Impossible = Possible
The dotted red lines show the impossibility proof at work. In figures that are impossible (Figures 1, 2, 6, 8, and 9), the dotted red lines fail to converge at a point, or in the case of Figure 9, the dotted red lines show that the two creases between the two blue surfaces do not align.
Figure 3 is possible because the intersection of the three lines of the planes does converge at a point. Figure 4 is possible, but only if the three blue areas that seem to be continuations of the same surface jog so they do not lie on a single plane. Figure 5 is possible if the center hole that seems to be floating in space is in fact a hole in the back wall of the well, shown here in blue. Figure 7 is possible if the two short segments that jut out to the right are at a much steeper angle than they appear to be. The Huffman impossibility proof does not work on Figure 7, because there is no cycle of three surfaces that intersect in pairs, as there is in Figure 8.
David Huffman coined the term "improbable figure" for a drawing such as Figure 5, which, though it appears impossible, actually can be built in three dimensions. He also discovered figures that appear to be possible but are not, such as Figure 6. If we assume three planes meet at each vertex, then the left- and right-most edges must belong to the same planar surface that forms the back of a truncated triangular pyramid. But the lines of intersection between sides of the pyramid don't converge at a point, as shown by the dotted lines. Black = White 1.
lead = top = capital = gold 2.
hot = biting = cold 3.
slow = moderate = steady = fast 4.
matter = weight = power = energy 5.
necessity = want = fancy = invention 6.
true = perfect = ideal = unreal = false 7.
small = mean = formidable = mighty = large 8.
positive = cold = impersonal = disinterested = negative 9.
solid = thick = fat = oily = smooth = fluent = liquid
There may be better solutions. The shortest possible chain is a single word that is its own antonym, such as "cleave" (which can mean "to adhere" or "to separate") and "overlook" ("to ignore" or "to observe carefully"). 1 = 2 1.
The mistake is in the last step. Since a
you cannot divide both sides by (a2
because that would be dividing by 0. 2.
The mistake is in the very first line. In order for an equation to be valid, both sides must have well-defined values. But if you add up the first few terms of the infinite series at right, you will see that its value oscillates between the values 1 and 2 and does not converge toward a single value the way a series such as 1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . converges toward the value 2. So we've already erred when we say that the series 1 + 1 - 1 + 1 - 1 + . . . equals itself.
1 + 1 = 2
1 + 1 - 1 = 1
1 + 1 - 1 + 1 = 2
1 + 1 - 1 + 1 - 1 = 1
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