Can anyone help with this puzzle that is confusing me?

Question

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. Explain how to cut the pancakes to make a total of five pieces so that each person can get the same amount. Explain such reasonings!

Answer 1

The key is
a^2 + b^2 = c^2
So the large one is half of the total
You cut that in half.

You cut up the middle to make up the difference of the small one.
eg. 345, you cut up the 5 in half, you cut not quite a quarter from the 4.

Answer 2

A pythagorean triple is an a^+b^=c^ (^=squared) such as 3, 4, 5, so assume the pancakes are 3, 4, and 5 inches in diameter. You'll have to figure out the total volume of pancake (2-pi-r?) and then divide by 4 to see how much each person gets. Working backward from there may reveal an easy way to cut 5 equal pieces.

Answer 3

To cut three pancakes of equal sizes into five equal pieces is impossible to share four ways. Each person would have 3/4 of a pancake so no pancake could be left whole. As there are three you would have at least six pieces. I would suggest cutting 1/4 from each. That leaves three pancakes of 3/4 size and three pieces of 1/4 size, six in total.

However the clue is in the Pythagorean triple.

This means that the pancakes are not of equal size, except for thickness, and when placed next to each other would have diameters of the proportions 3 : 4 : 5 so the areas of each would be calculated using the formula Pi r squared. So the answer is that the smallest pancake would equal 1/4 of the total area of the pancakes leaving two pieces to be cut from the two larger ones. You would then have five pieces. One whole one, two that have been pared and two pieces that make up the same area as the smallest. The amounts to be cut off the two larger ones would, for the second largest 1/3 the area of the smallest and off the largest 2/3 the area of the smallest.

Another solution is to get the smallest pancake and place it on top of the others and then cut around it. You would then have three pancakes the same size as the smallest and two pieces that, in area, would equal the size of the others. Five pieces all for four people.

Answer 4

Trying the simplest (best known) Pythagorean triple, i.e. 3,4,5 for the diameters, we have areas of 9, 16 and 25 x Pi/4. We can now ignore the Pi/4 as a constant and work with the simple ratios:

The total area is 9 + 16 + 25 (= 2 x 25) = 50
so each of 4 equal shares is 12.5.

If you look at the areas of the 3 given pancakes, you see immediately that the largest one (25) must be cut in half. Then an area of 3.5 must be cut off the 2nd one (16) and added to the smallest (9), giving 16 - 3.5 = 12.5 and 9 + 3.5 = 12.5

5 pieces and 4 equal shares. QED


[When cutting the 2nd pancake you need to cut a segment with area of 3.5xPi/4 from the whole 16xPi/4. i.e. a segment with subtended angle = 360x3.5/16 = 78.75 degrees]
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Answer 5

if diameters are a, b and c with a < b < c, then
a^2 + b^2 = c^2

if A_n is the area of pancake's top surface with diameter of n, then
A_a = π(a/2)^2
A_b = π(b/2)^2
A_c = π(c/2)^2

from a^2 + b^2 = c^2,
A_a + A_b = A_c

clearly both sides of the equation hold as 1/2 of the whole. so A_c /2 will give you 2 of the 4 portions. then cut away an area of (b^2 - a^2)/2 from A_b. give the larger of A_b cut to one and A_a plus the smaller A_b cut to another.

Answer 6

The key is
a^2 + b^2 = c^2
So the large one is half of the total
You cut that in half.

You cut up the middle to make up the difference of the small one.
eg. 345, you cut up the 5 in half, you cut not quite a quarter from the 4.

Answer 7

Why is ther only four people and u want it divided into 5 pieces? does someone get 2 servings? Can i?

Answer 8

Why don't they just order 2 more pancakes. Math is stupid.

Answer 9

errrrm
everybody gets 3/4 each

Answer 10

I'm confused! You said four people, and then you said to divide it into five pieces. What??