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Earlier today, I asked you the following four riddles:
1. You have a 3 x 3 x 3 inch cheese cube that you want to cut into 27 smaller cubes of 1 x 1 x 1 inch, as shown below. If you have a straight knife, what is the minimum number of slices you need to do it? You are allowed to rearrange the pieces after each slice.
Does it make a difference if you have a really, really elastic cheese?
Solution
With a hard cheese, the minimum number of slices is 6, which are the slices along the dotted lines in the photo. It's two across each set of parallel faces. Consider it as follows: if you plan to cut the cube into 27 smaller sub-cubes, these subcubes will be 9 on the lower horizontal slice, 9 on the central horizontal slice, and 9 on the horizontal slice. higher. The sub-cube in the center of the middle slice requires six slices because it has six faces and each of these faces must be cut in a different slice. (All other sub-cubes have a face that is part of an outer face of the original cube, so these sub-cubes require fewer slices, but the middle one requires six slices).
In the case of a really very elastic cheese, it was possible to fold the cube so that only one slice could cut two sides (these faces would be parallel once the cheese rolled out), then theoretically you only need three slices: one slice for each set of parallel faces. You will need to fold the cheese in three different (perpendicular) ways for each of these slices. Messy!
2. A cheese cube is divided into 27 identical "sub-cubes", as above. A mouse starts at one of the sub-cubes in the corner and makes its way into the cheese (without ever crossing the same sub-cube more than once). Whichever sub-cube the mouse is in, it can only move in a sub-cube horizontally or vertically. he.
Show a way through the cheese that goes through each sub-cube and ends in the central sub-cube, or show that such a path is impossible.
Solution
It is impossible. Imagine that the 27 sub-cubes are colored as in a 3-D checkerboard, so that each white cube is only vertically and horizontally adjacent to the black cubes, and vice versa. That the corner cubes are all black. The cube therefore contains 14 black sub-cubes and 13 white sub-cubes. The subcube of the center will be white.
If the mouse can move only horizontally or vertically, the sub-cubes through which it passes must alternate in black and white. Thus, if the first cube visited is black, all the odd cubes that it visits will be black and all the even cubes will be blank. It is therefore impossible for the 27th cube to be white. The central cube (white) can not be the final cube visited.
3. A cheese cube sits on a horizontal table. If you cut it in half with a vertical cut, halfway and parallel to two of the sides, the cross section of the cut will be a square.
How do you cut a cube into two equal parts, so that the cross-section of the slice is a hexagon?
Solution
The fact that you can get a hex in a cube is a nice surprise. You cut it like this:
Reader Rachel Baker made one of butter (she was short of cheese):
4. How to cut a cheese cube into six pieces of identical volume with only three cuts?
Solution
The nicest symmetrical method is to reduce the diagonals, as Rachel Baker shows here with a parmigiano reggiano cube. The slice 1 is vertical from the diagonal on the upper face to the diagonal on the underside.
Slices 2 and 3 are located between the diagonals of the other pairs of faces. Each of the three diagonals must meet at one of the vertices of the cube.
The six pieces are all tetrahedra – four-sided objects where each side is a triangle, here isosceles triangles. The pieces have an identical volume but they are not, they have two orientations. The impressive cut of Rachel is worth the price of one of my books today.
Rachel is not the only solution. A simpler method is demonstrated here by Ed Southall. Easy nerdy!
Ed Southall
(@Solvemymaths)
In fact, Ed has come up with many creative solutions. Just as there is more than one way to peel a cat, there is more than one way to slice a cheese cube.
Ed Southall
(@Solvemymaths)
Ed Southall
(@Solvemymaths)
Ed Southall
(@Solvemymaths)
Ed is familiar with geometry and canapés: he is the author of the fantastic Geometry Snacks.
I hope you enjoyed the riddles of today. I'll be back in two weeks.
I put a puzzle here every two weeks on a Monday. I am always looking for great puzzles. If you want to suggest one, write to me.
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