No Way!!! The brownie cutting methods described in the “official answer” to “Cutting the Holey Cake” are totally bogus, and do not necessarily give an equal division.
First of all, on the “hard answer” cutting the brownie on a horizontal plane, parallel to the bottom, would not do it. Any brownie, or other cake, would not be perfectly flat. There would be some tilt, and rounding of the top… so cutting horizontally would not result in the true equal cutting in half. And besides, as pointed out, one gets the rounded top half, while the other gets the flat bottom, which do not look at all alike, so there would definitely be a “kiddie fight” no matter if it is equal or not.
Second method, using the centers of the inner and outer rectangle. This would only work if the two shapes are PERFECT rectangles. I dare say that hubby cutting a quick “rectangle” would not get it perfectly, and the geometric division method does not give exact halves.
Besides, both of these methods are much more difficult than my two suggestions, which result in perfectly equal division, completely independent of shape of the brownie or the “sampling hole” in the middle.
Method #1: Get a straightedge, such as a yardstick. Place it horizontally, on a table, with one edge up. Place the brownie on it, so that it balances, remaining horizontal. The balance line is the equal-cut line.
Method #2: Put a string through the brownie hole. Hold both ends of the string together, and lift the brownie by the string. Where the string touches the brownie, is the equal-cut line. Just extend the line to the bottom half of the brownie.
Neither of these “methods” are physically possible. I believe they are correct with respect to geometry, and would be fine if the object was a solid material like plywood or sheet metal, etc. But this puzzler is about a pan of brownies, ie, cake.
Any brownie which would remain intact when balanced on an edge, or when held from a string, would surely be inedible.
In the OP’s defense, I vaguely recall somewhere in my past having seen brownie results which might have withstood the proposed test methods, and in despair I may have tried to gnaw into the thing, but I’m sure that attempt failed. ;~)
Balancing on a straight-edge does not work. Being balanced does NOT mean both sides weigh the same. Balancing is a function of weight AND distance. If the halves are not the same shape mirrored, they will not weigh the same. How far the hole is from the balance line affects the difference between the balance and the weight.
Playing the “perfection” card is useless, as these are inconsistant BROWNIES. Besides, these are for brothers, where both halves need to be bigger than the other.
When I make brownies, they’re flat on top, not domed. The trick is to shake the pan after you pour the mix in, which makes it self-level. I tend to make thinner brownies (so that you can cut a larger piece as a base for vanilla ice cream, of course) so you don’t get the doming effect you get from thicker ones or cakes.
I think the brothers are right (for once) geometrically speaking. If you draw a line that goes through both the center of the small and the large rectangles it divides both in half. The resulting pieces will have equal-area holes cut out of them, and will be equal area overall. This is not the balancing problem that cropped up on the previous gas tank puzzler.
On the radio, the answer was said to have come from a mathematician. Clearly, this is not the case: a mathematician would have recognized that this is a three-dimensional problem with an infinite number of solutions, not just the two (one with a vertical cut and the other with a horizontal cut) that were given.
As a 3-D shape, the center of the brownie (as a rectangular prism), is the point that is halfway between the top and bottom, left and right, and front and back. The center of the missing piece is found the same way. Looking from above, these centers are just where the centers of the brownie and the hole were described in the original solution, but in 3-D, they are inside the brownie, halfway between the top and the bottom.
Now, any straight (planar) cut that passes through the center of the brownie and the center of the hole will divide the brownie in half. The two solutions given, a vertical cut along the line connecting the centers, and a horizontal cut at half the thickness, are two examples, but with careful aim, one could tilt the knife at any angle, make a straight cut that passes through the two center points, and still have two equal halves.
If anyone can supply me with a laser-guided compound miter saw, I’d be willing to try some of these other solutions.
@AETA - You’re correct, but maybe the mathematician simplified it for C&C!
And @RickNY - balancing doesn’t work, as others have said. Take a ‘T’ shape, with the horizontal and vertical parts equal in area. If you balance it across the vertical part parallel to the horizontal part of the ‘T’ the two areas won’t be equal.
A of remaining brownie = A of pan minus A of hole.
On either side of line constructed as Ray says, since the line divides each by 1/2, each area is 1/2 A of pan and 1/2 A of hole respectively
So A of remaining brownie on either side of line is 1/2 A of pan minus 1/2 A of hole
and by the distributive property
A of remaining brownie of either side of line is 1/2 * (A of pan minus A of hole)
So the line divides the remaining brownie area equally into two parts.
this wouldn’t be limited to a rectangular hole to work. There are many shapes for the hole that would work. For example, a circle would work. As would a parallelagram I expect. The shape of the hole just needs enough symmetry so that any line through a particular point divides it into two equal parts, irrespective of the orientation of the line. I wouldn’t be surprised if there were a mathematical term from the field of topology for shapes with that property.
@texases - Have you ever heard a mathematician simplify anything?
@GeorgeSanJose - I think the mathematical term is from group theory: C2: any 2D shape which is unchanged after a 180 degree rotation will have the property that a cut through the center of rotation will divide the shape into two identical shapes with (obviously) equal areas.
Never mind mathematics. What you need for this puzzle is a bit of parenting experience. Give the knife to the older kid and tell him or her to cut it in half, and that the other child will get to choose which piece he or she gets. The brownie will be cut perfectly in half every time.
@AETA
Media analyst, who happens to be an ex-physicist, listens to show and immediately realizes, like AETA, that there are infinite solutions, a half circle’s worth, of which the above two are but special cases.
I don’t think it is enough to be a mathematician or a physicist: The kind of person who is likely to think of these other solutions (though I can’t say I thought of them or looked for them, they sort of just jumped out at me) is much more a geometer with familiarity with thinking in > 2D.
That can work even better than “identical” division if the kids have different ideas of the best parts of the brownie. Say one just LOVES the chewy, crusty edges and the other CAN’T STAND them. The kid with the knife could cut a big rectangle out of the middle, and they’re both happier than they would have been with a simple straight cut.
(There are ways to generalize that to more than two people. That’s the extent of my knowledge; “Google University” will undoubtedly reveal more.)