NeoPets Riddles (Lenny Conundrums) and Answers Here

i just gave a full, and correct, explanation for this riddle. simple math really, easy.

the right answer if 2500
goodnight all

oh, youre right, eliz.
i guess it would be 2500 i think.

I just draw it up to get a feeling of how it's supposed to look like and to get an aprox. value and I got something of "less than 1850" Then I saw lem's post and I was going along the same way with my thinking, but wasn't quite there yet with my math. So thank you for posting. I guess I was not that bad with my aprox. then.

Let's take a pizza and cut diagonally with respect to original 50x50 box....

Let's take a pizza and cut diagonally with respect to original 50x50 box.
We now have half a box free of pizza. So we can shrink it from two sides until it touches the corners. Let's figure out by how much:

The box diagonal is sqrt(2x50^2) = 70.71068
From corner of box to corner of pizza = (70.71068-50)/2 (there are 2 corners) = 10.35534

thats a diagonal of remaining square at each corner. It's side is sqrt(10.35534^2/2) = 7.32233

So we take this much out of each side, now side of a box is 50-7.322233=42.67767

And the area of this box is 42.67767^2 = 1821.383 which rounds to 1821.

lem wrote:

Let's take a pizza and cut diagonally with respect to original 50x50 box....

Another method:

Take the half-circle, drawn with the straight side as a diagonal (rather than horizontal or vertical) line. Draw a square around this, with the squares sides touching the ends of the straight side and two points on the curved portion. Very crudely, the picture looks something like this:

Code:+----------*-------+
| * * |
| 50 * * |
| * *
| * *
|* *
* *|
| * * |
| * * |
+------***---------+

From the midpoint of the diameter (the straight side of the half-circle, labelled "50" above), draw a square by extending a vertical line down to the bottom of the large square and an horizontal line across to the right side of the large square.

Since the sides drawn go from the center to the larger square, where the circle is tangent (and thus perpendicular) to the larger square, this small square is indeed a square (all four corners are ninety degrees), and the sides must (being radii) have length 25.

From the horizontal line in the smaller square, draw a vertical up to the point where the straight side of the half-circle intersects the larger square. This forms a 45-45-90 triangle with hypotenuse 25 (being half of the straight side with length 50). Then the height of the triangle must be 25/sqrt[2].

Adding, we get that the side of the larger square must be 25 + 25/sqrt[2], so the area must be (25 + 25/sqrt[2])^2, or about 1821.3834764....

Assuming that this is the configuration that the "Lenny" authors intended we get an area of 1821, when rounded to the nearest whole number.

Please check my work.

Eliz.

lem and stapel both have good solutions. The one remaining assumption that needs to be addressed is that the best configuration is when the cut edge of the half pizza is turned at 45 degrees to the edges of the square box.

Assume the half pizza is turned so that the cut edge is somewhere between 0 and 90 degrees from the horizontal. (0 degrees means it is at the top with the round part below, while 90 degrees means the cut edge is on the left side with the round part to the right.) In either of these cases, we will require a 50x50 box to match the length of the cut edge (and the original whole pizza will fit in the same box).

For any intermediate position, let A be the left/bottom end of the cut edge, and B the right/top end. Let P be the center point of the cut edge. For any angle, the closest possible positions of the left and top sides of the box will be where they touch A and B respectively. Let a be the distance from P to that left line and b be the distance from P to that top line. Then a^2 + b^2 = 25^2 = 625. (^ indicates exponentiation)

The minimum distance for the bottom and right sides of the box will be 25 cm from P (the points where the semi-circle reaches out the farthest). So the minimum width of the box will be a + 25 and the minimum height will be b + 25.

Since we are required to have a square box, we need the maximum of these two values: max(a+25, b+25). Now a starts at 25 at 0 degrees and decreases to 0 at 90 degrees, while b goes from 25 down to 0. What we want to find is the minimum value of max(a+25, b+25), or equivalently, the minimum value of max(a, b), where a and b are within this range. a^2 + b^2 = 625 describes a circle of radius 25 and the minimum of max(a,b) in the first quadrant of the circle will occur when a=b, at 45 degrees. Hence, the appropriate position of our half pizza is with the cut edge turned 45 degrees and the solutions given do provide a minimal size for the square pizza box.

And yes, I know it is 2 days after the new Lenny came out and I'm just getting around to it, but life is busy. Someday my avatar will come.