I needed this illustration to understand. Thank you, astro!
Now a more interesting problem would be how to do those four cuts so that the areas of the slices are the least unequal.
I can suggest one measure that has the advantage of simplicity, even if it’s not the ‘best’. Difference between the area of the largest piece and the area of the smallest piece.
The best anthologies of Martin Gardner’s works that are relevant to this thread are The Colossal Book of Short Puzzles and Problems and The Colossal Book of Mathematics. There are a lot of other anthologies of his works, but they are shorter and not quite as consistently good. The two I list collect the best of the Mathematical Games columns.
Yeah that and the first sentence of the OP states:
and the fifth sentence states:
Yeah, but my cuts are straight. What’s the problem?
Chronos’ solution works in the 2d plane, as well. Just rearrange the pieces in the plane so that one straight cut goes through all of them on each cut.
Because that’s literally “thinking outside the box”
Ahah! You are all missing out on another interpretation of the puzzle. “Cuts” can be a verb, as well as a noun. If I interpret “4 straight cuts” as a verb, I can slice the pizza (without folding it, or making cuts in 3D) into as many arbitrary pieces as I would like…
… by using a multi-bladed knife with parallel blades!
I prefer cutting my pizzas into 4 slices instead of 8, because I can’t eat 8 slices.
This thread still going?
Imho, the answer as given by Gardner is correct. Pizza, cut horizontally, would not be pizza anymore. On the other hand, other things can be cut horizontally and still be what it is, e.g. cake.
I misread your solution, my apologies.:smack:
This does makes sense, you could get 16 pieces and satisfy all the conditions.
Most likely, Martin Gardner meant it as a problem to be solved without rearranging the pieces .
I think you mean the same thing as my first suggestion, max area - min area, unless I don’t quite understand what you’re saying.
But I’m still interested in the problem how to make the pieces as nearly equal as possible and can zero difference be achieved. Anyone have any detailed thoughts on that?
If it were possible to cut the pie into eleven equal pieces, each line (slice) would divide the pie into integer-sized pieces. (There can be no fractions because it is impossible to recombine the fractions into a whole piece.)
So let’s divide an imaginary pie having an area of 11 into four pieces, using two of our four slices.
We have two slices remaining. Any single region may be sliced into four pieces by having those two slices intersect within that region. So the largest piece that we can have at this stage has area 4, if we want to have eleven equal pieces at the end. However, our two slices can only slice a single region into 4 pieces – there can’t be two regions with area 4 at this stage. (Slicing into 4 requires that the two remaining slices intersect, and in flat 2-d geometry, two non-collinear lines can only intersect at a single point.)
So the groups of four integers that total 11, having a maximum value of 4 for a single entry, are:
4, 3, 3, 1
4, 2, 2, 3
and 3,3,3,2
These are the possible areas of our pie pieces at this point.
Looking at the last possibility - 3,3,3,2 - it should be obvious that this can’t be sliced into 11 using only two slices. One of the (straight line) slices would have to intersect all four pieces, bisecting the smallest (area 2) and removing exactly 1 from each of the other 3 areas. This is impossible.
The first option - 4,3,3,1 - is similarly flawed. Our single (straight) slices would have to intersect each of the three largest pieces twice, and would have to intersect in the largest piece, dividing it into 4 equal sized pieces. Geometrically, this won’t work. Straight lines cannot accomplish this subdivision.
Similarly, the final option - 4, 2, 2, 3 - requires that the lines which split one area into 3 equal parts somehow intersect to split the region on the opposite side of the circle into 4 equal parts, while each at the same time transsecting an area of 2 exactly in half.
It’s not rigorous … but when you sketch the little diagrams for each of them, it should be obvious that division into 11 equal parts with 4 straight lines is impossible.
As to the best-case division, I’m not certain how to approach it.