# When logic takes a leave of absence

If:        2 = 6
3 = 12
4 = 20
5 = 30
6 = 42
Then:      9 = ?

The logical course of action (though in a moment I’ll show why it’s actually not) is to look for a pattern, right?  And if we look at the numbers on the right we see that every time the number on the left is multiplied with a number one value higher: 2×3=6, 3×4=12, 4×5=20, 5×6=30, 6×7=42.  This gives us the formula $y=x \times (x+1)$ or $y=x^2+x$, and if $x=9$ then the answer is 90.  This is surely the most mathematical approach and pretty much refutes all the other possible answers.

Some people claim it’s 72, and they vehemently defend that answer.  They claim if you follow the pattern, 4x5=20, 5x6=30, 6x7=42, then 9 must be multiplied by 8 to give 72.  They are making the argument that one cannot assume the presence of 7 and 8 in the left-hand column and if 6 is multiplied by 7 then the next number, 9, must be multiplied by 8.  It’s a reasonable argument, but not as convincing as the first one if you understand anything about number sequences.

The third most common answer is 56.  These guys focused on the right-hand column. 6+6=12, 12+8=20, 20+10=30, 30+12=42.  With every new line the number added increases by a value of two, so to 42 we must add 14 giving us 56.  The big problem with this one is of course that it completely disregards the left-hand side of the equation which is a transgression of a central principle of arithmetic – two sides of an equation should always be in perfect balance.

Then there’s the bunch who insist the answer is 54 after they multiply 9 with 6, the number that appears before it in the puzzle.  This solution does not follow any type of pattern whatsoever, nor does any of the other answers supplied.

The thing is, one version of the puzzle claims ninety percent of people will get it wrong, and as more than 50% of people say it’s ninety, there must be some trick to this, right?

I believe there is.  I believe none of these answers are correct, not even 90, and it all comes down to one little symbol.  Visit Page 3 below for my answer.

## 15 thoughts on “When logic takes a leave of absence”

1. Priya says:

Thank you for this!!! I was going insane for the past 10 minutes because no one had made that comment, and I thought all the geeks in the world had left me 😦

It’s ridiculous how important this problem is to so many people. Some poor souls actually believe they are geniuses because they have “solved” this problem.

1. Ah, but doesn’t the fact that we also obsess about this make us as sad as the rest of them? After all, we consider ourselves geniuses for spotting the error no one else does, so are we really any better? 😛

2. BobB says:

Using your logic, Kokkie, there is no answer at all because there is no “else” portion of the puzzle.

1. With a strictly literal interpretation that’s true, but do read the comments by @hpgross and @H_N_O below. They both offer very interesting perspectives on this problem.

1. Priya says:

I must say I agree with Bob’s interpretation. Even with hpgross’s perspective, whether or not they meant to do so the writers of the problem did leave out the function symbol, so the only real answer I can think of for this problem is “true” (since if given the statement p –> q and p is false, then the full statement is often regarded as true).

3. I got 90 on the assumption that the if-then sequence was defined by a multiplier that was 1 higher than the indicated number. But there are all sorts of ways, indeed, of looking at this & I probably shouldn’t think too hard about it, my head might explode… 🙂

1. My issue remains with that = symbol. Unless you ignore that the whole thing backfires. I must say I like hpgross’s argument that the answer can actually be anything.

4. hpgross says:

The major problem of course is that the people who developed this problem wanted it for lay people to show how smart they were to one another, but didn’t want to use mathematical notation which would have been very useful for this problem.

Namely, if they had used either function notation e. g. f(2)=6 or sequence notation e. g. a_2=6 then this problem would have been much more interesting.

As it stands however, you are partially correct, in that when given the statements above, you quickly get in to nonsense. If 2=6, then 2-2=6-2 or 0=4. But if 0=4 then 0+0=4+4 or zero is now also every multiple of 4, and we are now working in the integers mod 4. But then we also get 3=12. 12=4+4+4=0+0+0=0, so 3=0. If 3=0 and 4=0, then 1=4-3=0-0=0, and then all integers can equal anything. So everyone who answered any answer were in one sense correct, in that based on the syntax 9 could be equal to at minimum any integer.

If they were to use function notation or sequence notation, the typical answers would be f(x)=x^2+x or a_n=n^2+n. Given x=9 or n=9 we get the “expected” answer of 90. But it is also true that while it fits a pattern for those terms, it could literally be any sequence.

For example, if I define a function to be f(x)=x^2+x + (x-2)(x-3)(x-4)(x-5)(x-6) then:
f(2)=6
f(3)=12
f(4)=20
f(5)=30
f(6)=42
f(7)=116
f(8)=432
f(9)=1350

This seems insane, but it satisfies the initial conditions.

1. Next time I see this I’m going to give the answer 1350 and when people ask my how I get that I’m giving them this formula 😀

I had to read your comment a few times to get what you’re saying (high school math was a long time ago), but you make a very good point. Depending on interpretation of the riddle the answer could really be anything. This actually ties in nicely to the follow-up post that’s coming tomorrow.

A better puzzle would be to give the completed sequence as you did at the end of your comment and then ask people to extrapolate the function, but that will be above the heads of most laypeople, even ones who aced high school math 😉

1. H_N_O says:

Apparently, it took a while for the riddle to reach me in Germany. Hope you are still interested..

In my search for interesting views on this I find your take on this problem to be a very refreshing one. Great! And I also like the wizarding hpgross does in his comment above. That’s both way better than all the 108, 90, 72, 60, 56 stuff I was reading. However, in general I tend towards your approach not looking for patterns in the numbers but looking for the logical structure (the ‘=‘ and the if-then).

Well, maybe one could take this problem one step further. What do you think of the following two thoughts?

1) As you pointed out, there is a If-Then-Structure holding all the statements together. And as you also stated, the if-part is false (as it is safe to assume that 2 does not equal 6, 3 not 12 and so on). In formal logic, conditionals like ‘If p, then q’ (or more accurately ‘p->q’) are only false, if the antecedent (‘p’) is true, and the consequent is false (‘q’). In all other three cases ‘p->q’ is true (see e.g. en.wikipedia.org/wiki/Material_conditional ). As in our example ‘p’ is false (because 2 does not equal 6), it does not matter, whether ‘q’ (the then-part: then 9= ??) is true or false: The conditional is true anyway. So, logically speaking, we can enter whatever we like: be it 9, 1350, 42, or PI – nothing can make the conditional false..

2) As hpgross illustrated, answering the riddle depends on what kind of pattern we see, and adding a premise or two (like for example 7=116 and 8 =432) might change the inference completely. So this might be a case of non-monotonic logic ( en.wikipedia.org/wiki/Non-monotonic_logic ). Whereas in the case of monotonic logic adding premises does not alter the inference, in most real life problems it does. If it rains and if I am outside, then I will get wet. But if it rains and if I am outside and if I use an umbrella, then I won’t get wet. But if it rains and if I am outside and if I use an umbrella and if the umbrella has wholes in it, then I will get wet. And so on. Observing form this perspective, the formal stuff with the ‘=‘ and the ‘->’ becomes more or less irrelevant. And – as you already pointed out – we might explain (maybe more sociologically or anthropologically than mathematically), how it comes that people find soooo many answers to this problem – because (secretly) adding premises changes everything..

So, does some of that resonate with you? Did I miss some loopholes? Did logic take its leave?

2. Your comment is very welcome. This post has suddenly received an insane amount of traffic this past week and I had actually reopened the comments in the hope that someone would take the bait.

Don’t pay too much heed to that thing about the equal sign. That was me taking a hammer to the problem, and percussive maintenance is never delicate work 😉

As @hpgross and now you have pointed out, multiple answers to this problem is definitely possible, both using formal logic and sound mathematics. But I think the problem is most people are using neither (at least not consciously) to try and solve the problem. Which isn’t really their fault – very few people even know of formal logic, not to mention study and understand it, and most forsake Math the moment the graduate from high school and won’t even think to apply it to real life (a shame, really). But it does mean that in most attempts to solve this problem logic does indeed take its leave, quietly and via the back door.

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