Here is a good shape problem I heard about recently:

There are 3 buns with sprinkles on the top that 4 people want to share. The buns have a radius of 3 inches, 4 inches and 5 inches, and although the people know where the center of each bun is, they don't know anything else about the buns, and all they have is a knife to divide the buns.

What is the fewest number of pieces required to let each person have the same area of bun? (Note that each cut must be from top to bottom; horizontal cuts would result in uneven sprinkle distribution. The cuts don't need to be straight.)

I re-thought the phrase "The cuts don't need to be straight." I first went looking for overlaps of segments but that was too complicated.

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I thought that a few people had already mentioned making circular cuts but wasn't sure how many and as I didn't have the time I composed the following as my response only to find that Game had posted the solution.

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Anyway here is my interpretation:

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I am disregarding Pi for the sake of simplicity; it is  a constant equally applied to all calculations and so can be eliminated.

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The 3” bun, 4” un and 5” bun therefore have respective areas of 9, 16, and 25 units being different from each other by 7 and 9 units in that sequence.

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Now the total area is 50 units.

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Required is 4 equal shares, ie: 12.5 units.

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Two shares are readily formed by dividing the 5” bun diametrically.

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As the 3” bun needs a share of the 4” bun, a 5^th piece needs to be created.

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Using a diameter of the 5” bun, mark two diametrically opposite points on the circumferences of both the 3” and  4” buns [maybe their respective diameters could be marked as such an action would not cut out any extra pieces].

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Now placing the 3” bun atop the 4” bun, with both centres directly above each other (we still know where the cetres are) and aligning the diameter markings, and “annulus” of the 4” bun will be noticed.  It’s area is 7 units, being the difference between 9 and 16; but we only want half of that.

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Making 1” cuts at the ends of the 4” bun’s diameter, and joining them by using the circumference of the 3” bun, half of the annulus of the 4” bun will be removed.

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And, Voila!  We have 4 equal shares of 12.5 units, the latter two being the 3” bun along with the half annulus and the 4” bun with half of the annulus removed. And we have a minimum of 5 pieces.

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This has all been accomplished by using only parts of the buns and measures and templates, and the knife as the given cutting instrument.

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(Yes, it does take a certain amount of guesswork when one sets out to remove a 7/32 sector from the 4” bun)

Posted by brianjn on 2004-06-14 03:27:31