An interesting and related problem, from the Sydney University Maths Competition [0] 2007 [1] goes as follows:
The sisters Alice, Bess, and Cath are fighting over a triangular pizza, which may be
imagined as a triangle PQR.
Their father David proposes the following procedure for sharing it between the four
of them. Alice will select a point A on the edge PQ, then Bess will select a
point B on the edge PR, then Cath will select a point C on the edge QR.
David will then cut the pizza along the lines AB, BC, and AC, and take the centre
piece ABC for himself, leaving three corner pieces (some possibly empty, if
endpoints of edges have been chosen).
The sisters will then either all take the corner piece to the left of the point
they selected, or all take the corner piece to the right of their point; Alice
(as the eldest) will get to choose left or right.
As everyone knows, each sister will make her choices purely to maximize the area
of her own share, except that Alice and Bess, if their own shares are unaffected,
will act to the advantage of the youngest sister Cath.
If they all reason perfectly, what will they do?
The constraints on this problem lead to an interesting result, but the fun is in teasing out a logical argument to why that result is the case, so I will leave it for those who want to try it.
Applying the concept from this problem to the one at hand, try the following scheme:
Alice picks an arbitrary point A on the pizza.
Bess picks a second point B, and Cath then picks a point C.
Cut the pizza from the centre to those points. Alice then picks the piece either to the left or right of her point A. Everyone else has to take the piece in the same direction of their point.
Under this scheme, if Cath makes a bad choice she will get the smallest piece, and either Alice or Beth gets a bigger piece then the other depending on which way Cath's choice was bad.
If Beth makes a poor choice, assuming Cath makes the best choice, she will always get the smallest piece.
If Beth and Cath conspire against Alice they can force her to have a tiny piece, however one of them will have an even smaller piece. If each instead tries to get the most for herself then it will work out roughly even.
Comments
An interesting and related problem, from the Sydney University Maths Competition [0] 2007 [1] goes as follows:
The constraints on this problem lead to an interesting result, but the fun is in teasing out a logical argument to why that result is the case, so I will leave it for those who want to try it.[0] http://www.maths.usyd.edu.au/u/SUMS/
[1] http://www.maths.usyd.edu.au/u/SUMS/sums2007.pdf
Applying the concept from this problem to the one at hand, try the following scheme:
Alice picks an arbitrary point A on the pizza.
Bess picks a second point B, and Cath then picks a point C.
Cut the pizza from the centre to those points. Alice then picks the piece either to the left or right of her point A. Everyone else has to take the piece in the same direction of their point.
Under this scheme, if Cath makes a bad choice she will get the smallest piece, and either Alice or Beth gets a bigger piece then the other depending on which way Cath's choice was bad.
If Beth makes a poor choice, assuming Cath makes the best choice, she will always get the smallest piece.
If Beth and Cath conspire against Alice they can force her to have a tiny piece, however one of them will have an even smaller piece. If each instead tries to get the most for herself then it will work out roughly even.