokes between squares) since all squares so far share two sides and a corner. To fill in these borders, another NxN square would do us no good, so add progressively smaller squares starting from NxN square and moving to the corner opposite the existing 1x1 square. You will find that the last 2 borders will require again a 1x1 square just as it did for the 2x2 grid. Edited October 30, 2009 by Egghead Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 I got 8 squares. I will use the following grid to depict all the corners for the 8x8 grid to explain my answer: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1: 0, 4, 24, 20 2: 6, 8, 18, 16 3: 0, 2, 12, 10 4: 12, 14, 24, 22 5: 0, 1, 6, 5 6: 3, 4, 8, 9 7: 18, 19, 24, 23 8: 15, 16, 21, 20 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 I got 8 squares. I will use the following grid to depict all the corners for the 8x8 grid to explain my answer: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1: 0, 4, 24, 20 2: 6, 8, 18, 16 3: 0, 2, 12, 10 4: 12, 14, 24, 22 5: 0, 1, 6, 5 6: 3, 4, 8, 9 7: 18, 19, 24, 23 8: 15, 16, 21, 20 Your grid is a 5x5 grid, the op is for an 8x8 grid Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 First guess. Assuming they have to be squares, not rectangles. 15 The squares corner to corner on a 8x8 grid (0 based) (0,0)(8,8) (0,0)(7,7) (0,0)(6,6) (0,0)(5,5) (0,0)(4,4) (0,0)(3,3) (0,0)(2,2) (0,0)(1,1) (1,1)(8,8) (2,2)(8,8) (3,3)(8,8) (4,4)(8,8) (5,5)(8,8) (6,6)(8,8) (7,7)(8,8) Using your method I made a picture. (I like to be able to see the answers) Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 (edited) I got 8 squares. I will use the following grid to depict all the corners for the 8x8 grid to explain my answer: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1: 0, 4, 24, 20 2: 6, 8, 18, 16 3: 0, 2, 12, 10 4: 12, 14, 24, 22 5: 0, 1, 6, 5 6: 3, 4, 8, 9 7: 18, 19, 24, 23 8: 15, 16, 21, 20 Your grid is a 5x5 grid, the op is for an 8x8 grid Also, your grid is missing 8 borders; 2 on each of the four corners of the grid. Borders missing are between cells 0,1 ; 0,5 ; 3,4 ; 4,9 ; 15,20 ; 19,24 ; 20,21 ; and 23,24. Adding these borders with your method would require the addition of four more squares which brings the total to 12 squares which is more than the (2*(5-1))+1 = 9 squares. Edited October 30, 2009 by Egghead Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 33... assuming you go by EXACTLY what he said on the first post of this thread he is asking for the minimum amount of squares required to complete a 8x8 grid... so i came up with this.. _ _ _ _ _ _ _ _ | _ | 1 | _ | 2 | _ | 3 | _ | 4 | _ _ _ _ _ _ _ _ | 5 | _ | 6 | _ | 7 | _ | 8 | _ | _ _ _ _ _ _ _ _ | _ | 9 | _ | 10| _ | 11| _ | 12| _ _ _ _ _ _ _ _ | 13| _ | 14| _ | 15| _ | 16| _ | _ _ _ _ _ _ _ _ | _ | 17| _ | 18| _ | 19| _ | 20| _ _ _ _ _ _ _ _ | 21| _ | 22| _ | 23| _ | 24| _ | _ _ _ _ _ _ _ _ | _ | 25| _ | 26| _ | 27| _ | 28| _ _ _ _ _ _ _ _ | 29| _ | 30| _ | 31| _ | 32| _ | and 33 would be the outer line for the whole grid.. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 lol that grid did not come out how i had planned but im sure you all get the idea.. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 lol.. sands beat me to it.. i guess im in second place.. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 squares.pdf Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 30, 2009 Report Share Posted October 30, 2009 33... assuming you go by EXACTLY what he said on the first post of this thread he is asking for the minimum amount of squares required to complete a 8x8 grid... so i came up with this.. _ _ _ _ _ _ _ _ | _ | 1 | _ | 2 | _ | 3 | _ | 4 | _ _ _ _ _ _ _ _ | 5 | _ | 6 | _ | 7 | _ | 8 | _ | _ _ _ _ _ _ _ _ | _ | 9 | _ | 10| _ | 11| _ | 12| _ _ _ _ _ _ _ _ | 13| _ | 14| _ | 15| _ | 16| _ | _ _ _ _ _ _ _ _ | _ | 17| _ | 18| _ | 19| _ | 20| _ _ _ _ _ _ _ _ | 21| _ | 22| _ | 23| _ | 24| _ | _ _ _ _ _ _ _ _ | _ | 25| _ | 26| _ | 27| _ | 28| _ _ _ _ _ _ _ _ | 29| _ | 30| _ | 31| _ | 32| _ | and 33 would be the outer line for the whole grid.. You can make the same 8x8 grid with much fewer total squares drawn. Quote Link to comment Share on other sites More sharing options...
0 plainglazed Posted October 30, 2009 Report Share Posted October 30, 2009 14 - draw all squares from each of the four corners of the eight by eight grid as follows: four 7 x 7 unit squares, four 6 x 6 unit squares and four 5 x 5 unit squares; finally draw two 4 x 4 unit squares starting from opposite corners of the eight x eight grid Quote Link to comment Share on other sites More sharing options...
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A 2x2 grid can be achieved by drawing three squares on a sheet. How many squares (in minimum) are needed to get a 8x8 grid?
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