Stop me if you’ve seen this before:
You might think whoever designed this problem fudged the dimensions so that the figures only appear duplicated; maybe the pieces above aren’t identical to the pieces below. What happens is rather that the red and green triangles’ hypotenuses only appear to linearly flow into each other. Some points get coverage only in one figure but not the other, so you’ll know some trickery is afoot if you look long enough.
Crucially, this is all possible even if the top and bottom shapes contain the exact same pieces (with no subtle deformations between the two figures), best illustrated by changing them to rectangles:
What’s the meaningful difference between this rearrangement and the one with the triangles? Answering that question in detail may help you understand what’s really at play. After all, if you move some pieces around on a board, why wouldn’t you create empty space where pieces used to be? What assumptions are you making about the original figure that cause you to feel a loss of sensemaking? For more elaboration on the resolution of this problem, follow the footnote.1
Here’s another; start with b=a and follow steps 1-6:
For those who’ve never encountered this, the trick is that step (4) uses division by 0, which is undefined, so the next steps fail.
If you’re a practitioner of errorology, the first “paradox” certainly has more meat on the bone. But it basically reduces to the second one; both try to trick you by downplaying a key detail which would sever the connection between two ostensibly equivalent things. Based on your starting assumptions, the map of reality doesn’t quite match up with the land, because it’s just a lie by omission about those assumptions.
Sometimes when I’m being lied to in the real world, before I even have an awareness about it, I feel a subtle collapse in sensemaking. Similar to a math problem, a story also contains a series of logically dependent steps. If one part is omitted or distorted, the whole story might just seem off. If you’re attuned to the logical connections between things, being told a lie might result in confusion, because other contingent details may no longer line up with the official facts.
I direct your attention now to the Fox News debate controversy. Some brief background: recently, Trump backed out of a planned debate with Harris on ABC, instead saying he would only debate her on Fox a week earlier than scheduled (with some rule differences). This has had the usual life cycle of a story which the left and right use to taunt and bludgeon each other, distracting us from the important details.
Here’s a more thorough timeline:
July 21: Joe Biden withdrew his candidacy. Trump had originally agreed to debate Biden on ABC on September 10, fueling speculation about whether he would still debate the new nominee.
July 24: Fox invited both candidates to a debate on September 17.
August 2: After some waffling, Trump withdrew from the ABC debate, claiming to have made an agreement with Fox to debate Harris on September 4 instead:
August 3: Fox eventually reports on this supposed agreement.
August 8: It’s reported that both candidates have agreed to debate on ABC on September 10 as scheduled, with Trump announcing that other debates have been confirmed. Some reported on Trump’s claims more credulously than others.
While the surrounding discussion has focused heavily on Trump’s cowardice and antics, the real story is one of gross journalistic malpractice. No matter how you interpret these events, this represents a new ethical low for Fox. I say this as someone invested in the preservation of Fox’s integrity, since I read them periodically as part of my information diet.
I think it should be scandalous for a presidential candidate to unilaterally negotiate the terms of a debate with a friendly news network. The previous terms were seemingly changed at his behest, without the Harris campaign’s consultation. This legitimizes Trump’s tantrum against one of Fox’s rivals, while blurring the lines between reporting on events and influencing them2.
But I’m not even sure that’s what happened – if so, why was it Trump who broke the story? I’m not normally conspiracy-minded, but I think Fox hadn’t finalized any agreement with Trump. I think he was just bullsh*tting (or bluffing?), and they haven’t repudiated for fear of embarrassing him. Beyond the fact that none of the details make any sense, Fox have been evasive in their reporting around the Trump announcement. Here they refer to the debate that would be held on the 4th, not one that will be, with this additional sleight of hand:
Former President Trump announced Friday night that he has agreed to Fox News' proposal for a debate against Vice President Kamala Harris.
Trump said the debate would be held on Sept. 4 in Pennsylvania …
Um, yes, but what is Fox saying? If you read the rest of the article, or this one produced the same day, there’s a conspicuous absence of confirmation or denial of any supposed agreement. Carefully read them to verify my assessment, as if you’re reading the fine print of a contract.
I highlight this deceptive snippet from another more recent article:
FOX News Media sent formal letters to theTrumpand Harris campaigns last month proposing a debate to take place in September in Pennsylvania. Trump said Thursday he would debate on Fox on Sept. 4.
Fox is playing shell games with us. That day “in September” on which the debate was proposed to be held was September 17. Juxtaposing it with September 4 in this context without further specificity would imply that they are the same day. The fact that September 4 is not the date that Fox originally proposed is a dastardly omission, given that September 4 occurs before September 17, due to the monotonic nature of time. The fishiness of this story is owed in large part to the sudden date change, so concealing that detail only fuels further suspicion of Fox’s complicity.
There’s a kind of cagey legal speak that criminals use when they want to signal innocence, but without saying false statements that could put them in further legal jeopardy. When I read these Fox articles, I get the overpowering impression that they were written with the conscious goal of not technically lying. To avoid running afoul of journalistic standards, Fox can’t knowingly publish a false story; but to avoid the wrath of their preferred candidate, they’d have to withhold embarrassing truths about him. The bizarre editorial decisions above are the consequence of trying to satisfy both of these competing priorities, I’m sure of it. Unless a whistleblower validates my theory someday, we may never know for certain, but come on, just see for yourself.
1
A starting point would be to observe that the red and green hypotenuses have different slopes (0.375 and 0.4 respectively), so their endpoints aren’t collinear (ie, they don’t fall on the same line). The difference is clearest when viewing the space around the endpoints (specifically, the coordinates (6,3) and (9,12)) and comparing them to their counterparts in the other shape. Interestingly, for all the above values, we assumed a certain level of “niceness” – ie, that points that overlap with “nice” integer coordinates are precisely where they seem, and not off by some imperceptible amount. The endpoint that looks to be at (6,3) is assumed to be exactly at (6,3), as opposed to (5.998,3.001) or something like that. We can’t actually justify this: the figure appears to show a 1x1 square-shaped hole after rearrangement, but by my calculations, the missing area should only be 47/48 square units. Interestingly, the reason we know that some kind of fudginess has occurred is because translation is an area-preserving transformation. So, metamathematically, the problem actually reduces to establishing a very high likelihood that the shapes weren’t perfectly duplicated, leading to a barely perceptible loss of 1/48 square units of area. I suppose you could take the faith-based view that gasp, maybe it really is a paradox, but this shouldn’t be on the table – you don’t need an extraordinary explanation for a loss of 0.0208333… square units between two similar looking pictures. Deep down, this problem exposes the way we think of graphs as platonic objects, when they can be constructed with the same human fallibility as anything else in physical space.
2
Since 2020, the Presidential Debate Commission has been sidelined, allowing news networks to just “invite” candidates to whatever debates they schedule. One possible circumvention would be if Fox had “invited” both candidates and independently decided on rules that Trump coincidentally prefers. I think we would still view this as suspect, especially given that the date had been pushed forward to a week before the already scheduled “next” debate, just as Trump was wavering on attending that debate.
