Yesterday I posted this math teaser:

128 people came up with answers. I said there were two, depending on where one puts the parentheses in the last equation, but the mathies say that there is a convention: one does the multiplication first, and then the addition. The only trick in the piece was the last line: the kid is wearing two sneakers and holding two cones of whatever that stuff is. (What is it?)

Here is my answer, which I think is correct if you use the “multiply first convention”

6 sneakers = 30, ergo 1 sneaker = 5

Two boys + two sneakers = 20. Two boys + 10 = 20, ergo one boy = 5

4 cones plus one boy = 13. 4 cones + 5 = 13; ergo 4 cones = 8, so that one cone = 2.

In the last picture, we have one sneaker plus (one boy with two cones and two sneakers) times one cone.

Ergo 5 + (5 + 4 + 10) X 2 is the solution. That is 5 + 19 X 2

Using the multiplication rule first, that works out to 5 + 38 = 43.

If you put the parentheses in the last equation around (sneaker plus boy with cones and sneakers) X cone, you’d get 24 X 2 or 48. But the mathies say that this is wrong under the convention.

So the correct answer is 43. (I hope I didn’t screw up!)

Thyroid Planet was the first to post the correct answer(s) 29 minutes after the contest started, saying “48 or 43”.

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## 16 Comments

Thank you for the recognition, and :

Well done everyone! I think everyone who posted anything should also get recognition for taking on math problems, I think we all learned something new!

I certainly did – when you’re told to look carefully before starting, do it!

I got 43 too, but I didn’t answer because I was sure I was wrong somewhere: the answer MUST be 42.

“What is it?”

Someone here said it is licorice flavored popcorn. Since the tweet came from England, that would be a good guess. The English love licorice, or liquorice as they spell it.

I have lived in England for 53 years. I have never seen liquorice flavoured popcorn in my entire life.

Sounds like something they’d have in Blackpool. But you would know.

It’s a seaside treat alright, but coastal Maine not Blackpool.

https://www.coastalmainepopcorn.com/product-p/swt_lic.htm

Looks right, based on assumptions fairly clearly stated.

However I would continue to maintain that other answers can be correct also, including ‘no unique answer’, depending on assumptions. To repeat, it’s not so clear that a variable wearing red shoes is the same variable as the one wearing black socks!

Or that the red shoes are themselves a variable, but the black socks aren’t. Etc…

Joking, but maybe variables wearing other ones can be finessed into a new proof of Godel’s incompleteness! There’s a lot of amusing stuff there:

E.g. consider ‘the smallest positive integer that cannot be described in less than twenty-five syllables.’ (Count the number of syllables just written.) George Boolos, an MIT philosopher/mathematician, unfortunately dead at a too young age, had used that one for another incompleteness proof.

Maybe I’ve got too much time on my hands.

Or (and we are not told), is there another trick?

If a left shoe has a different value to a right shoe, the first line could be:

6+4 + 6+4 + 6+4 = 30

and so on for 7+3 etc.

Maybe the two cones have different values as well?

This would make the last line come to a different answer.

It is the wee small hours of Monday morning down here, so all relevant cautions apply . . .

I had this same thought. Nowhere does it state that left and right shoes are the same value.

And a BOGO on cones perhaps?

No, this is not sour grapes because I didn’t notice the difference in the boy’s picture on the last line.

The reason to use standard notation and conventions is so that different people in different places and times will all arrive at the same value. In this case, the non-standard notation creates quite a bit of ambiguity, and so there are a fair number of entirely reasonable answers (beyond simply dealing with the order of operations). I rather like the following solution (which I present mostly to be contrarian—I tried to post something similar yesterday, but it either failed to get through review, or I messed up when posting):

Let x be the value of one shoe, y be the value of the child, and z be the value of a newspaper cone. The unknown at the end is $w$. When two symbols are written next to each other or otherwise concatenated, the convention is to multiply. Thus, for example, the child with a pair of shoes and two cones is x^2 y z^2. With this convention, the four equations can be written using more conventional notation as

3x^2 = 30,

2y + x^2 = 20,

2z^2 + y = 13, and

x + x^2y z ^2 = w.

By back substitution, this equation has four sets of solutions:

x = ±sqrt(10),

y = 5,

z = ±2, and

w = ±sqrt(10) ± 400.

In any event, I kind of like this kind of problem, and may steal the idea for my precalc class. It might be worthwhile for students to deal with this kind of ambiguity.

What’s a good reference to know which topics are important in pre-calculus these days? I’ve never taught that (and long retired and not USian), but I assume it is the stuff not usually taught in secondary (high) school, but needed for calculus.

I did teach a lot of ‘beginning’ calculus over the years, the quotes because they had begun in secondary school, but it seemed best to start from the beginning again. Solving a system of linear equations didn’t seem necessary for calculus, however crucial elsewhere. (I realize yours here wasn’t linear.)

The standard precalculus curriculum, which is taught either to high school students or as remediation in college, consists of some basic algebra (algebraic manipulation, solving systems of equations, etc), analysis of functions (what do the graphs look like? what kinds of symmetries do they have? how can the graphs of functions be transformed? how do you spot minima and maxima? etc), and a cursory introduction to trigonometry and exponential functions. A lot of precalc classes also attempt to introduce notions of increasing and decreasing functions, as well as limits.

On the other hand, a colleague and I have been developing (and teaching) a new precalc course for our institution over the last several years. This curriculum is somewhat nontraditional—we emphasize mathematical reasoning and writing quite a lot more than in the traditional curriculum (this particular exercise, or something similar, would be good for discussing ambiguous problems, and emphasizing the need for clear definitions and notations), and deemphasize a lot of topics which really aren’t appropriate in a *pre*calculus class (such as limits, increasing and decreasing functions, etc).

I’ll also note that the algebra to solve systems of equations does show up in a few places in the calculus curriculum. For example, the standard curriculum for integral calculus typically involves a section on partial fraction decomposition, which can be attacked via systems of equations. Precalculus also often serves as a gateway to college mathematics, and students are expected to know how to solve systems in other classes, such as linear algebra and differential equations.

Thanks.

I had somehow forgotten the integral calculus partial fractions use of linear systems. Retired too long I suppose.

And why does the kid have that weird facial expression?

Since it’s obviously a trick question asking you to think ‘outside the box’, you might as well go all the way with it. If a pair of shoes was $10, would you really pay $5 for a single, unmatched shoe (just using dollars for convenience)? I think a shoe store would be lucky to give away unmatched shoes. And I’m guessing that’s $10 for brand new shoes. In the last row, those are now second hand shoes.

And what are we paying for the person for? I’m assuming we’re not buying him, but rather his services of some sort. Is he really going to charge more for his services just because he put shoes on?

And as others have pointed out, there are often BOGO deals on food, so there’s no telling how much 1 vs. 2 cones are. And are we buying the cones off of this guy, or is he just showing up to work with his lunch already in hand?