# Why is the answer to "the most annoying math puzzle" not 8? **URL:** \ **Category:** In My Humble Opinion\ **Created:** [January 8, 2023, 12:14pm UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874 "2023-01-08T12:14:25Z")\ **Posts on this page:** 20\ **Page:** 6
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 8, 2023, 11:08pm UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/101 "2023-01-08T23:08:58Z")
> [@LSLGuy](#): > > One of the posters here has done a fine job of explaining it over the last 100 posts or so. And I’m sufficiently trusting to take their word for it. Was it one of the posters who also hasn’t watched the video? If you haven’t watched it, that’s fine. But then I don’t think you are equipped to say either what the video presenter thinks _or_ who in this thread has fairly explained what the video presenter thinks. ---
**Author:** ![Novelty\_Bobble](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@Novelty\_Bobble](https://boards.straightdope.com/u/Novelty_Bobble)\ **Post date:** [January 8, 2023, 11:10pm UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/102 "2023-01-08T23:10:10Z")
> [@LSLGuy](#): > > One of the posters here has done a fine job of explaining it over the last 100 posts or so. And I’m sufficiently trusting to take their word for it. Having seen those hundred posts you should be fully aware that not everyone is understanding it or interpreting it in the same way. Whose word are you taking and why? I’d be fascinated to know if it is someone who’s views you already agree with because, as it happens, the latter part of the video (that you won’t watch) talks about exactly that sort of cognitive error as well. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 8, 2023, 11:12pm UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/103 "2023-01-08T23:12:07Z")
The OP did leave some detail out of their text summary of the question (that’s what makes it a summary, after all), but I don’t think it’s missing anything essential. The summary starts out with > [@griffin1977](#): > > Traffic junctions are judged based on how how many major accidents happen at them (which cause serious injury or death) and how many minor accidents (that cause only minor injuries or none at all). OK, so we’re judging junctions. I think it’s already pretty clear that junctions with fewer accidents (of whichever kind) are better than junctions with more accidents. That’s the basis on which we’re judging them. So if we want an “equivalent” intersection, it’s one we would judge the same way based on that criterion. And then if you watch the video for more details, you find that an agency is trying to choose between two designs. Again, it should be clear that they’re trying to get the design with fewer accidents. ---
**Author:** ![Novelty\_Bobble](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@Novelty\_Bobble](https://boards.straightdope.com/u/Novelty_Bobble)\ **Post date:** [January 8, 2023, 11:12pm UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/104 "2023-01-08T23:12:48Z")
> [@Riemann](#): > > Way to deflect, but I was terse simply because you were repeating a point that had already been made and discussed, ignoring the discussion that had already taken place. I’m not deflecting, I apologise if I was retreading old ground for you. I thought it was a relevant point to make, for others if not for you. We can leave it there then? ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 8, 2023, 11:47pm UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/105 "2023-01-08T23:47:36Z")
> [@Novelty\_Bobble](#): > > We aren’t the majority who unthinkingly go for an “obvious” answer. Well–to some extent that’s exactly what I did, because I also immediately put the problem into that category of math word problems where the words are just labels to make it more memorable and don’t actually have any meaning. “Major accidents” and “minor accidents” could have just been gloops and glorps as far as I interpreted it. Another aspect which makes the whole setup weird is how backwards it is. It’s presented as if the designers of layout B can somehow adjust the minor accident count to make the two equivalent, for some unknown end. I’d find the setup to be much clearer had the presenter said something like: “Suppose the DoT had found that the two layouts were equivalently safe for this location. Given that conclusion, what number would you have expected to show up in that box?” ---
**Author:** ![Novelty\_Bobble](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@Novelty\_Bobble](https://boards.straightdope.com/u/Novelty_Bobble)\ **Post date:** [January 9, 2023, 12:05am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/106 "2023-01-09T00:05:52Z")
Well that line of reasoning isn’t typical of the mindset that the question was designed to illuminate and it can be hard to put ourselves in the minds of others. How can it be that what is blindingly obvious to us is not apparent to others? > [@Dr.Strangelove](#): > > It’s presented as if the designers of layout B can somehow adjust the minor accident count to make the two equivalent, for some unknown end I was imagining that the first three numbers had come back from the modelling and only the final one was still to be generated. The big question being then what would that need to be to make B as good/bad as A? ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 9, 2023, 12:36am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/107 "2023-01-09T00:36:58Z")
> [@Novelty\_Bobble](#): > > How can it be that what is blindingly obvious to us is not apparent to others? Everybody has a slightly different approach to the problem and sees it in a different light. For instance, I suspect one thing that led me down the path I did is this: as soon as he started drawing the 2x2 grid, with two different possibilities and two different measurements, I immediately thought of [Simpson’s Paradox](https://en.wikipedia.org/wiki/Simpson%27s_paradox). I.e., the puzzle was about how road B could be safer even though it did worse on both metrics. A few moments later I realized it wasn’t about that, but the impression was still set, which is to say that we’re looking at independent situations (Simpson’s Paradox kicks in when there is another confounding variable). ---
**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\ **Post date:** [January 9, 2023, 1:15am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/108 "2023-01-09T01:15:32Z")
> [@Novelty\_Bobble](#): > > yes, when given to Yale students the vast majority gave the wrong answer and “8” was the most popular of the wrong answers. FWIW I rewatched now to see if there was some citation available, to see if the way it was presented to those college students was as garbled and unclear as his was. If it’s there I missed it on rewatch. To my read this is very meta. He very much wants to see something that confirms the cognitive bias he knows exists. I know it exists too. He then sees in this “experiment” something that is not there, precisely because he is subject to the cognitive bias he wants to see, just like shared the graph on support for same sex marriage uncritically. To some degree though he conflates two things. The point of the cognitive reflection test is how often we jump using broad heuristics and the degree we do or don’t stop to think if that answer makes any sense and rethink. But he conflates that with confirmation bias, accepting that which supports our existing beliefs as true fairly uncritically. ---
**Author:** ![ASL\_v2.0](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/asl_v2.0/32/3525_2.png) [@ASL\_v2.0](https://boards.straightdope.com/u/ASL_v2.0)\ **Post date:** [January 9, 2023, 1:37am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/109 "2023-01-09T01:37:30Z")
> [@Dr.Strangelove](#): > > You have to read “road layout” in two different ways (proposed road, and possible design) for that to work. But I read it as the DoT looking at road layouts individually, and had two independent cases they were looking at. But that’s not how the actual video frames it. It’s clear enough from the opening minute of the video that it is not two separate cases of data from two separate sections of extant road, but a choice between proposed layout A and proposed layout B for the same section of road. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 9, 2023, 1:55am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/110 "2023-01-09T01:55:19Z")
> [@ASL\_v2.0](#): > > But that’s not how the actual video frames it. It’s clear enough You can say that all you want, but I interpreted it a different way, and it was certainly not “clear enough” IMO. Looking at Brady’s animations again, I guess I can see they’re supposed to represent the same location, but I wasn’t looking at closely at them, and they aren’t part of Harford’s verbal description in any case. ---
**Author:** ![peccavi](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/peccavi/32/388_2.png) [@peccavi](https://boards.straightdope.com/u/peccavi)\ **Post date:** [January 9, 2023, 1:55am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/111 "2023-01-09T01:55:40Z")
There seems to me to be a thread of “this is important to understand because its not an abstract question, in the real world, you need to be able to understand how to judge these two cases” Well in the real world, these two layouts are already equivalent. A thousand accidents resulting in hospitalization or death is completely unacceptable and neither layout should be considered for implementation without major changes. They are equivalent in _suitability for implementation_. (and just reading the description of how the problem is defined, this seems a perfectly cromulent interpretation of the term equivalent). ---
**Author:** ![ASL\_v2.0](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/asl_v2.0/32/3525_2.png) [@ASL\_v2.0](https://boards.straightdope.com/u/ASL_v2.0)\ **Post date:** [January 9, 2023, 2:00am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/112 "2023-01-09T02:00:33Z")
The words are enough. He describes it as a choice being evaluated by the DOT for the same spot of road. One of which _will_ cause 2000 major and 16 minor accidents, the other of which _will_ cause 1000 major and X minor accidents. The _will_ is important as it further emphasizes that this is not a comparison of historical data from two different sections of road, but two options for the same section of road which the DOT is evaluating. That’s not the only cue, but it helps to resolve the ambiguity. As I said, _clear enough_. ETA: And specifically he says “gonna” and “are predicted” ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 9, 2023, 2:12am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/113 "2023-01-09T02:12:13Z")
> [@ASL\_v2.0](#): > > He describes it as a choice being evaluated by the DOT for the same spot of road He never says they’re the same spot of road. He says: > just imagine: the Department for Transport is looking at a road layout and they’ve got two different possibilities for the road layout and they’re trying to figure out which one is going to cause the fewest accidents Already the description is basically nonsense. Is there one road layout or two? And are they looking at “the fewest accidents” or some other metric like the financial or social cost of those accidents? If he had _actually_ said “the same spot of road” or “are proposing to build a road in a given location” or something equivalent, I’d have interpreted it differently. But nothing directly implies that they’re the same. ---
**Author:** ![ASL\_v2.0](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/asl_v2.0/32/3525_2.png) [@ASL\_v2.0](https://boards.straightdope.com/u/ASL_v2.0)\ **Post date:** [January 9, 2023, 2:15am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/114 "2023-01-09T02:15:09Z")
“…two different possibilities for **the** road layout…” seems clear enough to me. Note the use of the definite article “the.” ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 9, 2023, 2:38am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/115 "2023-01-09T02:38:27Z")
Ok, but he initially says **a** road layout. Since he clearly intended to say “two different possibilities for the [proposed] road layout,” **a** layout doesn’t make sense. He switches between an empty plot intended to contain a layout vs. the layouts themselves. At any rate, having to narrowly parse the grammar of his question to interpret it the “correct” way should perhaps indicate that it was poorly phrased. If, as **Andy\_L** suggested, suppose the question had been “Person A has 2000 one-dollar bills and 16 nickels; person B has 1000 one-dollar bills and some number of nickels. How many nickels is that if A and B have an equivalent amount of money?” Do you think a similar number of people would get that wrong? If not, then the problem is with the phrasing of the question. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [January 9, 2023, 2:47am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/116 "2023-01-09T02:47:42Z")
> [@Bootb](#): > > Are you sure? “Equivalent” can mean many different things–mathematical, financial, time lost, cost to the taxpayers, etc. But the point of the video was that many people gave the answer eight. Can you come up with a plausible use of the term equivalent that would say a location that has two thousand major accidents and sixteen minor accidents is equivalent to a location that has one thousand major accidents and eight minor accidents? The point was that a lot of people were thinking _“A is to B as C is to D, so what’s the value of D?”_ when they should have been thinking _“A plus B is equal to C plus D, so what’s the value of D?”_ ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [January 9, 2023, 2:48am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/117 "2023-01-09T02:48:09Z")
> [@Dr.Strangelove](#): > > If, as **Andy\_L** suggested, suppose the question had been “Person A has 2000 one-dollar bills and 16 nickels; person B has 1000 one-dollar bills and some number of nickels. How many nickels is that if A and B have an equivalent amount of money?” Do you think a similar number of people would get that wrong? If not, then the problem is with the phrasing of the question. And if the question had been instead “Person A has 2000 one-dollar bills and 16 nickels; person B has 1000 one-dollar bills and some number of nickels. How many nickels is that if A and B have an equivalent **pattern** of money?”, the right answer would be very different. Once we admit / agree the two different questions give two different answers, it’s time to move on to the actual question posed: “Person A has 2000 one-dollar bills and 16 nickels; person B has 1000 one-dollar bills and some number of nickels. How many nickels is that if A and B are equivalent?” Equivalent in what way? Aye, there’s the rub. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 9, 2023, 2:56am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/118 "2023-01-09T02:56:15Z")
> [@LSLGuy](#): > > Equivalent in what way? Aye, there’s the rub. Well, part of the rub. Even getting to the point where the definition of equivalence matters requires a specific parsing of the problem statement. But I agree that even with the right interpretation, you still have the problem of equivalence. Especially since Harford explicitly says they’re “trying to figure out which one is going to cause the fewest accidents”! By that metric, exactly 1016 is the right answer. I feel like there’s an alternate universe where most people gave an answer of 100,000 or whatever, and the response was “Idiots! I defined equivalence such that it doesn’t distinguish between major and minor. The answer is 1016.” ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [January 9, 2023, 3:02am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/119 "2023-01-09T03:02:56Z")
> [@Little\_Nemo](#): > > Can you come up with a plausible use of the term equivalent that would say a location that has two thousand major accidents and sixteen minor accidents is equivalent to a location that has one thousand major accidents and eight minor accidents? We have already done that, _numerous_ times. Why are you asking again? > [@Riemann](#): > > Since the predominant characteristic of A is its bizarre ratio of major to minor accidents, a plain English interpretation of this ambiguous question is that “equivalent” means “has the same bizarre properties as A” - for example, equivalent in the sense that it is similarly perched on the edge of a cliff. > [@Riemann](#): > > And, again, a perfectly sensible plain English interpretation of “equivalent” is “shares the same atypical and unexplained characteristic as”. > [@Riemann](#): > > And this is not just sophistry… this was genuinely my first thought - that the ratio of major:minor incidents is so weird that the most important challenge is to explain it, which might mean finding “equivalent” junctions _with a similar atypical ratio_ in order to see what they have in common with the first junction. > [@Dr.Strangelove](#): > > the way I interpreted it initially, the layouts were in different locations with different conditions, and the question of equivalence was about what would make the two situations have the same strange ratio. > [@Riemann](#): > > In the real world, if a junction layout has a very strange ratio of major to minor accidents, the first thing I would want to do would be to try to understand that. If the explanation were not obvious, I would look at other junction layouts to see how they compare. > > And since his explanation of the DoT’s objective made no sense in the first place (major and minor accidents carry equal weight), what can you do but ignore _exactly_ what he said (which was plainly wrong) and try to apply common sense to the situation? ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [January 9, 2023, 3:37am UTC](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874/120 "2023-01-09T03:37:56Z")
> [@Riemann](#): > > > [@Little\_Nemo](#): > > > > Can you come up with a plausible use of the term equivalent that would say a location that has two thousand major accidents and sixteen minor accidents is equivalent to a location that has one thousand major accidents and eight minor accidents? > > We have already done that, _numerous_ times. Why are you asking again? I asked for a _plausible_ use of the term equivalent. You didn’t produce one. The problem was presented as an engineering question (albeit a hypothetical one). I can see an engineer being asked to design a system on the basis of which one has fewer accidents. And I can see being asked to design a system based on which one has fewer serious accidents. Or maybe a design based on some other factor like cost or traffic flow or how quickly it could be constructed. But an engineer being asked to design a system so it would have a specified ratio of major and minor accidents? With the specific ration being a 250:1 ratio favoring major accidents? That is not even remotely plausible. [Previous page](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874.md?page=5) [Next page](https://boards.straightdope.com/t/why-is-the-answer-to-the-most-annoying-math-puzzle-not-8/977874.md?page=7)