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.)

 1. Cut the 5" bun in half (we still know where its centre is).<o:p></o:p>

2. Lay the radius of one half of the 5" bun across the 4" bun such that it forms a chord with the 4 " circumference.<o:p></o:p>

3. Mark the endpoints of this chord.<o:p></o:p>

4. Make two cuts from the the centre of the 4" bun to these points; this sector, along with the 3" bun will form the 4th share.<o:p></o:p>

5. And there are also 5 pieces (declared once before to be the minimum).<o:p></o:p>

AND the Proof [??]<o:p></o:p>

1. A triangle is formed having 2 sides of 4 units and an included angle of 7/32 of 360 Deg  = 78.75 Deg.<o:p></o:p>

2. The angle is bisected forming two triangles. The Opposite side of one of these equals 4 * sin (78.75/2)<o:p></o:p>

3. The 3rd side of the larger triangle is therefore 8 * sin(78.75/2) = 5.075......<o:p></o:p>

NOT BAD, Hey?

BUT:<o:p></o:p>

Using a triangle of 2 4" radii and a 5" side, the sine of half the included radial angle is 0.625.<o:p></o:p>

The central angle is:<o:p></o:p>

2 * arsin(0.625) = 2 * 38.6821.....  = 77.36...<o:p></o:p>

<o:p></o:p>

Oops! There is a difference of approx 1.4 Deg between these two calculated positions.<o:p></o:p>

I hope that Gamer has not used a 4,4,5 triangle as the basis of the solution.<o:p></o:p>

<o:p></o:p>

Maybe it's hint time again.<o:p></o:p>

Posted by brianjn on 2004-05-16 04:34:06