# AI: Ever Logical Or Not?

I’m going to assume that for the most part, we are all fairly logical people. I mean, that’s sort of the reason we’reÂ hanging out on this site. Sure we have a lot of fun — we get raunchy, drunk,Â and naked sometimes —Â and there’s pie every Wednesday afternoon, but developing and maintaining strong reasoning skills is really whatÂ it’s all about.

However, we doÂ have the capacity to turn it off if we want. We don’t have to beÂ logical all the time.

Now,Â I’m not suggesting we have to be one way or the other, but even among skeptics, there are those who think logically on a more consistent basis than others.Â AndÂ there are thoseÂ of usÂ who let flights of fancyÂ interplay withÂ our logical sides all the time.

Are you the hardcore logician? Are you Spock? Or do you surrender to flights of fancy in any good measure? What form do those flights take? Would you rather solve a logic problem or write a poem? Would you rather investigate a claim or tell some jokes? Would you rather do math problems or read a fantasy book?

And yourÂ AI ExtraÂ for today is the following puzzle:

I randomly choseÂ 2 numbers between 1 and 40, inclusive. The numbersÂ aren’t necessarilyÂ distinct, and their order doesn’t matter.

I gave the sum of these two numbers toÂ Elyse, and the product of these two numbers to Carrie.

CarrieÂ turns toÂ Elyse and says: “I don’t know if the two numbers are the same”
ElyseÂ replies: “I KNOW they aren’t the same”
CarrieÂ : “Do you know what the numbers are?”
Elyse: “No”
Carrie: “In that case, I now know what the numbers are”
Elyse: “Now I do too.”

What are the two numbers?

(There is no word play here. There is a real answer you can reach. Also there is only 1 answer. no one is lying, andÂ we’re referring to whole numbers between 1 and 40, as in 1, 2, 3….40).

The Afternoon Inquisition (or AI) is a question posed to you, the Skepchick community. Look for it to appear Tuesdays, Thursdays, Saturdays, and Sundays at 3pm ET.

### Sam Ogden

Sam Ogden is a writer, beach bum, and songwriter living in Houston, Texas, but he may be found scratching himself at many points across the globe. Follow him on Twitter @SamOgden

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1. The two numbers are 2 and Two.

The sum of the two numbers is 4, and the product of the two numbers four.

I am not a spock, but I wish I were. I’d love to be able to supress all emotion. There are also a couple of things which aren’t quite emotions which I’d like to suppress also, but, because I’m human, I’m not.

Besides, Data was so much cooler!

2. I don’t see the contradiction. I think you’re setting up a non-empirical dichotomy. I enjoy solving logic problems and writing poems. I enjoy invesigating claims and telling jokes. I would rather not do a math problem or read a fantasy book.

3. @infinitemonkey: I haven’t figured out the answer yet, but the second statement is “I KNOW they aren’t the same,” so the answer can’t be two and two.

4. Now, Iâ€™m not suggesting we have to be one way or the other, . . .

It is not 2 and 2.

5. I am the complete opposite. I am a skeptic because I am naturally illogical. I have to work hard to stay logical.

6. The product must be even; the sum must be odd, since Carrie couldn’t tell if the two numbers were the same (meaning an even product), but the sum being odd means they must be different, since they are whole numbers.

The numbers are six and seven. Their product, 42, is even, while their sum is odd.

I think.

That sad, I am generally not very logical. I have an interesting array of superstitions (I tend to utter a very general prayer to Fortune when I need something to go my way), but I can use logic as a tool if I consciously pick it up.

Of course, I might go orangutan with logic… instead of using it like the screwdriver you handed me, I might decide to pry the top off something and poke at it.

EDIT: Dammit, I know I’m wrong. The product has to be a square for Carrie to not know if they are the same. I think now we’re looking at 3 and 12.

7. I conclude one number is 3,
And the other is 12.
From poetry I flee,
Into logic I delve.

8. Spock was a hero to me back when I was a kid in the mid-’60s and Star Trek was being broadcast for the first time. He contrasted with the screaming madness of my very “human” family, and gave me a hint as to how to detach from them and protect myself. I’ve got a strong Spock-ish, logical side to me which has saved my ass many times and helps me at work and in my personal life, but have retained and cultivated the more harmlessly irrational and ridiculous aspects of my personality and my thought processes. I just don’t make important life decisions based on them alone as I used to, when I was a New Ager and submerged in a subculture where intuition was celebrated, thought and rationality was denigrated, and the Universe was replete with “messages” it was giving me.

9. @Mark Hall: Your logic is right, especially the part about the product being a square. However, it can’t be 3 and 12, 9 and 4 or 1 and 36, because those aren’t unique solutions. The product must be a square that factors ONLY into X*X and Y*Z, and we know that neither of the numbers is X. My guess is that Y and Z are 1 and 4.

10. 5 and 20. The product is the perfect square 100. Carrie now knows the answer is 4 and 25, 5 and 20, or 10 and 10. Since Elyse knows the sum is odd, 10 and 10 is eliminated. Elyse now knows that the product is a perfect square but still does not know the two numbers. This means that the sum can yield two different perfect squares, either 5 and 20 or 9 and 16. Thus Carrie know knows the numbers are 5 and 20.

Sorry for the clumsy explanation.

11. I’m strongly logical and always have been (although as a child, I fairly frequently had faulty bases for my logical extrapolations), but I also have a wide whimsical streak. At this point in my life, intellectual domination tends to be the rule, especially when I’m dealing with new people or people in a business or academic environment. When I’m hanging out with long-term friends, I’m more likely to explore flights of fancy, indulge in some creative writing or randomly burst into song.

12. @Mark Hall: Carrie knows the product. If she says she can’t know if the two numbers are the same, it means the square root of the product is a whole number. That doesn’t imply that the product is even. If the numbers were, say, 5 and 5, we’d get an odd number, but knowing the product wouldn’t lead us to the answer.

13. The guessers so far are missing a hint from Carrie’s first statement – since she doesn’t know if the numbers are the same, that means the product is a square.

Elyse’s response that the numbers are different both denies that they are the same and says that one is odd and one is even.

So, we need to look at the combinations of odd and even numbers in the set [1, 40] that multiply to a square and where one and only one is odd. I think this is that set:
(1, 36)
(1, 16)
(1,4)
(4,25)
(4,9)
(9,36)
(9,16)
(16,25)
(3,12)
(5,20)
(12,27)

Of these, only (5,20) and (9,16) share a sum, which is necessary for Elyse not to know the value yet.

So, Carrie had either 144 or 100. If she had 144, then (9,16) is the only combination of odd and even in the bounds that give her number. That excludes (9,16), because she wouldn’t need to ask Elyse anything at that point.

If she had 100, then both (4,25) and (5,20) would have been possibilities, requiring her to ask.

So, the pair is (5,20).

14. The numbers are 5 and 20.

After the first two questions, we know their product must be a square and their sum must be odd. We also know that among all pairs of numbers that share these properties, their sum appears more than once, orElyse would know the numbers. The only such sum is 25, as figured out by a quick computer program. This occurs with 9,16 and 5,20. However, 9*16=144 only occurs as the product of one of these pairs, 9, 16. So the fact that Carrie only knew after Elyse said she didn’t know tells us that the pair is 5, 20.

15. If the product was prime, Carrie would know the answer. She doesn’t, so it’s not prime.
Carrie thinks it could have a root, so the product is even.
Elyse has the sum, and knows the two numbers aren’t the same, so the sum is odd.
If the sum is odd, then one of the numbers is necessarily even, and the other is necessarily odd.
So the two numbers are necessarily distinct…

I don’t see how 3 and 12 don’t work. The product is even, it has a square and it is not prime. The sum is odd, and the numbers are each even, odd and distinct. I don’t think it’s the only answer, but it is a good one as far as the problem goes…

EDIT: oooh, I see we’ve figured it out since I refreshed the page… :D

16. nichole, the problem with (3,12) is that its the only pair with and even square product but a sum of 15. If those were the numbers, then the last interchange between Carrie and Elyse wouldn’t work.

17. Fairly logical but I have a big appreciation of the aesthetic and my emotional response to different art forms. I can also be quite the romantic when it comes to appreciating nature and the fundamental pleasures derived from basic human functions like eating, drinking, sex, or hanging out with other homo sapiens.

And Iâ€™d rather shovel snow than solve number puzzles.

18. Dangit, I forgot Elyse would then know what Carrie knew about the product being a square, which makes 1 and 4 wrong. Boo math :P

19. I’m known to enjoy logic puzzles, why else would I subscribe to a monthly puzzle magazine, and I like math problems, even if I suck at solving them.

While I don’t like to write, I love to read, and I read mostly fiction, favorite genres include fantasy and science fiction.

So I’m something in between, trying to be logical and practical in real life, with a few flights of fancy thrown in, but I really enjoy going off to fantasy worlds in books and games.

20. Are you the hardcore logician? No, I can be logical one minute and be looking in the freezer for my keys that I lost because maybe they might have jumped in there. (okay, I don’t really think they jumped in there, but I do look)

Are you Spock? I am not Spock. I first heard of Leonard Nimoy as the host of In Search Of and then as the singer of The Ballad of Bilbo Baggins and wasn’t a fan of Star Trek until Next Generation came around.

Or do you surrender to flights of fancy in any good measure? While I find it hard to desribe there are certain work that just “do it” for me. There’s something about the design or the dialogue or the feel.

What form do those flights take? Anything by Hayao Miyazaki (esp. Spirited Away, I adore that movie), The Triplets of Belleville, Dark City, I love to lose my self in a good book or movie or video game.

Would you rather solve a logic problem or write a poem? A poem probably, I don’t really like logic problems but I like the more visual versions like sudoku, kenken, and paint-by-numbers.

Would you rather investigate a claim or tell some jokes? I would investigate the claim the make jokes about it.

Would you rather do math problems or read a fantasy book? Don’t really like fantasy too much, and I prefer word games to math problems so I’d say it’s a wash.

21. No, I can be logical one minute and be looking in the freezer for my keys that I lost because maybe they might have jumped in there.

I laughed out loud at this. I thought I was the only one.

I once found my keys in my medicine cabinet. I still don’t know how they got in there or why I thought to look there.

22. I won’t answer the riddle as I see the answer has been posted, but as for the other question…

I am mostly logical, but by no means Spock. I couldn’t go through life without logic, without a view of the world that is as objective as it can be, given the limitations of perception, to guide my decisions. But at the same time, that objective view tells me that we are creatures of feeling and fancy. That ultimately, what we want most in life are fellings (happiness, fulfilment, love), that feelings matter in our decisions, influence our actions, and almost always determine our reactions. Fancy is a big part of what has moved us out of the caves and into the internet age.

So yeah, I have a healthy sense of humor, love a good fantasy story (I have a real thing for alternative versions of known stories from unusual points of view), daydream and take long walks in the rain (yes, singing and dancing sometimes). Life should be guided by logic, but it should also be enjoyed.

23. Nice work lawnboy

I like logic problems and fantasy both. Fantasy is great fun as long as you know it aint real.

24. I’m pretty logical. I pretty much find it creeping into my thoughts even when I don’t want it there. that being said, I was unaware of the pie. Where did you say we go for the pie?

25. I have logically determined that I got the wrong answer due a lack of sleep and Sam being a doody-head. ;-)

26. You’re a God Damn Mother Fucking Liar, Sam.

After hours, I still don’t know the answer.

I bet Carrie doesn’t either.

27. Elyse, are you saying that you and Carrie were both just bluffing the whole time?

28. So I guess I am TOO logical, because, just like in real life, I did not catch the hint in clue #1. I am terrible at catching hints and understanding other’s motivations.

29. I am somewhat obsessive about things making sense, but I’m not Mz. Amazing Logician. I totally wish I was, seeing as how I obsess over things that don’t make sense to me. But I abhor tedium, which I find in things like math problems and many logic puzzles. I think I enjoy investigating a claim for a bit, but ultimately there is only so long I last before it becomes tedious. I love dancing and reading…and I only like reading about science and/or the fantastical. So it’s one or the other there; no middle ground. I have a fondness for rules, as long as they make sense, and a fondness for seeing where randomness leads.

It all makes sense in my head, I swear.

30. @Sam Ogden: Things like keys should have a beeper attached that you can set off by phoning them, so you can find them after the gremlins have hidden them. (‘course, then you have to go on Twitter to get somebody to phone you so you can find your phone so you can phone your keys.)

Is it true that to get the Wednesday pie you have to be raunchy, drunk and naked? Is there enough pie to accomodate that?

31. I’m probably overthinking this (so, yes, that answers Sam’s question), but I think perhaps having Elyse say â€œI KNOW they arenâ€™t the sameâ€ after Carrie says â€œI donâ€™t know if the two numbers are the sameâ€ might not mean what I said above.

If Elyse knew that before Carrie said anything, then it does mean that the sum of the numbers is odd. However, if she knows that only because of the information from Carrie that the product is a perfect square, then I don’t think we can be sure what she knows.

Sam, does the ALL CAPS on KNOW mean that she knew that before Carrie said anything? Or does it not matter?

32. â€œElyse: â€œNow I do too.â€â€

No, I don’t think she does and nor do we. Carrie is the only one who knows whether the product is 100 or 144 and so she is also the only one who can distinguish between the possibilities {9,16} and {5,20}.

33. There’s a problem with your puzzle: The sum doesn’t necessarily have to be odd. If the sum were 26, 34, 50, or 52, Elyse would still know that the numbers aren’t the same, given that Carrie doesn’t.

For example:
-Carrie sees a product of 144.
-Elyse looks at a sum of 26.
-Carrie doesn’t know whether the numbers are the same: there are a lot of options, including 12 and 12.
-Elyse now knows that the numbers are not the same. If they were (13 and 13), Carrie would have seen a product of 169 and known immediately what the numbers were.

Therefore, the sum does not have to be odd, and the ladies are wrong in saying that they know what the numbers are, as there are still quite a few possibilities.

34. @Skepotter: IPv6… The world is going to run out[1] of IPv4 addresses sometime this or next month. (These are the 96.30.27.149 type numeric addresses that names like skepchick.org translate to.) This will force the introduction of IPv6 addresses which have enough bits[2] to number all the grains of sand in the universe (more or less), so your keys will have an IP address and you’ll be able to ping them and they’ll tell you (via GPS) where they are. Still waiting for your hovercraft? :-)

[1] for some value of “run out”, i.e. it will probably be at least a year before all the big chunks of addresses are distributed, and several years for all the dribs and drabs.

[2] 128 bits = 1.7*10^38, if my math skills haven’t totally deserted me. Grains of sand on Earth are variously estimated between 7.5*10^18 to 10^20, according to random googling.

35. @Bechamel:

There’s nothing wrong with the problem, only with your logical reasoning.

Elyse knows definitively that the two numbers are not the same- “I KNOW they arenâ€™t the same,â€ she states. The only way for her to know this is if the sum is odd and therefore one number is odd and one number is even.

Your example is also impossible. If Carrie gets a product of 144, then Elyse cannot get a sum of 26 (however a sum of 25 would be possible).

36. Oh, I’m quite logical, and have been the Spock-type (what my old shrink called “brain-on-a-stick”) in my time. But having grown a bit and experienced the world… emotions are real and important to human life. Therefore, ignoring emotions is not logical.

37. I most certainly have flights of fancy and fantasy, but they are just that, fantasy, and I know that is what they are. It is the same escapism you get watching a good movie or reading a good book, except I get to be the hero or the star or the millionaire, or whatever I want to be.

But interestingly enough, even in my fantasies, I find myself unable to break with reason and rationality. Any fantastic event I may be involved in within my own little world must at least be plausible. No super powers, no magic, but things that probably have the tiniest likelihood of happening, but could happen given the right set of fairly unlikely circumstances. For example, I could imagine myself as a master guitar player. In reality, if I spent every waking moment practicing and taking lessons with the best teachers in the world, after years I just might get that good, but the chances of that actually happening are slim to none because the reality is that I won’t put in that kind of time and effort.

39. Damn, I just can’t walk away from a puzzle. The numbers are 7 and 28.
Carrie doesn’t know if the numbers are the same or not so the product must be a square – 4, 9, 16, 25, 36, 49, …
Elyse gets a sum which she knows the numbers can’t be the same, therefore an odd number.

The maximum possible sum is 79 and lowest possible is 1.
Most square numbers have too many combinations of the sum of two numbers
The squared number can’t be a prime number otherwise the answer would be obvious.
The squared number can’t be an even number which is divisible by too many numbers e.g. 16
The squared number can’t be too high or two low otherwise there are too obvious, nor can it be divisible by too many numbers or it is too difficult to identify.
If Carrie received the number 196 the options are 14 x 14 or 7 x 28
but Elyse implies she gets an odd number e.g. 7 + 28 = 35
Once Carrie works out the answer Elyse can interpolate the answer as well?

40. Damn, there seem to be several other combinations that work, back to the drawing board!

41. While I now see that lawnboy and trotter’s explanations make sense, I would suggest that Elyse and Carrie must be math geniuses to work it out at the speed suggested by their dialog!

42. Carrie turns to Elyse and says: â€œI donâ€™t know if the two numbers are the sameâ€
[I take this to mean that she does not know if the numbers are the same and does not know if they are different.] If the numbers are the same then the product (AxB) MUST be a perfect square (N^2). But the product of non-identical numbers MAY also be a perfect square. Therefore Carrie has observed that the product is a perfect square but it is one that can also be obtained from different numbers.
Elyse replies: â€œI KNOW they arenâ€™t the sameâ€
This simple means that A+B is ODD therefore one number (let it be A) is ODD and the other is EVEN.
It follows that AxB is EVEN therefore N is even.
As B is ODD, N^2 and hence N must have an ODD factor.
As A and B are different they cannot both be 40 therefore N cannot be 40.
Possible values of N are: 6,10,12,14,18,20,22,24,26,28,30,34,36,38
Factorize each possible value of N (and hence of N^2). For each value of N, possible values of A may be calculated by multiplying various combinations of ODD factors only.
Thus for N=6 the factors are 2×3 and hence for N^2 they are 2x3x2x3. A may be 3 or 3×3=9. The value of B is simply (N^2)/A. In this case for N=6 A=3 & B=12 OR A=9 & B=4
We can draw up a table listing all possible values of A & B (values of A or B >40 are invalid and are not included)
N…………………A ………………B……………A+B
6=2×3………….3,9…………….12,4……….15,13
10=2×5………..5,25…………..20,4……….25,29
12=2x2x3…….3,9…………….X,16……….25
14=2×7………..7……………….28………….35
18=2x3x3…….3,9,27………..X,36,12…..45,39
20=2x2x5…….5,25…………..X,16……….41
24=2x2x2x3…3,9…………….X,X
26=2×13………13……………..X
28=2x2x7…….7……………….X
30=2x3x5…….3,5,9,15,25…X,X,X,X,36..61
34=2×17………17……………..X
36=2x2x3x3…3,9,27………..X
38=2x2x7…….7……………….X
(That looks better in a fixed width font)
Carrie : â€œDo you know what the numbers are?â€
Elyse: â€œNoâ€
Elyse could not tell the numbers from their SUM which means A+B=25 so N=10 or 12.
Carrie: â€œIn that case, I now know what the numbers areâ€
Carrie could not tell the numbers from their PRODUCT before Elyse replies. (But Elyse does not know this until Carrie has solved it.) This must be because N=6,10 or 18. Carrie now knows the answer: 5&20 and Elyse can reply:
â€œNow I do too.â€

43. Perhaps this reads a little more clearly:

Carrie turns to Elyse and says: â€œI donâ€™t know if the two numbers are the sameâ€

[I take this to mean that she does not know if the numbers are the same and also does not know if they are different.] If the numbers are the same then the product (AxB) MUST be a perfect square (N^2). But the product of non-identical numbers MAY also be a perfect square. Therefore Carrie has observed that the product is a perfect square but it is one that is also the product of two different numbers.

Elyse replies: â€œI KNOW they arenâ€™t the sameâ€

This simple means that A+B is ODD therefore one number (let it be A) is ODD and the other is EVEN.
It follows that AxB is EVEN so N^2 is EVEN and therefore N is EVEN.
As B is ODD, N^2 and hence N must have an ODD factor.
As A and B are different they cannot both be 40 therefore N cannot be 40.
Possible values of N are: 6,10,12,14,18,20,22,24,26,28,30,34,36,38

Factorize each possible value of N (and hence of N^2). For each value of N, possible values of A may be calculated by multiplying various combinations of ODD factors only.
Thus for N=6 the factors are 2Ã—3 and hence for N^2 they are 2x3x2x3. A may be 3 or 3Ã—3=9. The value of B is simply (N^2)/A. In this case for N=6 A=3 & B=12 OR A=9 & B=4.
We can draw up a table listing all possible values of A & B (values of A or B >40 are invalid and are not included or are marked by an X)

Nâ€¦â€¦â€¦â€¦â€¦â€¦â€¦A â€¦â€¦â€¦â€¦â€¦â€¦Bâ€¦â€¦â€¦â€¦â€¦A+B
6=2Ã—3â€¦â€¦â€¦â€¦.3,9â€¦â€¦â€¦.â€¦.12,4â€¦â€¦â€¦.15,13
10=2Ã—5â€¦â€¦â€¦..5,25â€¦â€¦â€¦…20,4â€¦â€¦â€¦.25,29
12=2x2x3â€¦â€¦.3,9â€¦â€¦â€¦..â€¦.X,16â€¦â€¦â€¦.25
14=2Ã—7â€¦â€¦â€¦..7â€¦â€¦â€¦â€¦.â€¦.28â€¦â€¦â€¦â€¦.35
18=2x3x3â€¦â€¦.3,9,27â€¦â€¦â€¦..X,36,12â€¦..45,39
20=2x2x5â€¦â€¦.5,25â€¦â€¦â€¦â€¦..X,16â€¦â€¦â€¦.41
24=2x2x2x3â€¦3,9â€¦â€¦â€¦â€¦â€¦.X,X
26=2Ã—13â€¦â€¦â€¦13â€¦â€¦â€¦â€¦â€¦..X
28=2x2x7â€¦â€¦.7â€¦â€¦â€¦â€¦â€¦â€¦.X
30=2x3x5â€¦â€¦.3,5,9,15,25â€¦X,X,X,X,36..61
34=2Ã—17â€¦â€¦â€¦17â€¦â€¦â€¦â€¦â€¦..X
36=2x2x3x3â€¦3,9,27â€¦â€¦â€¦..X
38=2x2x7â€¦â€¦.7â€¦â€¦â€¦â€¦â€¦â€¦.X

Carrie : â€œDo you know what the numbers are?â€

[Carrie does not know what they are but Elyse does not know this yet]

Elyse: â€œNoâ€

Elyse could not tell the numbers from their SUM which means A+B=25 so N=10 or 12.

Carrie: â€œIn that case, I now know what the numbers areâ€

Carrie could not tell the numbers from their PRODUCT before Elyse replies.
This must be because N=6,10 or 18. Carrie now knows the answer: 5&20 [and so do we].

Elyse: â€œNow I do too.â€

The important information that Elyse has just received from Carrie is that Carrie did not know the answer before she found out that Elyse did not know either. Elyse’s reasoning and her reply are of no relevance to us because we had already worked out the answer when Carrie did.

44. Buh. I -am- quite the hardcore logician, or I at least try to be, but at the same time I do like writing and understand the need to suspend disbelief — which I can do, but at the same time I can somehow enjoy a movie while still making random comments about it.

That being said, I’m currently trying to write a personal-ish fanfiction (okay, it will probably wind up being the equivalent of fanfiction heaven/hell depending on how many different universes I can work in) where the entire basis of the story remains mostly unexplained in favor of character development — of course, the characters themselves (read: me, my best friend and a now-ex-friend) aren’t terribly interested in where they get their powers either because now they have them and can mess around with literally no risk of repercussion.

As for the math-related problem, not even going to try at this hour. Maybe some other time.

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