bonanova Posted February 18, 2008 Report Share Posted February 18, 2008 Warmup question: How many squares are there on a chessboard? Edited: Grand prize: How many rectangles are there on a chessboard? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 18, 2008 Report Share Posted February 18, 2008 How many squares are there on a chessboard? I like this one for it makes my head dizzy when i try to visualize it but the answer is ... 204 Say your chess board has all 1 unit squares. Now if you count the edge for all 1 unit square there would be 8 on each side => 64 squares in total. Now count for the 2 unit squares. 7 on each side => 49 in total. 3 unit square. 6 on each side => 36 in total. and so on. So Total Number = 1+4+9+16+25+36+49+64 = 204! Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 18, 2008 Report Share Posted February 18, 2008 square of size 8: 1 (1X1) square of size 7: 4 (2X2) square of size 6: 9 (3X3) square of size 5: 16 (4X4) square of size 4: 25 (5X5) square of size 3: 36 (6X6) square of size 2: 49 (7X7) square of size 1: 64 (8X8) in total : 204 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted February 18, 2008 Author Report Share Posted February 18, 2008 That was the warm-up. Now for the Grand prize. [OP edited:] How many rectangles are there on a chessboard? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 18, 2008 Report Share Posted February 18, 2008 When you say rectangles, does that include squares also? Squares are rectangles... <_< Quote Link to comment Share on other sites More sharing options...
0 unreality Posted February 18, 2008 Report Share Posted February 18, 2008 squares: 1 * 8x8 4 * 7x7 9 * 6x6 etc x^2 * 9-x by 9-x written out fully: 1 * 8x8 4 * 7x7 9 * 6x6 16 * 5x5 25 * 4x4 36 * 3x3 49 * 2x2 64 * 1x1 1+4+9=14+16=30+25=55+36=91+49=140+64=204 204 squares on a chessboard as for the rectangles, this includes squares, right? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 18, 2008 Report Share Posted February 18, 2008 1x1: 64 1x2 and 2x1: 112 1x3 and 3x1: 96 1x4 and 4x1: 80 1x5 and 5x1: 64 1x6 and 6x1: 48 1x7 and 7x1: 32 1x8 and 8x1: 16 2x2: 49 2x3 and 3x2: 84 2x4 and 4x2: 70 2x5 and 5x2: 56 2x6 and 6x2: 42 2x7 and 7x2: 28 2x8 and 8x2: 14 3x3: 36 3x4 and 4x3: 60 3x5 and 5x3: 48 3x6 and 6x3: 36 3x7 and 7x3: 24 3x8 and 8x3: 12 4x4: 25 4x5 and 5x4: 40 4x6 and 6x4: 30 4x7 and 7x4: 20 4x8 and 8x4: 10 5x5: 16 5x6 and 6x5: 24 5x7 and 7x5: 16 5x8 and 8x5: 8 6x6: 9 6x7 and 7x6: 12 6x8 and 8x6: 6 7x7: 4 7x8 and 8x7: 4 8x8: 1 Grand Total: 1296 Am I right? Did I make any mistakes or miss anything? Note: I figured this out by myself, so please do not take any credit. Quote Link to comment Share on other sites More sharing options...
0 unreality Posted February 18, 2008 Report Share Posted February 18, 2008 after a few seconds of thinking I came up with the following formula: w = width of the rectangle h = height of the rectangle wa = width of the bigger rectangle ha = height of the bigger rectangle the formula for the amount of rectangles (squares included) of w width and h height is: (wa+1-w)(ha+1-h) In this case wa and ha are both 8, cuz the chessboard is 8x8, so this simplifies the equation to: (9-w)(9-h) Order matters, so all numbers 1-8 will be used for both w and h, which means adding together the 64 multiplication problems. (9-1)(9-1) + (9-1)(9-2) + (9-1)(9-3) + ...etc fortunately I made a program that did that for me lol I got 1296 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 18, 2008 Report Share Posted February 18, 2008 ... 1296 Rectangles Rectangles of Width 8 : Num of Rectangles X1 = 1*1=1 X2 = 1*2=2 X3 = 1*3=3 X4 = 1*4=4 X5 = 1*5=5 X6 = 1*6=6 X7 = 1*7=7 X8 = 1*8=8 ...... In total 36 rectangles of width 8 units and varying height Rectangles of Width 7 : Num of Rectangles X1 = 2*1=2 X2 = 2*2=4 X3 = 2*3=6 X4 = 2*4=8 X5 = 2*5=10 X6 = 2*6=12 X7 = 2*7=14 X8 = 2*8=16 ...... In total 72 rectangles of width 7 units and varying height Rectangles of Width 6 : Num of Rectangles X1 = 3*1=3 X2 = 3*2=6 X3 = 3*3=9 X4 = 3*4=12 X5 = 3*5=15 X6 = 3*6=18 X7 = 3*7=21 X8 = 3*8=24 ...... In total 108 rectangles of width 6 units and varying height .... In total 144 rectangles of width 5 units In total 180 rectangles of width 4 units In total 216 rectangles of width 3 units In total 252 rectangles of width 2 units In total 288 rectangles of width 1 units Therefore the number of rectangles is 36+72+108+144+180+216+252+288 = 1296 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 19, 2008 Report Share Posted February 19, 2008 I just did a little research and agree. Quote Link to comment Share on other sites More sharing options...
0 unreality Posted February 19, 2008 Report Share Posted February 19, 2008 bonanova was my formula correct? (in my most 3 up from this one) come to think of it, were we all right? You never confirmed it Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted February 19, 2008 Author Report Share Posted February 19, 2008 Bingo!!!! Confirmed. Quote Link to comment Share on other sites More sharing options...
0 unreality Posted February 19, 2008 Report Share Posted February 19, 2008 Quote Link to comment Share on other sites More sharing options...
Question
bonanova
Warmup question: How many squares are there on a chessboard?
Edited:
Grand prize: How many rectangles are there on a chessboard?
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