 Posers and Puzzles

1. 14 Apr '08 14:05
Can you cut a chess board into four rectangles so they have the ratios 1:1, 1:2, 1:3 and 1:4 respectively?
You cannot cut squares in parts.
2. 14 Apr '08 15:44
I think you cannot do that. Maybe my calculations are wrong:
I suggest letting x be the largest part of the board. Then we can rite the following equasion:
x + 1/2x + 1/3x + 1/4x = 64
Multiply the equasion by 12:
12x + 6x + 4x + 3x = 768
x = 30.72
3. 14 Apr '08 15:45
Originally posted by FabianFnas
Can you cut a chess board into four rectangles so they have the ratios 1:1, 1:2, 1:3 and 1:4 respectively?
You cannot cut squares in parts.
Lets see...

To cut the board into 4 rectangles, we must make 3 paralle cuts or 2 perpendicular cuts.

The first option is impossible since the 1:1 rectangle is a square.

The second option is impossible since the rectangle having no common edge with the 1:1 rectangle is a square, but 1:2, 1:3, 1:4 are not squares.

So there is no solution, even if cutting squares in parts is allowed.
4. 14 Apr '08 16:021 edit
Well, a square is a special kind of rectangle with the ratio 1:1. We do agree on this, don't we?
5. 14 Apr '08 16:12
Oh, I misunderstood the question. I thought you meant that the rectangles relate to each other as 1 to 1, 1 to 2 and so on by their square.
6. 14 Apr '08 16:21
Easier to say with a diagram, but split into the following blocks:

a1->d8 (2:1)
e1->f8 (4:1)
g1->h6 (3:1)
g7->h8 (1:1)

I got there by looking for +ve integer solutions to a^2 + 2b^2 + 3c^2 + 4d^2 = 64.
7. 14 Apr '08 16:38
* *|* * * * * *
* *|* * * * * *
-----------------
* * * * * * * *
* * * * * * * *
-----------------
* * * * * * * *
* * * * * * * *
* * * * * * * *
* * * * * * * *
8. 14 Apr '08 16:41
I think I used a simlar method, the areas of the simpler ratios were 1,2,3 and 4. I then multiplied these by 4 to make a total of 64 (doubling the sizes). This was a slight hunch because 64=2^6 so I assumed trebling would not help.
9. 14 Apr '08 16:56
Totally correct! Well done.
(and nice graphics, deriver69 !)
10. 14 Apr '08 18:21
Can you cut the chessboard into four congruent shapes, each of which contains exactly one of the following squares: a1, b2, c3, d4?
11. 14 Apr '08 18:31
Originally posted by Jirakon
Can you cut the chessboard into four congruent shapes, each of which contains exactly one of the following squares: a1, b2, c3, d4?
12. 14 Apr '08 20:08
Originally posted by David113
Lets see...

To cut the board into 4 rectangles, we must make 3 paralle cuts or 2 perpendicular cuts.

The first option is impossible since the 1:1 rectangle is a square.

The second option is impossible since the rectangle having no common edge with the 1:1 rectangle is a square, but 1:2, 1:3, 1:4 are not squares.

So there is no solution, even if cutting squares in parts is allowed.
Shame:'(
I forgot the third way, which gives a solution🙁
13. 14 Apr '08 21:45
Originally posted by mtthw
[fen]pRRRRRRR/pRrrrrrr/pRrPPPPr/pRrPppPr/pRrrRpPr/pRRRRpPr/ppppppPr/PPPPPPPr[/fen]
I like mine better.. 😛

14. 18 Apr '08 05:00
Originally posted by mtthw
[fen]pRRRRRRR/pRrrrrrr/pRrPPPPr/pRrPppPr/pRrrRpPr/pRRRRpPr/ppppppPr/PPPPPPPr[/fen]
Great. A swastika.
15. 18 Apr '08 05:01
Originally posted by Guych
I think you cannot do that. Maybe my calculations are wrong:
I suggest letting x be the largest part of the board. Then we can rite the following equasion:
x + 1/2x + 1/3x + 1/4x = 64
Multiply the equasion by 12:
12x + 6x + 4x + 3x = 768
x = 30.72
You're calculating areas. The problem is about ratios.