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### Chain email

Posted: Thu Feb 25, 2010 5:07 pm UTC
I got this chain email recentrly with this puzzle, and the answer is escaping me.
At first i thought that each number was treated as a different number but then realised that it wouldnt work.
Then i thought it might be in something other than base 10, but i am not smart enough to work it out.
Anyways...

If:

2 + 3 = 10
7 + 2 = 63
6 + 5 = 66
8 + 4 = 96

Then:

9 + 7 = ????

### Re: Chain email

Posted: Thu Feb 25, 2010 5:11 pm UTC
It's a pattern.

(a + b) * a = answer
(2+3) * 2 = 5 * 2 = 10
(7+2) * 7 = 9 * 7 = 63
(6+5) * 6 = 11 * 6 = 66
(8+4) * 8 = 12 * 8 = 96
(9+7) * 9 = 16 * 9 = 144

### Re: Chain email

Posted: Thu Feb 25, 2010 5:15 pm UTC
This has got to be the stupidest overloading of the '+' operator since I don't know when. We should all remember that "communicating badly and then acting smug when you're misunderstood is not cleverness."

EDIT: I should have remembered my usual mantra of "Is this worth cluttering my egosearch?"

### Re: Chain email

Posted: Thu Feb 25, 2010 5:23 pm UTC
Briareos wrote:This has got to be the stupidest overloading of the '+' operator since I don't know when. We should all remember that "communicating badly and then acting smug when you're misunderstood is not cleverness."

EDIT: I should have remembered my usual mantra of "Is this worth cluttering my egosearch?"

It's not communicating badly, since it's obvious from the statement of the puzzle that something is not being given it's standard meaning. The task then is to figure out what is not being given its standard meaning, and what the new pattern is.

### Re: Chain email

Posted: Fri Feb 26, 2010 12:15 am UTC
skeptical scientist wrote:
Briareos wrote:This has got to be the stupidest overloading of the '+' operator since I don't know when. We should all remember that "communicating badly and then acting smug when you're misunderstood is not cleverness."

EDIT: I should have remembered my usual mantra of "Is this worth cluttering my egosearch?"

It's not communicating badly, since it's obvious from the statement of the puzzle that something is not being given it's standard meaning. The task then is to figure out what is not being given its standard meaning, and what the new pattern is.

But why then use +? why not, for example, |, or $, or ¬? It's given an alternate meaning not ever used. It's like saying this post has words, but they aren't in their standard meaning. The actual meaning is "The puppy jumped onto the 16:21 space elevator to station 5477/2, and bit the leg of the man in red". ### Re: Chain email Posted: Fri Feb 26, 2010 1:31 am UTC It's a little brain teaser puzzle. Why care so much about what particular symbol is used? That's somewhat missing the forest for trees. ### Re: Chain email Posted: Fri Feb 26, 2010 1:38 am UTC kernelpanic wrote: skeptical scientist wrote: Briareos wrote:This has got to be the stupidest overloading of the '+' operator since I don't know when. We should all remember that "communicating badly and then acting smug when you're misunderstood is not cleverness." EDIT: I should have remembered my usual mantra of "Is this worth cluttering my egosearch?" It's not communicating badly, since it's obvious from the statement of the puzzle that something is not being given it's standard meaning. The task then is to figure out what is not being given its standard meaning, and what the new pattern is. But why then use +? why not, for example, |, or$, or ¬? It's given an alternate meaning not ever used. It's like saying this post has words, but they aren't in their standard meaning. The actual meaning is "The puppy jumped onto the 16:21 space elevator to station 5477/2, and bit the leg of the man in red".

Yes, except here it is possible to figure out the meaning from context; in fact, that's the whole point of the puzzle.

IMO, the difference between a valid puzzle and a 169-style puzzle is whether it is designed to be solved. A 169 puzzle is designed so that it won't be solved, and then the inventor can act smug when their audience fails an unfair test. This puzzle, on the other hand, is clearly meant to be solved, so the objection in comic 169 does not apply.

### Re: Chain email

Posted: Fri Feb 26, 2010 5:32 pm UTC
kernelpanic wrote:But why then use +? why not, for example, |, or $, or ¬? It's given an alternate meaning not ever used. Because it's a puzzle aimed at general readers, most of whom are probably more familiar with word-based puzzles, and are probably unfamiliar with a lot of mathematical notation. Sure, you could write it using standard f(u, v) style notation, but that might scare off the general readers. Doing it this way may screw with the minds of the more mathematically inclined, but the puzzle creator may consider that a good thing, in that it forces us to think outside the box. ### Re: Chain email Posted: Mon Mar 01, 2010 3:29 pm UTC These problems have an infinite number of solutions, so you could guess a solution and solve a set of 4 equations, say a polynomial: $f(x,y)=\frac{1}{9}(512-13x-214y+41xy)$ In which case the answer will be: $(9+7)=f(9,7)=\frac{1480}{9}$ Equally valid, less beautiful. ### Re: Chain email Posted: Mon Mar 01, 2010 4:48 pm UTC kernelpanic wrote:But why then use +? why not, for example, |, or$, or ¬? It's given an alternate meaning not ever used. It's like saying this post has words, but they aren't in their standard meaning. The actual meaning is "The puppy jumped onto the 16:21 space elevator to station 5477/2, and bit the leg of the man in red".

Actually, the + is a HUGE clue. If you look at what a+b is in each case, you might notice that a+b is always a factor of the answer, and then when you ask which factor, you solve the problem. I happened to get this email a few days ago, and I solved it in about 2 minutes. It probably would have taken 3 or 4 had all the ' + ' been ' * '.

odulwa wrote:These problems have an infinite number of solutions, so you could guess a solution and solve a set of 4 equations, say a polynomial:

$f(x,y)=\frac{1}{9}(512-13x-214y+41xy)$

In which case the answer will be:

$(9+7)=f(9,7)=\frac{1480}{9}$

Equally valid, less beautiful.

This is why mathematics is an art. Yes there are technically infinitely many solutions, but there is an 'obvious' elegant answer.

### Re: Chain email

Posted: Mon Mar 01, 2010 11:31 pm UTC
odulwa wrote:These problems have an infinite number of solutions...
Equally valid, less beautiful.

Equally valid, but incorrect. Yes, these puzzles always have infinitely many "valid" solutions, but if they are well constructed, there will be exactly one solution which is obviously the right one once you find it, and all the other solutions will be obviously wrong. This puzzle is one of the well constructed ones.

### Re: Chain email

Posted: Tue Mar 02, 2010 12:24 am UTC
Let N be the set of natural numbers and let [imath]\left( {N} , + \right)[/imath] be a magma with $+:{N} \times {N}\longmapsto {N},\ \left( a,b \right) \mapsto a \cdot \left( a \oplus b \right)$
([imath]\oplus[/imath] stands for the 'usual' +)

With [imath]+[/imath] being defined like that it shouldn't be an issue.

(Edit: I first wrote it is a semigroup but since [imath]+[/imath] is not associative it's only a magma.)

### Re: Chain email

Posted: Tue Mar 02, 2010 3:26 am UTC
Just to anger / calm both of you, you could use the six-pointed star ([imath]*[/imath]), which in Fraleigh's Abstract Algebra means "binary operator"...

skeptical scientist wrote:
odulwa wrote:These problems have an infinite number of solutions...
Equally valid, less beautiful.

Equally valid, but incorrect. Yes, these puzzles always have infinitely many "valid" solutions, but if they are well constructed, there will be exactly one solution which is obviously the right one once you find it, and all the other solutions will be obviously wrong. This puzzle is one of the well constructed ones.

And you could just say "this represents a function that maps these two numbers to the third, therefore I could name any number in the last set as the solution, and it is correct merely by definition," but as s.s. stated, that's not the point. Use Occam's Razor on this: the simplest explanation is typically the correct one. It really is much simpler to find the function which takes 2 & 3 and maps it to 10 by way of algebraic manipulation, than to create any number of operators that create the functions you want them to.
Besides, this was a chain email from someone who probably wasn't too fussy about proper notation.

### Re: Chain email

Posted: Tue Mar 02, 2010 3:58 am UTC
Wouldn't the simpliest solution often be discarded as the 'trivial' solution though?

Do any high level mathematicians (or for that matter, anyone highly advanced in any field) actually start chain letters?

I have yet to get any emails with conundrums pertaining to category theory (or some such thing; I don't speak high level maths ), although perhaps thats because it wouldn't propagate nearly as fast due to the majority deleting it as general nonsense when they got it.

### Re: Chain email

Posted: Tue Mar 02, 2010 7:33 am UTC
Dopefish wrote:Wouldn't the simpliest solution often be discarded as the 'trivial' solution though?

Sometimes but not always. One generally uses "trivial" to refer to something that is either obvious or so simple it might be overlooked, or else something which is always a solution to a general type of problem, when one is interested in things which are only solutions to a particular instance.* I would not describe the simplest solution as being trivial in general (sometimes the simplest solution to a problem is still quite subtle) and I wouldn't describe the solution to this puzzle as being trivial.

However, trivial solutions are still solutions, and are not discarded simply because they are trivial.

______
*For example, the differential equation f"=-f has the "trivial" solution f=0, but also the nontrivial solutions f=sin and f=cos. (In fact, any linear combination of sin and cos is a solution.) Here we call the zero solution "trivial" because it is always a solution to differential equations of this type (linear & homogeneous) but sin and cos are solutions specific to this particular differential equation, and not, say, the differential equation f''=f+f'.

### Re: Chain email

Posted: Tue Mar 02, 2010 8:16 am UTC
skeptical scientist wrote:
Dopefish wrote:Wouldn't the simpliest solution often be discarded as the 'trivial' solution though?
______
*For example, the differential equation f"=-f has the "trivial" solution f=0, but also the nontrivial solutions f=sin and f=cos. (In fact, any linear combination of sin and cos is a solution.) Here we call the zero solution "trivial" because it is always a solution to differential equations of this type (linear & homogeneous) but sin and cos are solutions specific to this particular differential equation, and not, say, the differential equation f''=f+f'.

It's trivial if it can be proven on less than ten A4 sheets of paper. Or so I've heard...

### Re: Chain email

Posted: Tue Mar 02, 2010 10:12 am UTC
Another vote for that the notation of this puzzle is stupid. In this case it might be okay, since it's just replacing the + operator (but I'd rather it be another operator), but I've seen puzzles in the form of
1 = 7
2 = 6
3 = 10
4 = 1
...
which really grinded my gears. Don't change the meaning of the equality sign! Use a colon or arrow instead! When I see that, I only think that the puzzle creator does not understand the equality sign, rather than that he's chosen a particular notation for a puzzle.

### Re: Chain email

Posted: Tue Mar 02, 2010 3:43 pm UTC
Timtu wrote:
skeptical scientist wrote:
Dopefish wrote:Wouldn't the simpliest solution often be discarded as the 'trivial' solution though?
______
*For example, the differential equation f"=-f has the "trivial" solution f=0, but also the nontrivial solutions f=sin and f=cos. (In fact, any linear combination of sin and cos is a solution.) Here we call the zero solution "trivial" because it is always a solution to differential equations of this type (linear & homogeneous) but sin and cos are solutions specific to this particular differential equation, and not, say, the differential equation f''=f+f'.

It's trivial if it can be proven on less than ten A4 sheets of paper. Or so I've heard...

In a rather advanced graduate seminar, an audience member asked the lectuerer if something was true. The lecturer tried in vain to prove the statement for an hour, until his time was up. The next week he came back and announced that it was trivial.

### Re: Chain email

Posted: Wed Mar 03, 2010 5:14 pm UTC
skeptical scientist wrote:
odulwa wrote:These problems have an infinite number of solutions...
Equally valid, less beautiful.

Equally valid, but incorrect.

Yeah, the "I could justify any answer" argument doesn't hold much water. I mean, the same could be said of learning a language as a baby: you only get finite input, so logically there are an infinite number of possible generalizations. And yet, just about every child that has ever lived has managed to pick the single "right" generalization that results in speaking intelligible, grammatical sentences.

### Re: Chain email

Posted: Wed Mar 03, 2010 6:50 pm UTC
Torn Apart By Dingos wrote:which really grinded my gears. Don't change the meaning of the equality sign!

Why not? Equivalence relations are all over the place, and it clutters things up to use a different symbol. Do you object to the use of "=" for natural, rational, real and complex numbers even though the relation changes? It's the most natural symbol to use when attempting to communicate the notion of "sameness" to a non-technical audience, just as "+" communicates the idea of a binary operation in the most straight forward manner.

### Re: Chain email

Posted: Wed Mar 03, 2010 7:45 pm UTC
Cleverbeans wrote:
Torn Apart By Dingos wrote:which really grinded my gears. Don't change the meaning of the equality sign!

Why not? Equivalence relations are all over the place, and it clutters things up to use a different symbol. Do you object to the use of "=" for natural, rational, real and complex numbers even though the relation changes? It's the most natural symbol to use when attempting to communicate the notion of "sameness" to a non-technical audience, just as "+" communicates the idea of a binary operation in the most straight forward manner.

I object to it in this case precisely because it's not an equivalence relation.

### Re: Chain email

Posted: Wed Mar 03, 2010 9:11 pm UTC
Torn Apart By Dingos wrote:I object to it in this case precisely because it's not an equivalence relation.

So the object is to the use of "=" as an assignment operator? It's used that way in most programming languages, and I think statements like "Solve for f(x) given x=5" are common enough to justify using it this way for a non-technical audience. If it's expressive and communicates the correct idea, I have no objection to the abuse of any common token for personal amusement.

### Re: Chain email

Posted: Thu Mar 04, 2010 9:59 pm UTC
Cleverbeans wrote:
Torn Apart By Dingos wrote:I object to it in this case precisely because it's not an equivalence relation.

So the object is to the use of "=" as an assignment operator? It's used that way in most programming languages, and I think statements like "Solve for f(x) given x=5" are common enough to justify using it this way for a non-technical audience. If it's expressive and communicates the correct idea, I have no objection to the abuse of any common token for personal amusement.

No. You've misunderstood my example. The particular puzzle I had in mind (which I've seen posted on this forum, but I'm not going to try to search for it), basically asked you to find f(n), given that
f(1)=4
f(2)=12
...
f(n-1)=3

which the OP wrote as
1=4
2=12
...
n-1=3
n=?

This is not an equivalence relation, and it's not assignment. Writing it this way doesn't help the reader, it is just confusing and perpetuates a bad understanding of the equality sign.

To go off on a tangent, I don't like the use of = as an assignment operator, nor as a replacement for "is/are" (like "me = happy"). This doesn't make a difference though because it's so commonplace I use it myself (the latter in the form of f(x)=O(x)).

### Re: Chain email

Posted: Thu Mar 04, 2010 11:41 pm UTC
Torn Apart By Dingos wrote:This is not an equivalence relation, and it's not assignment. Writing it this way doesn't help the reader, it is just confusing and perpetuates a bad understanding of the equality sign.

Seriously? Do you think anyone is confused about what equality is because of an unusual use in a puzzle? Do you really think that it "perpetuates a bad understanding of the equality sign" that some guy used it in an abnormal way as part of a riddle? Because to me this seems a tad hyperbolic.

### Re: Chain email

Posted: Fri Mar 05, 2010 1:02 am UTC
Wow. Brain teaser puzzles. Serious business.

### Re: Chain email

Posted: Fri Mar 05, 2010 11:41 pm UTC
skeptical scientist wrote:
Torn Apart By Dingos wrote:This is not an equivalence relation, and it's not assignment. Writing it this way doesn't help the reader, it is just confusing and perpetuates a bad understanding of the equality sign.

Seriously? Do you think anyone is confused about what equality is because of an unusual use in a puzzle? Do you really think that it "perpetuates a bad understanding of the equality sign" that some guy used it in an abnormal way as part of a riddle? Because to me this seems a tad hyperbolic.

Some students have a horrible understanding of equality, and I think this is because they've been exposed to improper use of it too many times. Seeing it used incorrectly in faux-mathematical settings should only make things worse.

I don't know why these semantic discussions always arise on this forum. To me, the notation is initially confusing, even though I easily understand its intent, and to non-mathematically trained people, it might actually be more intuitive. This does not change the fact that it is horrible notation and could've been written in a much better way.

### Re: Chain email

Posted: Sat Mar 06, 2010 5:01 am UTC
Well, I have no idea what you mean by students having a horrible understanding of equality. In my experience students understand equality just fine. The one thing I sometimes see is students using equals signs in places they shouldn't, like for the next step in an algorithm they've been taught, or other manipulations. Generally these are the manipulations they should be doing to solve the problem, but they want to use some symbol between steps, and aren't sure what to use, so they use equals signs. This is certainly a bad habit that I try to break students of, but I don't think it shows that they don't understand the concept of equality, but rather that they misuse the symbol as they have a limited mathematical vocabulary. Is that what you are referring to? In that case, I think the solution is to suggest some other more appropriate symbol (perhaps arrows, or short explanations using actual English words). In my experience, this usually solves the problem (because students really do understand equality!)

### Re: Chain email

Posted: Sat Mar 06, 2010 9:27 am UTC
Actually that was what I was referring to. Writing "a=b=2a=2b" (equality sign between equations) or "a=>2a/2" (arrow where there should be an equality sign). I've seen worse, but admittedly it's rare. You're right that it doesn't mean they don't understand equality, but I'd like to see the equality sign used correctly when it is used.

### Re: Chain email

Posted: Sat Mar 06, 2010 1:55 pm UTC
Torn Apart By Dingos wrote:Actually that was what I was referring to. Writing "a=b=2a=2b" (equality sign between equations) or "a=>2a/2" (arrow where there should be an equality sign). I've seen worse, but admittedly it's rare. You're right that it doesn't mean they don't understand equality, but I'd like to see the equality sign used correctly when it is used.

That bothers me too. Could we be considered mathematical notation nazis? On a related note, I am now founding the mathematical notation national socialistparty.