Replacing Tiles

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Posted · Report post

A rectangular floor is composed of whole square tiles. A diagonal line is drawn (corner to corner) and ruins some of the tiles.
(on a 2 x 5 floor, 6 tiles are ruined, on a 2 x 4, only 4 are ruined.)

2by5.gif2by4.gif

a) How many tiles are ruined on a 4 by 6 floor?
b) How about a 63 by 81 floor?
c) Generalize to an m by n floor.
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Posted · Report post

Assume the height H is greater than the width W.
Each column will have a number of damaged tiles
That number is CEILING (H/W).
Where CEILING means least integer greater than.
CEILING (3.5) = 4 and CEILING (2) = 2.
Multiply by the number of columns.

Ruined tiles = W x CEILING (H/W)

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Posted · Report post

m + n - gcd(m,n)

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Posted · Report post

a) How many tiles are ruined on a 4 by 6 floor?
b) How about a 63 by 81 floor?
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Posted · Report post

a) How many tiles are ruined on a 4 by 6 floor?
b) How about a 63 by 81 floor?

4x6

ceiling (6/4) = 2

2x4 = 8 ruined tiles

63x81

ceiling (81/63) = 2

2x63 = 126 ruined tiles

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Posted · Report post

a) How many tiles are ruined on a 4 by 6 floor?
b) How about a 63 by 81 floor?

4x6

ceiling (6/4) = 2

2x4 = 8 ruined tiles

63x81

ceiling (81/63) = 2

2x63 = 126 ruined tiles

6 x 4 should be 2

81 x 63 should be 135

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Posted · Report post

a) How many tiles are ruined on a 4 by 6 floor?
b) How about a 63 by 81 floor?

4x6

ceiling (6/4) = 2

2x4 = 8 ruined tiles

63x81

ceiling (81/63) = 2

2x63 = 126 ruined tiles

6 x 4 should be 2

81 x 63 should be 135

6x4 ruins at least the two corner tiles. Nothing else?

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Posted · Report post

a) How many tiles are ruined on a 4 by 6 floor?
b) How about a 63 by 81 floor?

4x6

ceiling (6/4) = 2

2x4 = 8 ruined tiles

63x81

ceiling (81/63) = 2

2x63 = 126 ruined tiles

6 x 4 should be 2

81 x 63 should be 135

6x4 ruins at least the two corner tiles. Nothing else?

i missed typed 6x4 ruins 8

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Posted · Report post

a) How many tiles are ruined on a 4 by 6 floor?
b) How about a 63 by 81 floor?

a) 4 + 6 - gcd(4,6) = 4+ 6 - 2 = 8

b) 63 + 81 - gcd(63,81) = 63 + 81 - 9 = 135

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