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bonanova
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clarification:

should we consider the circles separate when cutting? or should we treat the overlaps as part of the same cut?

(i.e. does this

post-19400-082184800 1298834502.jpg

require two cuts or one cut?)

also are we allowed non neat shapes? that is are we just looking for total area to be equal, or total area and nice complete circles?

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By four groups of equal area which do you mean?

Cut the shape into pieces A, B, C, D and E with the areas

A = B = C = D + E?

or

A = B; A = C + D; B = C + D; C = D + E

----

I can get the second scenerio using the edges of the circles as the cut line using circles with the radius of 1, 1, and sqrt(2). The first scenerio...I have no idea yet.

circles.bmp

C = E; A + B = C; A + B = E and A + D = B

*shrug*

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By four groups of equal area which do you mean?

Cut the shape into pieces A, B, C, D and E with the areas

A = B = C = D + E?

or

A = B; A = C + D; B = C + D; C = D + E

I can get the second scenerio using the edges of the circles as the cut line using circles with the radius of 1, 1, and sqrt(2).

The first scenerio...I have no idea yet.

circles.bmp

C = E; A + B = C; A + B = E and A + D = B

*shrug*

Like case 1.

Four disjoint groups of equal aggregate area.

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Is the shape one piece or three actual circles?

If one piece, are the circles drawn for us along with the centers?

Three actual circles.

I originally heard it as three pancakes to be divided equally among four

children for breakfast, by cutting them into five pieces.

One child obvious gets two pieces, the other

three get one each of the remaining pieces.

It's not that difficult - maybe an aha! will strike ... ^_^

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That makes it a bit easier! I was trying to work with one shape.

Total area = pi( x^2 + y^2 + z^2 )

Goal = pi( x^2 + y^2 + z^2 ) / 4

Since the diameters form a right triangle we know the radii(?) also form a right triangle.

so x^2 + y^2 = z^2

Goal => pi( z^2 + z^2 ) / 4 = > 2 * pi( z^2 ) / 4 => pi( z^2 ) / 2 or half of the biggest "pancake".

Now we have two of our fourths.

Next step is to cut the middle "pancake" so that the piece taken off when added to the small "pancake" will make them equal.

Best I can think of is using the smallest circle as a template place so the edges are touching. Cut along circumference until you reach the straight line made by the centers.

If six pieces were allowed we would just have to cut the two smaller circles in half.

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That makes it a bit easier! I was trying to work with one shape.

Total area = pi( x^2 + y^2 + z^2 )

Goal = pi( x^2 + y^2 + z^2 ) / 4

Since the diameters form a right triangle we know the radii(?) also form a right triangle.

so x^2 + y^2 = z^2

Goal => pi( z^2 + z^2 ) / 4 = > 2 * pi( z^2 ) / 4 => pi( z^2 ) / 2 or half of the biggest "pancake".

Now we have two of our fourths.

Next step is to cut the middle "pancake" so that the piece taken off when added to the small "pancake" will make them equal.

Best I can think of is using the smallest circle as a template place so the edges are touching. Cut along circumference until you reach the straight line made by the centers.

If six pieces were allowed we would just have to cut the two smaller circles in half.

That's it. ;)

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