The Philosopher's Tone: Mind Games: Part Three: Math is Hard

Part Three: Math is Hard

Carrying the one: He ain’t heavy, he’s my brother.

The third type of math puzzle that I see a lot of is just doing some simple arithmetic. Simple you say? Well, no, it turns out. And the discussions of the correct answers are endless. Generally, it breaks down into those that approach the problem from a grammatical perspective (even though they are written in mathematical notation), that is, reading left to right, and those that want to adhere to mathematical conventions. (“The Battle of English and Mathematics” anyone?) This topic is covered by a fairly decent article called How Social Media Proved That Math is a Language (https://roomtodiscover.com/math-is-a-language/), so I will try not to cover too much of the same ground. I say “fairly decent” and not “excellent” because the author talks about “PEMDAS” but never really gets around to explaining what that might be, which those of us who might not be up on the lingo would appreciate. He finally comes close to doing so about a third of the way in indirectly. PEMDAS is the acronym of the mathematical convention for the order of operations in equations. It stands for “(P)arenthesis, (E)xponents (M)ultiplication (D)ivision (A)ddition (S)ubtraction. Where the article excels is in navigating the ins and outs of the “simple” math involved.

Arithmetic

My inner imp: “The 69 argument has been around as long as there have been people.”

Actually, it is a drawing of one half of a pair of broken glasses…(also, “uniformed” or “uninformed”, you twit.)

The whole point here is that there are many ways of looking at things and how we proceed is determined by which set of rules we follow, if any. It turns out that even following the rules might yield different results.

Here follows a number of examples, beginning with the one from the article. I shall number them and provide the (potentially multiple) answers at the end of the article in Appendix I. (Damn…I am guilty of redundancy.)

See? I’m not the only one who obsesses about stuff like this.

Remember, just because someone put it in writing, doesn’t mean it’s true.

Two versions of the same problem. How much does the notation affect the outcome?

This is messed up. The answers are in a different order and one is using leading zeroes. It’s impossible to solve.

This might be the only one here that has only one answer. Okay, maybe two.

Algebra and Symbolic Logic (sort of)

There is another group of math puzzles which go a little deeper than just arithmetic, although some of them still require it. This example works like the ones above, but you need to perform an additional step: figure out what numbers you are working with by reverse engineering the clues provided by some pictorial substitutions.

The top three lines exist solely to define the terms of the fourth. It also answers the question of which came first, the chicken or the banana.

Some other examples.

Because nothing works without emojis anymore.

Oh! I know! I know! “Internet Explorer!”

Good twin…

10.

…evil twin. Fuck you. No doubt created by the same person that calculated the airspeed of an unladen swallow. (If you don’t go to the appendix for any other reason, you need to see this one to believe it.)

11.

This one, while being guilty of ambiguity, and therefore “faulty” imho, is ingenious nonetheless because of the depth of the symbolism. Details will be presented in the appendix.

12.

Spoiler alert for this next one. I’m giving the answer here because the story involves goes to my cred as a mathematician.

No explicit operations are notated in this example; however, some mental arithmetic is to understand its code. I have classified it as ambiguous, because according to the author “The Answer of this Math Puzzle for College Students is as below

TWISTER = 14

It is number of letters of the word on the left multiplied by 2.

The author did not count on the analytic tenacity and imagination of a polemicist like me. I used no such simple approach (actually the author’s suggestion never occurred to me). My reasoning was as follows. Assuming “AT” is either the sum or the product of 1, 2, or 3 (assuming is allowed at this stage, it is a thought experiment not a conclusion), I picked 2 to start off allowing either possibility. With the addition of “C” in the second line, “A” and “T” must be the number 2 and that makes “C” a 2 as well and the operation is addition not multiplication. (At this point I was not considering fractions, negative numbers, subtraction, or division). As I went down the list of equations, “2” seemed to work in every instance. The C from CAT carried down to the C from CROW, the R from CROW to BRAIN, and finally T from AT and CAT, W from Crow, I from BRAIN, the R from CROW and BRAIN all dropped down to TWISTER, all equaling 2. But there was no “S” and no “E”. There you have it. Since S and E were not definable, they would have to remain as variables and the equation could not be solved for a specific value. But whatever variables you used, the result could not be less than 10, the sum of TWITR.

That’s when I looked up the answer. Oh.

Classic example of not seeing the forest for the trees. Instead of counting the letters and multiplying by 2, I equated all the letters with 2 and added them; net result would have been the same had I equated the S and the E with 2. I simply—and simple-mindedly—approached the whole problem backwards.

13.

Then there’s this piece of work.

The answer is obvious in retrospect.

That’s it for the ambiguous puzzles. Thanks for reading along. Since I don’t want this to be a bad experience for you, leaving you no hope of ever finding fun and challenging math puzzles, Part Four is a robust sampling of mind games you can play that have only one solution.

Next-Part Four: Fun Time and Epilogue

Back-Part Two: Love Triangles