How many rectangles are there on a chessboard?
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Due Date
Each problem will have a two week deadline. You must blog your answers for 2 out of the 3 problems by:
Friday, November 20, 2009
Friday, November 20, 2009
Stuck?
Question 1: Chessboard Rectangles
- Try a small chessboard first.
- What systematic way of counting the rectangles will be best?
- Examine the method used to count the squares on a
chessboard and generalize.
Question 2: Quick and Toasty
- Watch out for ambiguity!
- It can be done in less than 140 seconds.
- Which time periods should you concentrate on minimizing?
Question 3: Cubed Cubes
- Can you find one arrangement?
- What do you KNOW? Are the conditions clear?
- What do you WANT? When are two large cubes different?
- What presentation can you INTRODUCE? Before using 8
coloured cubes try drawing a cube or diagrams of its faces or
try looking at a convenient box - it may help you visualize
the situation even if you do not touch it.
- INTRODUCE a way of describing or recording different
solutions.
- Try a small chessboard first.
- What systematic way of counting the rectangles will be best?
- Examine the method used to count the squares on a
chessboard and generalize.
Question 2: Quick and Toasty
- Watch out for ambiguity!
- It can be done in less than 140 seconds.
- Which time periods should you concentrate on minimizing?
Question 3: Cubed Cubes
- Can you find one arrangement?
- What do you KNOW? Are the conditions clear?
- What do you WANT? When are two large cubes different?
- What presentation can you INTRODUCE? Before using 8
coloured cubes try drawing a cube or diagrams of its faces or
try looking at a convenient box - it may help you visualize
the situation even if you do not touch it.
- INTRODUCE a way of describing or recording different
solutions.
there are 32 rectangles on a chess board
ReplyDeletethere are also 64 squares on a chess board
1 rectangle-2 squares
jeff
i think that 94 rectangles can fit on a chess board because i tried first with the small then got bigger. first i did rectangles one square the two etc. I added it all up and got 94!
ReplyDeleteIris
I think that there is 94 rectangles on the chess board because the chess board is 8by8, and you first do the 1by2 rectangles which is 4 in a row and 4x8=32. then you double the rectangel so its 4by2. and there is thats 8, and then do 4by 1, and there are 16. thendo then the bigger the rectangle gets the smaller the number so then divide by 2 you get 4(from 8), and then 2. add it up it = 94.
ReplyDeleteHA! THIS IS DUE ON MEH B-DAY!!!!!!!!!!! AND BTW DOES A SKWARE COUNT AS A RECTANGLE?!?!?!?!?!?!
ReplyDeleteI think there can be 164 rectangles on a chess board (using each square more than once, obviously), if not there can be 64. I used bigger and bigger rectangles, drawing them on my paper, and the reason (I think) I got different answers from others, is because they may have (not saying they did) forgotten that squares are also rectangles!
ReplyDeletek i got 54 rectangles NOT INCLUDING SKWARES and lets see... 139 INCLUDING SKWARES!
ReplyDeleteI got a much bigger answer. I got 169 rectangles. I did it systematically by listing possible rectangle sizes and their numbers in each size, as seen below.
ReplyDeletePossible Rectange Size - # of rectangles
1x2 - 32
1x3 - 10
1x4 - 16
1x5 - 11
1x6 - 10
1x7 - 9
1x8 - 8
2x3 - 10
2x4 - 8
2x5 - 5
2x6 - 5
2x7 - 4
2x8 - 4
3x4 - 4
3x5 - 3
3x6 - 2
3x7 - 2
3x8 - 2
4x5 - 2
4x6 - 2
4x7 - 2
4x8 - 2
5x6 - 1
5x7 - 1
5x8 - 1
6x7 - 1
6x8 - 1
7x8 - 1
There it is! After listing these I added up the numbers on the right hand side and got 169!
hey its ur friendly neighborhood gonthorian and im gonna do a rap for u guys after i anwer EXCITEMENT!!! (SIGH)
ReplyDeleteokay so for this question if u count all of them there are thousands but i came up with a ton of answers and i dont really think its neccesary to type for an hour after all im not dumbledor. ok so i got when u use just each rectangle as 2 squares it would be 32 but if u used all sizes and all lengths it would be alot more so i dont really know how to answer, you could say there are 8 different sizes in length but when u add width it is alot more but if u took 64 squares on a borad and divided it by 2 u get 32 and then start from there and as each size gets bigger add that many per the board and the number will get smaller and smaller so if u do this by LENGTH u would get ........ FRODO THE ELVE no im just joking you would get i uhhh uhh ok... maybe 512 for there are 64 squares 8 rows and 8 collums and 64 times 16 is 1024 yea..oh wait u ya now im gonna do a rap for u guys...so i uh guess ill go now... ok (bea boxing) this is about a game called chess yea and its the bessst you play all day and u play all night but your queens and pawns just constantly fight, i rap for u so u can understand that not all the pieces were superman some wpuld die and some would get lost some would frie and some would get mossed now anyways on with the show i hope that u got a tempoo and the answer seems to vary because u might just carry a different board around with u and then u will be super scrwd because well i dont meen to be rude but since i was i appologise a trillion times now again and again i thought and thought so then in the end i just took a shot i found out there are this many squares then my mom came in and said "no ONE CARES" i agreed then continued on and im sitting here just righting this song it is long but its the BOMB. ms dimitry is kool u think other wise and ur just a big fool. it annoys me i cant swear becuase i feel so square i hope u like my raps now continue with ur math blog. i know u are reading just sitting at home and this is starting to sound like a stupid frigen poem i like those too but i still think that im better than u and the king of the board meens alot but then again look at wut uve got 64 squares and 8 rows now what happens when someone THROWS that at u dont u feel so new like u dont understand just stick it to the man but when that dosnt work and ur friends call u a jerk and there aint no perks so i know u like my rap but GET BACK TO WORK and original pineapple production from tur friendly neighboorhood DUMBLEDOOR
I did this the very very very evry very very very very etc. hard way. But i couldn't figure out a different way so what i did was just ocunt them. I did this on a 8x8 chess board so i hope thats what u were talking about. I got 94.......I hope it's right. Well thats al for now! Plz gimmie feedback if I'm wrong. THNX!!!!
ReplyDelete64 for 2 squared rectangles vertically and horizantally +64 for 3 sqaured rectanles V & H +32 for 4 sqaured rectangles V & H +32 for 5 squared rectangles V & H +64 for 6 and 7 squared rectangles V & H + 32 for 8 squared rectangles V and H..........64
ReplyDelete+64
32
32
64
32
______
288
Hey everyone,
ReplyDeleteIt's Katty again! Using a excel spread sheet, I figured out that there is 1296 rectangles on a chess board (including squares). How I found that was by using patterns, and creating a table. For example, each rectangle has a length and a width that ranges from 1-8 so you combine all of them. For instance, there are 64 1 by 1 squares. There are 56 1 by 2 squares having 2 across and 1 as the up and down side. I hope I explained it well because it worked, but it was NOT easy to explain. See you all Monday!
~KATTY
I think that there can be 107 rectangles if you overlap.
ReplyDelete