I have a pet peeve, and I wanted to discuss it. Every now and then, a “math puzzle” floats around Facebook and it always frustrates me. To be clear, what follows is in no way an indictment of those that share or comment on these puzzles, more about the nature of the puzzles themselves. There are two main iterations I’ll discuss here, but the main flaw is the same. The puzzles are always unclear, require people to make assumptions, and anything other than the accepted answer is incorrect. To these puzzles, math is little more than a trick – a thorn in your side that pokes you unless you magically get it right.
Ill-Defined Symbols and PEMDAS
A sample of the first puzzle I want to talk about is above. Here, we have 4 equations and seemingly establish 3 symbols – a witch (with wand and broom), a wand, and a broom (and possibly a thicker or two brooms?). The final equation displays a witch without wand and broom, and also a double (or thicker?) wand symbol and asks for the answer.
The answer this wants you to get is probably 73. 3 witches = 45, so each witch is 15. 3 wands is 21, so each wand is 7. 4 brooms (if the middle symbol is 2 brooms) is 12, so each broom must be 3. That gives the bottom equation of 1 broom (3) + a which without a wand or broom (15 – 7 – 3 = 5) times two wands (2 * 7 = 14). 3 + 5 * 14 = 73. I kind of did this fast and loose, so forgive me if any of that is wrong. As I’m about to describe though, it kind of doesn’t matter if I got this right or not.
Our first issue is ill-defined symbols. Are those 2 distinct wands and brooms in their respective equations, or just thicker versions? Does it matter? I guess you could say 2 wands would just be twice the value of one wand, but we’re dealing with abstract symbols – none of these values are associated with units of any kind. If I said w + w = 20 and then asked you to solve for “v”, you’d rightfully be confused? But what is a “w” but 2 “v”s next to each other, right? It’s the same logic – an abstract variable to represent a value.
As for the witch without a broom and wand, how do we know to subtract the value of the wand and broom from the witch in general? Is the value of the witch representative of their magical powers, and the magical properties of the wand/broom represent a greater proportion of the witches overall value? Are we measuring the witch’s value as a person (or their magical powers are innate and not related to items) such that the value of the witch as a person is greater than those items? A witch’s value may be more in the form of “a * (person) + b * (wand) + c * (broom)”, where a, b, and c are coefficients applied to each component part of the “witch” variable to produce the overall value of the witch. The puzzle seems to assume all of these coefficients are equal to 1, and thus all have equal importance, but is that true for the value of the witch? I’ve seen similar puzzles adding or removing food items – bananas, fries, etc. Sure a small number of those would bring on a certain level of satiation and happiness, but surely the marginal value of adding more items decreases as more are added? The first 5 fries taste great, but after eating 1000 you maybe don’t want to see fries again?
This is a smaller thing, but these puzzles always seem to end with some form of a + b * c, which seems to confuse those that learned math before order of operations and just learned to evaluate left to right? My personal observation shows this to be a generational thing, but it can easily be fixed with using a set of parenthesis to denote the correct operational order (a + b) * c or a + (b * c). Order of Operations would already define the calculation to be the latter of the two, but why not make it as clear as possible? Maybe even rewrite it to be b * c + a to have even the left to right people make it work? Often in my work, if I’m doing some kind of calculation, I’ll use parenthesis or reorder things just to be certain that it will be evaluated in a specific way. Heck, SQL has an operand called “BETWEEN” which is used to check a value against an upper or lower bound. I can never remember straight if “BETWEEN” includes those upper or lower bounds (I believe it does), so I pretty much always write something like “a <= b and b <= c” rather than “b between a and c”. Those two things would do the same thing, but I want to know for certain how it will be evaluated. And again, real math would have units or some other clues as to what needs to be done. The equation seems to produce a value of X brooms + Y witch-wands (dual-wielding witches?) and that doesn’t make any sense.
Am I overthinking this, absolutely. But the puzzles themselves bring it on. I’m attempting to make the case that you can’t just change a symbol without defining the composition of that symbol. An individual has to make assumptions related to the symbol in order to solve the puzzle, and the puzzle itself purports to only have 1 “correct” answer – with no acknowledgement of the assumptions made to get there. Every day, I receive report or system requests that often lack the necessary info to complete (through not fault of the requester, who may simply be unaware of the questions that need to be answered). I take these requests, make assumptions given the nature of the request, and produce an output that I think best fits the need of the request. Even if the output is not what was desired, it is still “correct” based on the assumptions made – changing those assumptions produces a different output.
Going To The River
The puzzle above is similar – there is incomplete information requiring assumptions to be made and only a specific reading of the question produces the “correct” answer. For this one, I believe the assumption that is needed to be made is that the elephants see the same 2 monkeys (“Every elephant saw 2 monkeys going towards the river.”). The answer then comes out to be 5: 1 rabbit + 2 monkeys + 2 parrots.
The puzzle is phrased as being for geniuses only, and a battle between English and Mathematics – but it seemingly fails to provide enough information to be solvable. If the elephants were all in a group then yeah, it might make sense for them to all see the same 2 monkeys – but we know nothing of their proximity. Maybe the elephants were all in a group, but it was a whole group of monkeys that ran along – darting in between bushes and trees. Each elephant maybe saw 2 monkeys, but who is to say the monkeys seen by the first elephant weren’t already hidden in the bushes before the next couple came out to be seen by the subsequent elephant? I contend that the number of monkeys is any value between 2 and 12, since the question is unclear on whether or not the same monkeys were seen by each. Being from Southern Illinois, I certainly know of times where I’d be driving along with someone and see some deer off in a field – we may each see 3 deer but one of the ones I saw maybe dove off into a ditch or into the woods before the other person could see it – and that person in turn caught one I didn’t see. Maybe I shouldn’t use the analogy of me “seeing” deer while driving……..
Again, does any of this matter? Not really, no. But, the puzzle is intentionally ambiguous while saying there is only one true answer. It’s not a battle of English and Mathematics whenever insufficient information is provided. If there was some kind of play on words or something that was hiding a bit of crucial information, but which would cause the puzzle to be solved once uncovered, then sure – that’d be a battle of English and Mathematics.
Jesus, You Care A Lot About This
Ok, yeah. Fair point. I’ve always been good at math, and often saw myself in math classes (at least in HS) understanding a concept immediately, figuring out a way to write a program on my graphing calculator to do the work, and then going around to fellow classmates trying to help them understand. I’ve seen far too many people dismiss math as being not for them or being needlessly obtuse. I know of some that go so far as saying it’s not in their blood / genes.
I hate this perception of math. I think anyone can come to find and enjoy math – to see what it can tell us about the world. I’ve fallen down rabbit holes discovering math communicators on YouTube – with the likes of Matt Parker and Numberphile – or the insane physics descriptions of Randall Munroe’s What If blog or Rhett Allain’s work for Wired magazine.
And there’s certainly a place for puzzles. Presenting a problem where the answer isn’t immediate and asking for an answer gets us closer to real world problems. Most of our work is seeing a problem with an unclear solution, making assumptions to put together a solution, and then seeing how things look on the other side. The critical item isn’t necessarily the output but rather the quality of the assumptions and the process used to create the output. After all, if the process is solid, one should simply change a few minor things to produce a new/better output given corrected assumptions.
These puzzles don’t have that. The information is insufficient requiring the user to create assumptions when coming up with a solution. Once solutions are presented, often times incorrect answers are just met with a flat “No” – with no opportunity for learning or growth. And sure – that’s not the job of the people sharing these things on Facebook. I don’t expect them to be math educators that care about the learning process. These puzzles don’t facilitate learning. Rather, they perpetuate the idea that “If you get this, you’re a magical genius. If you don’t, math just isn’t for you.”
Again, absolutely no bad blood between me and anyone that shares these kinds of puzzles – they’re just some fun dumb thing on Facebook to distract us from our crazy world. If you do want some math puzzles, maybe check out the Matt Parker’s Maths Puzzles series – where a puzzle is presented and then something like 1-2 weeks later a solution video is posted describing the solution.
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