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PuzzleScot
Daedalian Member

 Posted: Tue Jan 19, 2010 11:31 am    Post subject: 1 This is from a Scottish Schools Maths contest: Three large pancakes, each of the same thickness, are to be shared equally among four people. The diameters of the pancakes form a pythagorean triple. Without measuring, show how to cut the pancakes to make a total of just five pieces so that each person gets exactly the same amount.
Zag
Tired of his old title

 Posted: Tue Jan 19, 2010 1:14 pm    Post subject: 2 Say their diameters are a, b, and c, from smallest to largest Their combined areas are A = pi * (a 2 + b 2 + c 2 ) Since they are a Pythagorean triple a 2 + b 2 = c 2 , so c is half of the total area. You can cut c pancake in half and give a half to 2 people and they will be getting the right amount. So the only question is how much of b pancake to cut off to make it the same quantity. Lay the a pancake on top of the b one, with edges tangent. Consider the portion of b you can see -- i.e. which is not covered by the a -- we want to cut away half of that. So cut around the a pancake from the point of tangencency to the point directly opposite, then cut across the rest of the b pancake along the continuation of that same diameter of a.Last edited by Zag on Tue Jan 19, 2010 1:33 pm; edited 1 time in total
PuzzleScot
Daedalian Member

 Posted: Tue Jan 19, 2010 1:30 pm    Post subject: 3 That's pretty much the same answer I arrived at. You have to assume that the diameters are easily available on each pancake, because I guess folding each of the 2 larger ones in half would break them! Then align the diameter of the smallest on the diameter of the medium one (doesn't have to be tangential as such, just entirely within!). Cut along the visible diameter of the medium sized one, and around one side of the smaller one. Nice little puzzle, I thought worth sharing...
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